Mathematics, Volume 12, Issue 4 (February-2 2024) – 118 articles

Community structure is a significant characteristic of complex networks, and community detection has valuable applications in network structure analysis. Non-negative matrix factorization (NMF) is a key set of algorithms used to solve the community detection issue. Nevertheless, the localization of feature vectors in the adjacency matrix, which represents the characteristics of complex network structures, frequently leads to the failure of NMF-based approaches when the data matrix has a low density. This paper presents a novel algorithm for detecting sparse network communities using non-negative matrix factorization (NMF). The algorithm utilizes local feature vectors to represent the original network topological features and learns regularization matrices. The resulting feature matrices effectively reveal the global structure of the data matrix, demonstrating enhanced feature expression capabilities. The regularized data matrix resolves the issue of localized feature vectors caused by sparsity or noise, in contrast to the adjacency matrix. The approach has superior accuracy in detecting community structures compared to standard NMF-based community detection algorithms, as evidenced by experimental findings on both simulated and real-world networks. Full article

Any triangle in an isotropic plane has a circumcircle u and incircle i. It turns out that there are infinitely many triangles with the same circumcircle u and incircle i. This one-parameter family of triangles is called a poristic system of triangles. We study the trace of the centroid, the Feuerbach point, the symmedian point, the Gergonne point, the Steiner point and the Brocard points for such a system of triangles. We also study the traces of some further points associated with the triangles of the poristic family, and we prove that the vertices of the contact triangle, tangential triangle and anticomplementary triangle move on circles while the initial triangle traverses the poristic family. Full article

This paper considers the proportional–integral–derivative (PID) control for continuous-time positive systems. A three-stage strategy is introduced to design the PID controller. In the first stage, the proportional and integral components of the PID control are designed. A matrix decomposition approach is used to describe the gain matrices of the proportional and integral components. The positivity and stability of the closed-loop systems without the derivative component of PID control are achieved by the properties of a Metzler and Hurwitz matrix. In the second stage, a non-negative inverse matrix is constructed to maintain the Metzler and Hurwitz properties of the closed-loop system matrix in the first stage. To deal with the inverse of the derivative component of PID control, a matrix decomposition approach is further utilized to design a non-negative inverse matrix. Then, the derivative component is obtained by virtue of the designed inverse matrix. All the presented conditions can be solved by virtue of a linear programming approach. Furthermore, the three-stage PID design is developed for a state observer-based PID controller. Finally, a simulation example is provided to verify the effectiveness and validity of the proposed design. Full article

Medical imaging is crucial for the detection and diagnosis of breast cancer. Artificial intelligence and computer vision have rapidly become popular in medical image analyses thanks to technological advancements. To improve the effectiveness and efficiency of medical diagnosis and treatment, significant efforts have been made in the literature on medical image processing, segmentation, volumetric analysis, and prediction. This paper is interested in the development of a prediction pipeline for breast cancer studies based on 3D computed tomography (CT) scans. Several algorithms were designed and integrated to classify the suitability of the CT slices. The selected slices from patients were then further processed in the pipeline. This was followed by data generalization and volume segmentation to reduce the computation complexity. The selected input data were fed into a 3D U-Net architecture in the pipeline for analysis and volumetric predictions of cancer tumors. Three types of U-Net models were designed and compared. The experimental results show that Model 1 of U-Net obtained the highest accuracy at 91.44% with the highest memory usage; Model 2 had the lowest memory usage with the lowest accuracy at 85.18%; and Model 3 achieved a balanced performance in accuracy and memory usage, which is a more suitable configuration for the developed pipeline. Full article

Traditional image steganography conceals secret messages in unprocessed natural images by modifying the pixel value, causing the obtained stego to be different from the original image in terms of the statistical distribution; thereby, it can be detected by a well-trained classifier for steganalysis. To ensure the steganography is imperceptible and in line with the trend of art images produced by Artificial-Intelligence-Generated Content (AIGC) becoming popular on social networks, this paper proposes to embed hidden information throughout the process of the generation of an art-style image by designing an image-style-transformation neural network with a steganography function. The proposed scheme takes a content image, an art-style image, and messages to be embedded as inputs, processing them with an encoder–decoder model, and finally, generates a styled image containing the secret messages at the same time. An adversarial training technique was applied to enhance the imperceptibility of the generated art-style stego image from plain-style-transferred images. The lack of the original cover image makes it difficult for the opponent learning steganalyzer to identify the stego. The proposed approach can successfully withstand existing steganalysis techniques and attain the embedding capacity of three bits per pixel for a color image, according to the experimental results. Full article

