Iterated Function Research Articles

Positive periodic solutions for revisited Nicholson's blowflies equation with iterative harvesting term

Nicholson's blowflies differential equation is one of the famous equations in the field of population dynamics. The first steps of the apparition of this equation began in the 1950s with the Nicholson experiments on the sheep blowfly Lucilia cuprina, the hot topic in Australia at that time and since then it has drawn more and more attention of many researchers. The main purpose of the current study is to investigate the existence of periodic positive solutions for Nicholson's blowflies differential equation with an iterative harvesting function by using Schauder's fixed point theorem. Finally, an example is employed to illustrate the effectiveness of our results.

Sustainable scheduling optimization of mixed-line production for large marine power components

The manufacturing of large marine power component including cutting, welding, shrink fitting, heat treatment is a typical industry with high-material consumption, high-energy consumption and high-environmental pollution. It is noted that green manufacturing is gradually emerging. Comprehensive optimization of energy consumption for multiple processes is an effective way to promote a cleaner production industry. To reduce carbon emission from the plant-wide production processes, a new integrated sustainable scheduling approach of mixed-line production with heterogeneous manufacturing technologies is presented in this paper. In this approach, three carbon emission formulations related to cutting, welding and heat treatment are established respectively by analyzing energy consumption and its composition. An optimization mathematical model with the goal of minimizing the total emission of carbon dioxide and the total makespan is then established. In order to solve the model, the genetic algorithm is adaptively improved. The niche and elite retention strategy are introduced in the selection operator, and the dynamic crossover operator is set to adjust the genetic parameters with its iteration and adaptive function values to improve the efficiency of the algorithm. Finally, the effectiveness of the proposed approach is validated by comparing with two types of models considering the makespan and carbon emission respectively. The results indicate that the proposed method satisfactorily tackles the mixed-line production scheduling problem of large marine power component manufacturing.

On the existence of the Hutchinson measure for generalized iterated function systems

We prove that each generalized (in the sense of Miculescu and Mihail) IFS consisting of contractive maps generates the unique generalized Hutchinson measure. This result extends the earlier result due to Miculescu in which the assertion is proved under certain additional contractive assumptions.

NONLINEAR RECURRENT HIDDEN VARIABLE FRACTAL INTERPOLATION CURVES WITH FUNCTION VERTICAL SCALING FACTORS

In this paper, we present a construction of new nonlinear recurrent hidden variable fractal interpolation curves. In order to get new fractal curves, we use Rakotch’s fixed point theorem. We construct recurrent hidden variable iterated function systems with function vertical scaling factors to generate more flexible fractal interpolation curves. We also give an illustrative example to demonstrate the effectiveness of our results.

Latent variable iterative learning model predictive control for multivariable control of batch processes

A latent variable iterative learning model predictive control (LV-ILMPC) method is presented for trajectory tracking in batch processes. Different from the iterative learning model predictive control (ILMPC) model built from the original variable space, LV-ILMPC develops a latent variable model based on dynamic partial least squares (DyPLS) to capture the dominant features of each batch. In each latent variable space, we use a state–space model to describe the dynamic characteristics of the internal model, and an LV-ILMPC controller is designed. Each LV-ILMPC controller tracks the set points of the current batch projection in the corresponding latent variable space, and the optimal control law is determined and the persistent process disturbances is rejected along both time and batch horizons. The proposed LV-ILMPC formulation is based on general LV-MPC and incorporates an iterative learning function into LV-MPC. In addition, the real physical input that drives the process can be reconstructed from the latent variable space. Therefore, this algorithm is particularly suitable for multiple-input, multiple-output (MIMO) systems with strong coupling and serious collinearity. Three studies are used to illustrate the effectiveness of the proposed LV-ILMPC .

An Iteration Function Having Optimal Eighth-Order of Convergence for Multiple Roots and Local Convergence

In the study of dynamics of physical systems an important role is played by symmetry principles. As an example in classical physics, symmetry plays a role in quantum physics, turbulence and similar theoretical models. We end up having to deal with an equation whose solution we desire to be in a closed form. But obtaining a solution in such form is achieved only in special cases. Hence, we resort to iterative schemes. There is where the novelty of our study lies, as well as our motivation for writing it. We have a very limited literature with eighth-order convergent iteration functions that can handle multiple zeros m≥1. Therefore, we suggest an eighth-order scheme for multiple zeros having optimal convergence along with fast convergence and uncomplicated structure. We develop an extensive convergence study in the main theorem that illustrates eighth-order convergence of our scheme. Finally, the applicability and comparison was illustrated on real life problems, e.g., Van der Waal’s equation of state, Chemical reactor with fractional conversion, continuous stirred reactor and multi-factor problems, etc., with existing schemes. These examples further show the superiority of our schemes over the earlier ones.

