Iterated Function Research Articles

Random iteration on hyperbolic Riemann surfaces

Let {f_nu }subset mathop {mathrm {Hol}}nolimits (X,X) be a sequence of holomorphic self-maps of a hyperbolic Riemann surface X. In this paper we shall study the asymptotic behaviour of the sequences obtained by iteratively left-composing or right-composing the maps {f_nu }; the sequences of self-maps of X so obtained are called left (respectively, right) iterated function systems. We shall obtain the analogue for left iterated function systems of the theorems proved by Beardon, Carne, Minda and Ng for right iterated function systems with value in a Bloch domain; and we shall extend to the setting of general hyperbolic Riemann surfaces results obtained by Short and the second author in the unit disk mathbb {D} for iterated function systems generated by maps close enough to a given self-map.

Dirichlet-type problems for n-Poisson equation in Clifford analysis

Iterated Dirichlet problem, also called as Riquier or Navier problem, and polyharmonic Dirichlet problem are studied for n-Poisson equation in Clifford analysis using iterated polyharmonic Green function and polyharmonic Green-Almansi type function appropriate for the boundary conditions of the problems.

Fixed Point Results of Dynamic Process D ˇ ϒ , μ 0 through F I C -Contractions with Applications

This article constitutes the new fixed point results of dynamic process D(ϒ, μ0) through FIC-integral contractions of the Ciric kind and investigates the said contraction to iterate a fixed point of set-valued mappings in the module of metric space. To do so, we use the dynamic process instead of the conventional Picard sequence. The main results are examined by tangible nontrivial examples which display the motivation for such investigation. The work is completed by giving an application to Liouville‐Caputo fractional differential equations.

On the asymptotic behavior of the Diaconis–Freedman chain in a multi-dimensional simplex

AbstractWe give a setting of the Diaconis–Freedman chain in a multi-dimensional simplex and consider its asymptotic behavior. By using techniques from random iterated function theory and quasi-compact operator theory, we first give some sufficient conditions which ensure the existence and uniqueness of an invariant probability measure and, in particular cases, explicit formulas for the invariant probability density. Moreover, we completely classify all behaviors of this chain in dimension two. Some other settings of the chain are also discussed.

Fractal set of generalized countable partial iterated function system with generalized contractions via partial Hausdorff metric

The primary interest of this article is to extend the fixed point result proved for a complete partial metric space (simply, pms) with a generalized contraction to a bivariate case. Taking this as a basic note, a generalized countable partial iterated function system (GCPIFS, for short) is introduced with the generalized contractions, using the partial Hausdorff metric. The attractor of this GCPIFS is proved as the fixed point of the generalized Hutchinson operator concentrated on the complete metric space (CBpm⁎(P⁎)×CBpm⁎(P⁎),Hpm⁎). A certain number of examples are given to show the existence of the theoretical results which we proved.

Some critical remarks on “Some new fixed point results in rectangular metric spaces with an application to fractional-order functional differential equations”

In this manuscript, we generalize, improve, and enrich recent results established by Budhia et al. [L. Budhia, H. Aydi, A.H. Ansari, D. Gopal, Some new fixed point results in rectangular metric spaces with application to fractional-order functional differential equations, Nonlinear Anal. Model. Control, 25(4):580–597, 2020]. This paper aims to provide much simpler and shorter proofs of some results in rectangular metric spaces. According to one of our recent lemmas, we show that the given contractive condition yields Cauchyness of the corresponding Picard sequence. The obtained results improve well-known comparable results in the literature. Using our new approach, we prove that a Picard sequence is Cauchy in the framework of rectangular metric spaces. Our obtained results complement and enrich several methods in the existing state-ofart. Endorsing the materiality of the presented results, we also propound an application to dynamic programming associated with the multistage process.

A unification of Geraghty type and Ciric type fixed point theorems

In the framework of metric spaces, we introduce the concept of Geraghty-Ciric type contractions and show the existence and uniqueness of the fixed point of such mappings. This result improves and unifies those obtained by Geraghty (Proc. Amer. Math. Soc. 40, 604-608 (1973)) and Ciric (Proc. Amer. Math. Soc. 45, 267-273, (1974)). Several technical lemmas are employed to ensure that a Picard sequence is a Cauchy sequence. In addition, two illustrative examples are provided to indicate the validity of the obtained results.