The time series of exchange rate fluctuations are characterized by non-stationary and nonlinear features, and forecasting using traditional linear or single-machine models can cause significant bias. Based on this, the authors propose the combination of the advantages of the EMD and LSTM models to reduce the complexity by analyzing and decomposing the time series and forming a new model, EMD-LSTM-SVR, with a stronger generalization ability. More than 30,000 units of data on the USD/CNY exchange rate opening price from 2 January 2015 to 30 April 2022 were selected for an empirical demonstration of the model’s accuracy. The empirical results showed that the prediction of the exchange rate fluctuation with the EMD-LSTM-SVR model not only had higher accuracy, but also ensured that most of the predicted positions deviated less from the actual positions. The new model had a stronger generalization ability, a concise structure, and a high degree of ability to fit nonlinear features, and it prevented gradient vanishing and overfitting to achieve a higher degree of prediction accuracy. Full article

Many properties of the structure and dynamics of complex networks derive from the characteristics of the spectrum of the associated Laplacian matrix, specifically from the set of its eigenvalues. In this paper, we show that there exist graphs for which the ratio between the length of the spectrum (that is, the difference between the largest and smallest eigenvalues of the Laplacian matrix) and its spread (the difference between the second smallest eigenvalue and the smallest one) is equal to the golden ratio. We call such graphs Golden Laplacian Graphs (GLG). In this paper, we first find all such graphs with a number of nodes $n\le 10$. We then prove several graph-theoretic and algebraic properties that characterize these graphs. These graphs prove to be extremely robust, as they have large vertex and edge connectivity along with a large isoperimetric constant. Finally, we study the synchronization properties of GLGs, showing that they are among the top synchronizable graphs of the same size. Therefore, GLGs represent very good candidates for engineering and communication networks. Full article

In nonlinear dynamics, there is a continuous exploration of introducing systems with evidence of chaotic behavior. The presence of nonlinearity within system equations is crucial, as it allows for the emergence of chaotic dynamics. Given that quadratic terms represent the simplest form of nonlinearity, our study focuses on introducing a novel chaotic system characterized by only quadratic nonlinearities. We conducted an extensive analysis of this system’s dynamical properties, encompassing the examination of equilibrium stability, bifurcation phenomena, Lyapunov analysis, and the system’s basin of attraction. Our investigations revealed the presence of eight unstable equilibria, the coexistence of symmetrical strange repeller(s), and the potential for multistability in the system. Full article

A new solution to the continuous-time bilinear quadratic regulator optimal control problem (CBQR) was recently developed using Krotov’s Method. This paper provides two theoretical results related to the properties of that solution. The first discusses the equivalent representation of the cost-to-go performance index. The second one breaks down this equivalence into smaller identities referencing the components of the performance index. The paper shows how these results can be used to verify the numerical accuracy of the computed solution. Additionally, the meaning of the improving function and the equivalent representation, which are the main elements in the discussed CBQR’s solution, are explained according to the derived notions. A numerical example of structural control application exemplifies the significance of these results and how they can be applied to a specific CBQR problem. Full article

In this paper, we investigate a single machine scheduling problem with a proportional job deterioration. Under release times (dates) of jobs, the objective is to minimize the total weighted completion time. For the general condition, some dominance properties, a lower bound and an upper bound are given, then a branch-and-bound algorithm is proposed. In addition, some meta-heuristic algorithms (including the tabu search ($TS$), simulated annealing ($SA$) and heuristic ($NEH$) algorithms) are proposed. Finally, experimental results are provided to compare the branch-and-bound algorithm and another three algorithms, which indicate that the branch-and-bound algorithm can solve instances of 40 jobs within a reasonable time and that the $NEH$ and $SA$ are more accurate than the $TS$. Full article