ANN and IFS based wearable hybrid fractal antenna with DGS for S, C and X band application

This paper presents the design and analysis of Wearable Hybrid Fractal Antenna (WHFA) with Defected Ground Structure (DGS) using Artificial Neural Network (ANN) and Iterative Function System (IFS). The proposed antenna is useful for ISM band (Industrial Scientific and Medical), Bluetooth, Satellite communication, and RFID (Radio Frequency Identification) applications. The dimensions of the defected ground plane varied to use optimization techniques for achieving the desired gain, bandwidth at the resonant frequency of the antenna. The comparison of the outcomes of two algorithms Feeds Forward Back Propagation Network (FFBPN) and ELMAN Back Propagation Network (EBPN) with the desired outcomes is done in terms of mean square error. The WHFA with Koch-Meander slot gives the 178 MHz, 4.75 MHz, and 334 MHz bandwidth, gain 8.61 dB, 3.92 dB, and 5.38 dB at resonant frequencies 2.53 GHz, 4.9 and 7.6 GHz respectively. The prototype of the proposed antenna made on a jean substrate with relative permittivity 1.7, height 1 mm, and loss tangent is 0.025. Simulated ANN and measured results are compared and found in agreement with each other.

Average frequencies of digits in infinite IFS\u2019s and applications to continued fractions and L\xfcroth expansions

The detailed investigation of the distribution of frequencies of digits of points belonging to attractors K of Infinite iterated functions systems (IIFS’s) is a fundamental and important problem in the study of attractors of IIFS’s. This paper studies the Baire category of different families of sets of points belonging to attractors of IIFS’s characterised by the behaviour of the frequencies of their digits. All our results are of the following form:a typical (in the sense of Baire) point xin K has the following property: the average frequencies of digits of x have maximal oscillation.We consider general types of average frequencies, namely, average frequencies associated with general averaging systems. These averages include, for example, all higher order Hölder and Cesaro averages, and Riesz averages. Surprising, for all averaging systems (regardless of how powerful they are) we prove that a typical (in the sense of Baire) point xin K has the following property: the average frequencies of digits of x have maximal oscillation. This substantially extends previous results and provides a powerful topological manifestation of the fact that “points of divergence” are highly visible. Several applications are given, e.g. to continued fraction digits and Lüroth expansion digits.

Relative pose estimation of cooperative target based on monocular vision

With the vigorous development of the global logistics industry, warehousing automation has become a popular applied research direction in computer vision. In this paper, the fork and the fork hole’s relative posture during the forklift picking process were taken as the research object. The relative position of the two is solved based on the Aruco cooperative logo PnP algorithm. In this paper, the initial values of the relative poses of the two are addressed by the linear algorithm DLT, and the relative poses are further optimized by Bundle Adjustment (BA). The iterative error observation function is constructed based on the visual pose estimation model, and the Gauss-Newton method is adopted. The least-square fitting iterative process is carried out many times, and the relative position and posture of the fork and the fork hole with high precision are finally obtained.

Mode Selection and Power Allocation in Multi-Level Cache-Enabled Networks

Moving contents proximity to the network edge and proactively caching popular contents at multiple infrastructures are promising directions for solving the backhaul congestion problem. This letter proposes and evaluates a multi-level cache-enabled network, where cache-hit users can fetch their data from the available cache at either small base-stations, unmanned aerial vehicles, or cache-enabled mobile device-to-device users. Cache-miss users, on the other hand, fetch their data from the central cloud via limited capacity backhaul links. This letter considers the problem of maximizing the network weighted-sum rate by jointly determining the users’ mode of operation and their transmit powers, subject to backhaul capacity and transmit power constraints. After showing how the association problem can be formulated as a generalized assignment problem, the letter solves the transmit power problem using an iterative function evaluation apprach. The resulting mode-selection and power-allocation (MSPA) iterative algorithm is then tested through numerical simulations, which suggest that, while being easily implementable, the proposed multi-level caching can substantially relieve the backhaul congestion, especially in dense-networks, and at low-backhaul capacity regimes.

TW-Co-MFC: Two-level weighted collaborative fuzzy clustering based on maximum entropy for multi-view data

In recent years, multi-view clustering research has attracted considerable attention because of the rapidly growing demand for unsupervised analysis of multi-view data in practical applications. Despite the significant advances in multi-view clustering, two challenges still need to be addressed, i.e., how to make full use of the consistent and complementary information in multiple views and how to discriminate the contributions of different views and features in the same view to efficiently reveal the latent cluster structure of multi-view data for clustering. In this study, we propose a novel Two-level Weighted Collaborative Multi-view Fuzzy Clustering (TW-Co-MFC) approach to address the aforementioned issues. In TW-Co-MFC, a two-level weighting strategy is devised to measure the importance of views and features, and a collaborative working mechanism is introduced to balance the within-view clustering quality and the cross-view clustering consistency. Then an iterative optimization objective function based on the maximum entropy principle is designed for multi-view clustering. Experiments on real-world datasets show the effectiveness of the proposed approach.