Bus-Privacy-Preserving Distributed Economic Dispatch Method for Power Systems Using Iteration Functions

The sufficiency of electricity market competition and the economy of power system operation require that the economic dispatch should not only preserve the bus privacy but consider network losses. The existing economic dispatch methods, however, can’t preserve the privacy of each bus from leakage and even complicate the solution when they reasonably consider network losses, due to the coupling complexity of network losses. To address these issues, this paper presents a novel distributed economic dispatch (DED) method based on KKT optimality conditions and iteration functions. It focuses on an economic dispatch model that contains bus active power balance equations and generator power limit inequalities. First, the model’s KKT optimality conditions with inequalities are simplified to equivalent pure equality-type optimality conditions. Then a set of iteration functions is created using the Taylor series, which overcomes the difficulty caused by the fact that unknown bus voltage phases are completely contained in the sine and cosine functions of optimality conditions. Finally, a fast fixed-point iteration procedure is proposed for bus agents to solve the economic dispatch model. The proposed method reasonably considers network losses by using the set of bus active power balance equations. It not only preserves the privacy of each bus from leakage but is straightforward in solution (does not complicate the solution). In addition, it is fully bus-level distributed. Numerical simulation results of systems with different sizes verified the effectiveness and advantages of the proposed method.

Robust Beamforming for Enhancing User Fairness in Multibeam Satellite Systems With NOMA

This paper proposes a downlink transmission scheme exploiting robust beamforming in conjunction with non-orthogonal multiple access for multibeam satellite systems to enhance the spectral efficiency and user fairness. Specifically, by employing the imperfect channel state information and considering the fairness among multiple satellite terminals (STs), we first formulate an optimization problem to maximize the sum <inline-formula><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula>-fair utility, while guaranteeing the transmit power budget and quality-of-service requirement of each ST. Since the original problem is nonconvex, we then adopt a discretization method to convert the channel uncertainty into deterministic forms, and propose an iterative penalty function algorithm combined with sequential convex approximation to obtain the optimal solution. Finally, simulation results are given to confirm the effectiveness and superiority of the proposed scheme over some existing works. It is also shown that our scheme can achieve a good balance between the system performance and user fairness.

Certain dynamic iterative scheme families and multi-valued fixed point results

&lt;abstract&gt;&lt;p&gt;The article presents a systematic investigation of an extension of the developments concerning $ F $-contraction mappings which were proposed in 2012 by Wardowski. We develop the notion of $ F $-contractions to the case of non-linear ($ F $, $ F_{H} $)-dynamic-iterative scheme for Branciari Ćirić type-contractions and prove multi-valued fixed point results in controlled-metric spaces. An approximation of the dynamic-iterative scheme instead of the conventional Picard sequence is determined. The paper also includes a tangible example and a graphical interpretation that displays the motivation for such investigations. The work is illustrated by providing an application of the proposed non-linear ($ F $, $ F_{H} $)-dynamic-iterative scheme to the Liouville-Caputo fractional derivatives and fractional differential equations.&lt;/p&gt;&lt;/abstract&gt;

An Adaptive Smoothness Parameter Strategy for Variational Optical Flow Model

The smoothness parameter is used to balance the weight of the data term and the smoothness term in variational optical flow model, which plays very significant role for the optical flow estimation, but existing methods fail to obtain the optimal smoothness parameters (OSP). In order to solve this problem, an adaptive smoothness parameter strategy is proposed. First, an amalgamated simple linear iterative cluster (SLIC) and local membership function (LMF) algorithm is used to segment the entire image into several superpixel regions. Then, image quality parameters (IQP) are calculated, respectively, for each superpixel region. Finally, a neural network model is applied to compute the smoothness parameter by these image quality parameters of each superpixel region. Experiments were done in three public datasets (Middlebury, MPI_Sintel, and KITTI) and our self-constructed outdoor dataset with the proposed method and other existing classical methods; the results show that our OSP method achieves higher accuracy than other smoothness parameter selection methods in all these four datasets. Combined with the dual fractional order variational optical flow model (DFOVOFM), the proposed model shows better performance than other models in scenes with illumination inhomogeneity and abnormity. The OSP method fills the blank of the research of adaptive smoothness parameter, pushing the development of the variational optical flow models.