In DWI, multi-exponential signal analysis can be used to determine signal underlying diffusion components. However, the approach is very complex due to the inherent low SNR, the limited number of signal decay data points, and the absence of appropriate acquisition parameters and standardized analysis methods. Within the scope of this work, different methods for multi-exponential analysis of the diffusion signal in the kidney were compared. To assess the impact of fitting parameters, a simulation was conducted comparing the free non-negative (NNLS) and rigid non-linear least square (NLLS) fitting methods. The simulation demonstrated improved accuracy for NNLS in combination with area-under-curve estimation. Furthermore, the accuracy and stability of the results were further enhanced utilizing optimized parameters, namely 350 logarithmically spaced diffusion coefficients within [0.7, 300] × 10⁻³ mm²/s and a minimal SNR of 100. The NNLS approach shows an improvement over the rigid NLLS method. This becomes apparent not only in terms of accuracy and omitting prior knowledge, but also in better representation of renal tissue physiology. By employing the determined fitting parameters, it is expected that more stable and reliable results for diffusion imaging in the kidney can be achieved. This might enable more accurate DWI results for clinical utilization. Full article

We provide a formal statistical theory of consistent estimation of the set of all arbitrage portfolios that meet the description of being a stochastic arbitrage opportunity. Two empirical likelihood ratio tests are developed: one for the null that a given arbitrage portfolio is qualified, and another for the alternative that the portfolio is not qualified. Apart from considering generalized concepts and hypotheses based on multiple host portfolios, the statistical assumption framework is also more general than in earlier studies that focused on special cases with a single benchmark portfolio. Various extensions and generalizations of the theory are discussed. A Monte Carlo simulation study shows promising statistical size and power properties for testing the null, for representative data dimensions. The results of an empirical application illustrate the importance of selecting a proper blocking structure and moment estimation method. Full article

Compared to traditional centrifugal pumps, magnetic pumps are widely used in industries such as chemical, pharmaceutical, and petroleum due to their characteristics of leakage-free operation and the ability to transport toxic and corrosive fluids. However, the efficiency of magnetic pumps is relatively low. Improving the efficiency of pumps helps to reduce energy loss and lower industrial costs. In this study, a magnetic pump was chosen as the research subject. The study aims to improve the efficiency and stability of the magnetic pump by optimizing the impeller blades based on the load curve. A combined approach of a numerical simulation and experimental verification was used to investigate the impact of the anterior loading point (AL), posterior loading point (PL), and slope (SL) in the blade loading curve on the pump’s performance. The slope, which had the most significant impact on pump performance, was selected as the dependent variable to analyze the internal pressure pulsation and main shaft radial force of the magnetic pump. The research found that the hydraulic performance test results of the magnetic pump were in good agreement with the simulation results. When efficiency is used as the optimization objective, the anterior loading point should be moved as far back as possible, and the posterior loading point should be moved as far forward as possible. Through the study of internal pressure fluctuations and radial forces within the pump, the radial force distribution is sequentially as follows: the anterior loading method, posterior loading method, and middle loading method at a rated flow rate. The maximum pressure pulsation amplitude was near the volute casing diffuser area. Compared to the original pump, the optimized magnetic pump showed a 5.05% improvement in hydraulic efficiency under the rated conditions. This research contributes to enhancing the performance and efficiency of magnetic pumps, making them more suitable for various industrial applications. Full article

We propose new multistart techniques for finding good local solutions in global optimization problems. The objective function is assumed to be differentiable, and the feasible set is a convex compact set. The techniques are based on finding maximum distant points on the feasible set. A special global optimization problem is used to determine the maximum distant points. Preliminary computational results are given. Full article

Entropy measures the randomness or uncertainty of a stochastic process, and the entropy rate refers to the limit of the time average of entropy. The generalized entropy rate in the form of delayed averages can overcome the redundancy of initial information while ensuring stationarity. Therefore, it has better practical value. A Hidden Markov Model (HMM) contains two stochastic processes, a stochastic process in which all states can be observed and a Markov chain in which all states cannot be observed. The entropy rate is an important characteristic of HMMs. The transition matrix of a homogeneous HMM is unique, while a Nonhomogeneous Hidden Markov Model (NHMM) requires the transition matrices to be dependent on time variables. From the perspective of model structure, NHMMs are novel extensions of homogeneous HMMs. In this paper, the concepts of the generalized entropy rate and NHMMs are defined and fully explained, a strong limit theorem and limit properties of a norm are presented, and then generalized entropy ergodic theorems with an almost surely convergence for NHMMs are obtained. These results provide concise formulas for the computation and estimation of the generalized entropy rate for NHMMs. Full article