On Hyperbolic Affine Generalized Infinite Iterated Function Systems

The aim of this paper is to provide alternative characterizations of hyperbolic affine generalized infinite iterated function systems. More precisely, we prove that, for such a system $${\mathcal {F}}=((X,\left\| .\right\| ),(f_{i})_{i\in I})$$ , among others, the following statements are equivalent: (a) $${\mathcal {F}}$$ is hyperbolic. (b) $$ {\mathcal {F}}$$ has attractor. (c) $${\mathcal {F}}$$ is strictly topologically contractive. (d) $${\mathcal {F}}$$ is uniformly point-fibred. In this way we generalize the result from the paper by Miculescu and Mihail (J Math Anal Appl 407:56–68, 2013). More equivalent statements are given for the particular case when I is finite and X is finite dimensional.

Multifractal formalism for generalised local dimension spectra of Gibbs measures on the real line

We extend the multifractal formalism for the local dimension spectrum of a Gibbs measure μ supported on the attractor Λ of a conformal iterated functions system on the real line. Namely, for α∈R, we establish the multifractal formalism for the Hausdorff dimension of the set of x∈Λ for which the μ-measure of a ball of radius rn centred at x obeys a power law rnα, for a sequence rn→0. This allows us to investigate the Hölder regularity of various fractal functions, such as distribution functions and conjugacy maps associated with conformal iterated function systems.

CT–scans of fractal and non fractal measures in the plane coded by affine homogeneous iterated function systems

An approximation method of measures in the plane is obtained by combining CT-like (computerized tomography) projections and a one–dimensional fractal approximation technique previously devised by the author. Both methods are based on sampling by orthogonal polynomials and involve operations on Jacobi matrices, to avoid the well known ill conditioning of moment methods.

A projection method based on self-adaptive rules for Stokes equations with nonlinear slip boundary conditions

In this work we propose and study a self-adaptive projection method for the numerical solution of Stokes equations under slip boundary conditions. We introduce the projection operator to formulate the slip constraint as a fixed point equation. To solve this problem we suggest a projection method by reformulating the nonlinear slip boundary conditions into an iterative form. To make the method more efficient, we find a self-adaptive rule that uses iterative functions to adjust the penalty parameter automatically. We show the convergence of the method in function space and give its application in detail. Finally, the numerical results are given to support our theoretical results.

Regenerasi Fungsi x<sup>2</sup>-9x-99 Dalam Pembangkit Bilangan Acak Berbasis CSRPNG Chaos

This study examines whether the function f(x)=x2-9x-99 can be used as a key generator in cryptography. The quadratic function is regenerated using the fixed point iteration method into an iteration function. The distribution of digits to the output of iterative function to generate a number of chaos. Randomization testing uses run test and monobit testing. Followed by cryptographic testing to get the correlation between ciphertext and key which will be used as a decision whether the resulting key is random or not. Based on research that has been done iteration function xi = (xi-12-9xi-1-99)/9 can generate CSRPNG Chaos random numbers with the correlation level closest to the value of 0.

On Some Recent Results Concerning F-Suzuki-Contractions in b-Metric Spaces

The purpose of this manuscript is to provide much simpler and shorter proofs of some recent significant results in the context of generalized F-Suzuki-contraction mappings in b-complete b-metric spaces. By using our new approach for the proof that a Picard sequence is b-Cauchy, our results generalize, complement and improve many known results in the existing literature. Further, some new contractive conditions are provided here to illustrate the usability of the obtained theoretical results.

FURTHER EXAMPLES OF PARAMETRIC ITERATIVE FUNCTION SYSTEMS FOR THE CONTINUUM GROWTH OF THE ATTRACTOR

We construct new examples of parametric iterated function systems converging to some fractal shapes. The main goal is the study of the continuous growth and the rate of change of the attractor of the corresponding parametrization.

Shrinking targets for non-autonomous systems

In the present work we establish a Bowen-type formula for the Hausdorff dimension of shrinking target sets for non-autonomous conformal iterated function systems in arbitrary dimensions and satisfying certain conditions. In the case of dimension 1 we also investigate non-linear perturbations of linear systems and obtain sufficient conditions under which the perturbed systems satisfy the conditions in our hypotheses.

Assouad dimension and local structure of self-similar sets with overlaps in [formula omitted

For a self-similar set in Rd that is the attractor of an iterated function system that does not verify the weak separation property, Fraser, Henderson, Olson and Robinson showed that its Assouad dimension is at least 1. In this paper, it is shown that the Assouad dimension of such a set is the sum of the dimension of the vector space spanned by the set of overlapping directions and the Assouad dimension of the orthogonal projection of the self-similar set onto the orthogonal complement of that vector space. This result is applied to give sufficient conditions on the orthogonal parts of the similarities so that the self-similar set has Assouad dimension bigger than 2, and also to answer a question posed by Farkas and Fraser. The result is also extended to the context of graph directed self-similar sets. The proof of the result relies on finding an appropriate weak tangent to the set. This tangent is used to describe partially the topological structure of self-similar sets which are both attractors of an iterated function system not satisfying the weak separation property and of an iterated functions system satisfying the open set condition.