Tunable reservoir computing based on iterative function systems

In this study, a performance-tunable model of reservoir computing based on iterative function systems is proposed and its performance is investigated. Iterated function systems devised for fractal generation are applied to embody a reservoir for generating diverse responses for computation. Reservoir computing is a model of neuromorphic computation suitable for physical implementation owing to its easy feasibility. Flexibility in the parameter space of the iterated function systems allows the properties of the reservoir and the performance of reservoir computation to be tuned. Computer simulations reveal the features of the proposed reservoir computing model in a chaotic signal prediction problem. An experimental system was constructed to demonstrate an optical implementation of the proposed method.

АЛГОРИТМ ФОРМУВАННЯ РАНДОМІЗОВАНОЇ СИСТЕМИ ІТЕРАЦІЙНИХ ФУНКЦІЙ ЗА СТРУКТУРОЮ КАНТОРА

This paper has been considered the results of the development of the randomized system of iterated functions (RSIF) formation algorithm from the existing fractal image of the “Fractal Dust” type (the Cantor set). The mathematical formulas and patterns for calculating the RSIF coefficients have been derived. This algorithm is to find the formulas of functions relative to the center of the first iteration of the fractal structure. This makes it possible to determine a randomized system of iterative functions from an existing fractal image. The construction algorithm does not use recursive functions and the entry of the loop into the loop, which allows without spending a lot of computing power, and is quite optimized. The algorithm will allow you to make direct and inverse transformations without involving additional software and hardware resources. The use of forward and inverse transformations will allow in the future to form a source data set for neural networks, which will form the basis of object recognition systems.

Penerapan Fungsi Hash dengan Algoritma RIPEMD-128 Pada Aplikasi Duplicate PDF Scanner

The development of the digital world era continues to grow and is increasing. Many industries have switched to digital technology such as photos, videos, music, and so on. Of course, this change brings many benefits in terms of speed and data access. Ease of accessing the internet also has a major impact on human life. The ease of accessing documents online is very helpful if you want to find and download pdf documents online. With this convenience, many of the same pdf files are downloaded more than once. This will definitely burden the storage space. To distinguish one pdf file from another, it is done by viewing and reading the contents of the pdf file. This is very inconvenient and time consuming. To overcome this, we need a practical way that can be used to search for the same pdf document so that it can delete duplicate pdf files. One way that can be used to overcome this problem is to use a hash function. The hash function in cryptography is a very important tool in various cryptographic applications, for example in digital signature formation, authentication, and so on. Since the discovery of the MD4 hash algorithm by Ron Rivest, many other hash algorithms have been developed based on the principles of this algorithm. One of them is the RIPEMD-128 algorithm. RIPEMD-128 is an iterative hash function that operates on 32-bit words. By using a hash function, the identity of the pdf file can be generated. If the pdf files are the same it will produce the same hash value as well. This will make it easier to group the same pdf documents so that it will speed up the process of deleting the same pdf files. With the pdf file scanner application, pdf files will be searched for all folders and will group pdf files based on the same hash value. So that it will be easier for users to delete the same pdf file.

Piecewise linear iterated function systems on the line of overlapping construction

In this paper we consider iterated function systems (IFS) on the real line consisting of continuous piecewise linear functions. We assume some bounds on the contraction ratios of the functions, but we do not assume any separation condition. Moreover, we do not require that the functions of the IFS are injective, but we assume that their derivatives are separated from zero. We prove that if we fix all the slopes but perturb all other parameters, then for all parameters outside of an exceptional set of less than full packing dimension, the Hausdorff dimension of the attractor is equal to the exponent which comes from the most natural system of covers of the attractor.

Підвищення ефективності стеганографічного методу приховування даних із застосуванням ітераційних функцій та додаванням шуму