We explore strong Nash equilibria in the max k-cut game on an undirected and unweighted graph with a set of k colors. Here, the vertices represent players, and the edges denote their relationships. Each player, v, selects a color as its strategy, and its payoff (or utility) is determined by the number of neighbors of v who have chosen a different color. Limited findings exist on the existence of strong equilibria in max k-cut games. In this paper, we make advancements in understanding the characteristics of strong equilibria. Specifically, our primary result demonstrates that optimal solutions are seven-robust equilibria. This implies that for a coalition of vertices to deviate and shift the system to a different configuration, i.e., a different coloring, a number of coalition vertices greater than seven is necessary. Then, we establish some properties of the minimal subsets concerning a robust deviation, revealing that each vertex within these subsets will deviate toward the color of one of its neighbors. Full article

The process of sorting or categorizing objects or information about these objects into clusters according to certain criteria is a fundamental procedure in data analysis. Where it is feasible to determine the distance metric for any pair of objects, the significance and reliability of the separation can be evaluated by calculating the separation/segregation power (SP) index proposed herein. The latter index is the ratio of the average inter distance to the average intra distance, independent of the scale parameter. Here, the calculated SP value is compared to its statistical distribution obtained by a simulation study for a given partition under the homogeneity null hypothesis to draw a conclusion using standard statistical procedures. The proposed concept is illustrated using three examples representing different types of objects under study. Some general considerations are given regarding the nature of the SP distribution under the null hypothesis and its dependence on the number of divisions and the amount of data within them. A detailed modus operandi (working method) for analyzing a metric data partition is also offered. Full article

This paper introduces the exponential time differencing (ETD) technique as a numerical method to efficiently solve vulnerable American options pricing. We address several challenges, including removing cross-derivative terms through appropriate transformations, treating early-exercise opportunities using the penalty method, and substituting fixed boundary conditions with corresponding one-sided finite differences. The proposed method is shown to be both accurate and efficient through numerical experiments, which also compare the results with existing methods and analyze the numerical stability and convergence rate. Full article

In this paper, we establish a structural inequality of the ∞-subLaplacian ${▵}_{0,\infty }$ in a class of the semi-simple Lie group endowed with the horizontal vector fields $X_1,\dots ,X_{2n}$. When $1<p\le 4$ with $n=1$ and $1<p<3+\frac{1}{n-1}$ with $n\ge 2$, we apply the structural inequality to obtain the local horizontal $W^{2,2}$-regularity of weak solutions to p-Laplacian equation in the semi-simple Lie group. Compared to Euclidean spaces ${\mathbb{R}}^{2n}$ with $n\ge 2$, the range of this p obtained is already optimal. Full article

Bipedal robots have long been a focal point of robotics research with an unwavering emphasis on platform stability. Achieving stability necessitates meticulous design considerations at every stage, encompassing resilience against environmental disturbances and the inevitable wear associated with various tasks. In pursuit of these objectives, here, the bipedal L04 Robot is introduced. The L04 Robot employs a groundbreaking approach by compactly enclosing the hip joints in all directions and employing a coupled joint design. This innovative approach allows the robot to attain the traditional 6 degrees of freedom in the hip joint while using only four motors. This design not only enhances energy efficiency and battery life but also safeguards all vulnerable motor reducers. Moreover, the double-slider leg design enables the robot to simulate knee bending and leg height adjustment through leg extension. This simulation can be mathematically modeled as a linear inverted pendulum (LIP), rendering the L04 Robot a versatile platform for research into bipedal robot motion control. A dynamic analysis of the bipedal robot based on this structural innovation is conducted accordingly. The design of motion control laws for forward, backward, and lateral movements are also presented. Both simulation and physical experiments corroborate the excellent bipedal walking performance, affirming the stability and superior walking capabilities of the L04 Robot. Full article

The greening trend in consumer markets and the marketization and financialization of carbon emission rights have begun to revitalize carbon assets. However, solitary efforts and the spillover of environmental protection effects still hamper enterprises’ enthusiasm for carbon emission reduction. To tackle this challenge, two vertical cooperation mechanisms, cost cooperation and alliance cooperation, are proposed. The mathematical models and solutions are developed for both of the two mechanisms, and their values and applicability are explored, respectively. In addition, the impact of fluctuations in consumer markets, financial markets, and carbon markets on cooperation is examined. The results show that both cooperation models effectively motivate enterprises to enhance carbon reduction and boost market demand. However, cost cooperation may result in inflated product prices and even weaken the profitability of the supply chain. In contrast, alliance cooperation can enhance product price performance and effectively increase supply chain profits. Concerning environmental performance, the initial market is better suited for alliance cooperation, whereas cost cooperation fits the mid-to-late market. The higher financing costs of the financial market and the trading price of the carbon market will strengthen the applicability of cost cooperation. This study offers managerial insights for collaborative decision-making in the context of a multi-market cross-section. Full article