The development of computer and digital technology contributes to the growth of information flows transmitted through open and closed communication channels. In many cases, this information is confidential, financial, or commercial in nature and is of value to its owners. This requires the development of mechanisms to protect information from unauthorized access. There are two fundamental areas of secure data transmission over the open communication channels – cryptography and steganography. The fundamental difference between them is that cryptography hides from others the content of the message, and steganography hides the very fact of the message transmission. This paper is devoted to steganographic methods of data concealment, which are less researched than cryptographic, but have significant potential for use in a variety of applications. One of the important characteristics of most methods is their effectiveness. In general, efficiency is assessed in the context of solving specific problems. However, the most common criteria for the effectiveness of steganographic methods are the amount of hidden data and the method of transmitting the secret key to the receiving party, which will not allow the attacker to intercept it. Because media files make up a significant portion of network traffic, a digital image is chosen as the stegocontainer. It is proposed to determine the coordinates of the embedding location on the basis of iterative functions. The advantage of their use is the compactness of the description of the coordinates of the pixels in which the data will be hidden. In addition, it is proposed to use the Diffie-Gellman algorithm to transfer the parameters of iterative functions to the receiving side. This method of key distribution makes the steganographic method less vulnerable to being stolen by an attacker. The second performance criterion is the amount of hidden data. The paper found that the moderate addition of multiplicative noise makes it possible to increase the amount of hidden data without significantly reducing the visual quality of the stegocontainer. To analyze the distortions in the image-stegocontainer, which are due to the influence of noise and modification of the lower bits of pixels, the method of a quantitative assessment of visual quality is used, which is based on the laws of visual perception. Keywords: steganographic data hiding; hiding efficiency; iterative functions; Diffie-Gelman algorithm.

Determination of the uranium elemental concentration in molten salt fuel using laser-induced breakdown spectroscopy with partial least squares-artificial neural network hybrid models

Accurate uranium (U) composition analysis of the molten salt nuclear fuel mixture is a necessary step in nuclear fuel quality control. Laser-induced breakdown spectroscopy (LIBS) is an optical emission spectrometry technique for elemental composition. The non-linear phenomena in LIBS signal are the drawback of this technique, especially for high atomic number (Z) element in a complex matrix. To overcome this limitation, a hybrid chemometrics model of the partial least squares-artificial neural network (PLS-ANN) was used to quantify the U composition in the salt fuel mixture of Molten salt breeder reactor. This work shows the optimisation of several parameters, viz., spectra pre-treatment procedure, factor number of PLS for ANN input, number of hidden neurons and layers in ANN, activation function, and number of ANN iterations. The collective advantages of data dimension reduction from PLS and the nonlinear processing ability from ANN, improved the accuracy of LIBS quantitative analysis for U from ≈7% using simple PLS to ≈4% precision using a hybrid model.

Cyclic generalized iterated function systems

In this article, we introduce the notion of cyclic generalized iterated function system (GIFS), which is a family of functions f 1 , f 2 , … , f M : X k → X , where each f i is a cyclic generalized φ -contraction (contractive) map on a collection of subsets { B j } j = 1 p of a complete metric space ( X , d ) respectively, and k , M , p are natural numbers. When B j , j = 1 , 2 , … , p are closed subsets of X, we show the existence of attractor of this cyclic GIFS, and investigate its properties. Further, we extend our ideas to cyclic countable GIFS.

Throughput Maximization of Network-Coded and Multi-Level Cache-Enabled Heterogeneous Network

One of the paramount advantages of multi-level cache-enabled (MLCE) networks is pushing contents proximity to the network edge and proactively caching them at multiple transmitters (i.e., small base-stations (SBSs), unmanned aerial vehicles (UAVs), and cache-enabled device-to-device (CE-D2D) users). As such, the fronthaul congestion between a core network and a large number of transmitters is alleviated. We consider the throughput maximization problem that optimizes jointly the network-coded user scheduling and power allocation, subject to fronthaul capacity, transmit power, and NC constraints. Given the intractability of the problem, we decouple it into two separate subproblems. In the first subproblem, we consider the network-coded user scheduling problem for the given power allocation, while in the second subproblem, we use the NC resulting user schedule to optimize the power levels. We design an innovative two-layered rate-aware NC (RA-IDNC) graph to solve the first subproblem and solve the second subproblem using an iterative function evaluation (IFE) approach. Simulation results are presented to depict the throughput gain of the proposed approach over the existing solutions.

Uniformly perfect and hereditarily non uniformly perfect analytic and conformal non-autonomous attractor sets

Conditions are given which imply that certain non-autonomous analytic iterated function systems (NIFSs) in the complex plane have uniformly perfect attractor sets, while other conditions imply the attractor is pointwise thin and thus hereditarily non uniformly perfect. Examples are given to illustrate the main theorems, as well as to indicate how they generalize other results. Examples are also given to illustrate how possible generalizations of corresponding results for autonomous IFS's do not hold in general in this more flexible setting. Further, applications to non-autonomous Julia sets are given. Lastly, since our definition of NIFS is in some ways more general than others found in the literature, a careful analysis is given to show when certain familiar relationships still hold, along with detailed examples showing when other relationships do not hold.