With the expansion of the Internet of Things (IoT) and artificial intelligence (AI) technologies, multilingual scenarios are gradually increasing, and applications based on multilingual resources are also on the rise. In this process, apart from the need for the construction of multilingual resources, privacy protection issues like data privacy leakage are increasingly highlighted. Comparable corpus is important in multilingual language information processing in IoT. However, the multilingual comparable corpus concerning privacy preserving is rare, so there is an urgent need to construct a multilingual corpus resource. This paper proposes a method for constructing a privacy-preserving multilingual comparable corpus, taking Chinese–Uighur–Tibetan IoT based news as an example, and mapping the different language texts to a unified language vector space to avoid sensitive information, then calculates the similarity between different language texts and serves as a comparability index to construct comparable relations. Through the decision-making mechanism of minimizing the impossibility, it can identify a comparable corpus pair of multilingual texts based on chapter size to realize the construction of a privacy-preserving Chinese–Uighur–Tibetan comparable corpus (CUTCC). Evaluation experiments demonstrate the effectiveness of our proposed provable method, which outperforms in accuracy rate by 77%, recall rate by 34% and F value by 47.17%. The CUTCC provides valuable privacy-preserving data resources support and language service for multilingual situations in IoT. Full article

Fault detection and localization are vital for ensuring the stability of data center networks (DCNs). Specifically, adaptive fault diagnosis is deemed a fundamental technology in achieving the fault tolerance of systems. The highly scalable data center network (HSDC) is a promising structure of server-centric DCNs, as it exhibits the capacity for incremental scalability, coupled with the assurance of low cost and energy consumption, low diameter, and high bisection width. In this paper, we first determine that both the connectivity and diagnosability of the m-dimensional complete HSDC, denoted by $HSD{C}_m\left(m\right)$, are m. Further, we propose an efficient adaptive fault diagnosis algorithm to diagnose an $HSD{C}_m\left(m\right)$ within three test rounds, and at most $N+4m(m-2)$ tests with $m\ge 3$ (resp. at most nine tests with $m=2$), where $N=m·2^m$ is the total number of nodes in $HSD{C}_m\left(m\right)$. Our experimental outcomes demonstrate that this diagnosis scheme of HSDC can achieve complete diagnosis and significantly reduce the number of required tests. Full article

The design of thin-walled structures is commonly based on the solutions of linear boundary-value problems, formulated within well-developed theories for elastic plates and shells. However, in modern appliances, especially in MEMS design, it is necessary to take into account non-linear mechanical effects that become decisive for flexible elements. Among the substantial non-linear effects that significantly change the deformation properties of thin plates are the effects of residual stresses caused by the incompatibility of deformations, which inevitably arise during the manufacture of ultrathin elements. The development of new methods of mathematical modeling of residual stresses and incompatible finite deformations in plates is the subject of this paper. To this end, the local unloading hypothesis is used. This makes it possible to define smooth fields of local deformations (inverse implant field) for the mathematical formalization of incompatibility. The main outcomes are field equations, natural boundary conditions and conservation laws, derived from the least action principle and variational symmetries taking account of the implant field. The derivations are carried out in the framework of elasticity theory for simple materials and, in addition, within Cosserat’s theory of a two-dimensional continuum. As illustrative examples, the distributions of incompatible deformations in a circular plate are considered. Full article

Statistical depth is widely used as a powerful tool to measure the center-outward rank of multivariate and functional data. Recent studies have introduced the notion of depth to the temporal point process, which exhibits randomness in the cardinality as well as distribution in the observed events. The proposed methods can well capture the rank of a point process in a given time interval, where a critical step is to measure the rank by using inter-arrival events. In this paper, we propose to extend the depth concept to multivariate spatial point process. In this case, the observed process is in a multi-dimensional location and there are no conventional inter-arrival events in the temporal process. We adopt the newly developed depth in metric space by defining two different metrics, namely the penalized metric and the smoothing metric, to fully explore the depth in the spatial point process. The mathematical properties and the large sample theory, as well as depth-based hypothesis testings, are thoroughly discussed. We then use several simulations to illustrate the effectiveness of the proposed depth method. Finally, we apply the new method in a real-world dataset and obtain desirable ranking performance. Full article

A significant part of agricultural farms in the Kingdom of Saudi Arabia (KSA) are in off-grid sites where there is a lack of sufficient water supply despite its availability from groundwater resources in several regions of the country. Since abundant agricultural production is mainly dependent on water, farmers are forced to pump water using diesel generators. This investigation deals with the increase in the effectiveness of a solar photovoltaic water pumping system (SPVWPS). It investigated, from a distinct perspective, the nonlinear behavior of photovoltaic modules that affects the induction motor-pump because of the repeated transitions between the current and the voltage. A new chaotic Horse Herd Optimization (CHHO)-based Maximum Power Point Tracking technique (MPPT) is proposed. This algorithm integrates the capabilities of chaotic search methods to solve the model with a boost converter to maximize power harvest while managing the nonlinear and unpredictable dynamical loads. The analytical modeling for the proposed SPVWPS components and the implemented control strategies of the optimal duty cycle of the DC–DC chopper duty cycle and the Direct Torque Control (DTC) of the Induction Motor (IM) has been conducted. Otherwise, the discussions and evaluations of the proposed model performance in guaranteeing the maximum water flow rate and the operation at MPP of the SPVWPS under partial shading conditions (PSC) and changing weather conditions have been carried out. A comparative study with competitive algorithms was conducted, and the proposed control system’s accuracy and its significant appropriateness to improve the tracking ability for SPVWPS application have been proven in steady and dynamic operating climates and PSC conditions. Full article

Since the emergence of blockchain, how to improve its transaction throughput and reduce transaction latency has always been an important issue. Hostuff has introduced a pipeline mechanism and combined it with a chain structure to improve the performance of blockchain networks. HCA has introduced a revoting mechanism on the basis of Hostuff, further reducing transaction latency, but it has also brought some problems. In HCA, if the leader is malicious, it would be possible to continuously call on the replica nodes to revote, which can lead to network congestion. This paper employs the global perfect coin technology to guarantee that every replica can obtain a globally consistent and the freshest candidate proposal during the Revote phase, thereby improving the robustness of the HCA protocol. The performance improvement of RHCA in attack scenarios has been verified through experiments. Full article

In this work, an innovative technique is presented to solve Emden–Fowler-type singular boundary value problems (SBVPs) with derivative dependence. These types of problems have significant applications in applied mathematics and astrophysics. Initially, the differential equation is transformed into a Fredholm integral equation, which is then converted into a system of nonlinear equations using the collocation technique based on Chebyshev polynomials. Subsequently, an iterative numerical approach, such as Newton’s method, is employed on the system of nonlinear equations to obtain an approximate solution. Error analysis is included to assess the accuracy of the obtained solutions and provide insights into the reliability of the numerical results. Furthermore, we graphically compare the residual errors of the current method to the previously established method for various examples. Full article

In selection and ranking, the partitioning of treatments by comparing them to a control treatment is an important statistical problem. For over eighty years, this problem has been investigated by a number of researchers via various statistical designs to specify the partitioning criteria and optimal strategies for data collection. Many researchers have proposed designs in order to generalize formulations known at that time. One such generalization adopted the indifference-zone formulation to designate the region between the boundaries for “good” and “bad” treatments as the indifference zone. Since then, this formulation has been adopted by a number of researchers to study various aspects of the partition problem. In this paper, a non-parametric purely sequential procedure is formulated for the partition problem. The “first-order” asymptotic properties of the proposed non-parametric procedure are derived. The performance of the proposed non-parametric procedure for small and moderate sample sizes is studied via Monte Carlo simulations. An example is provided to illustrate the proposed procedure. Full article

In this analytical study, a novel solving method for determining the precise coordinates of a mass point in orbit around a significantly more massive primary body, operating within the confines of the restricted two-body problem (R2BP), has been introduced. Such an approach entails the utilization of a continued fraction potential diverging from the conventional potential function used in Kepler’s formulation of the R2BP. Furthermore, a system of equations of motion has been successfully explored to identify an analytical means of representing the solution in polar coordinates. An analytical approach for obtaining the function t = t(r), incorporating an elliptic integral, is developed. Additionally, by establishing the inverse function r = r(t), further solutions can be extrapolated through quasi-periodic cycles. Consequently, the previously elusive restricted two-body problem (R2BP) with a continued fraction potential stands fully and analytically solved. Full article