In this paper, we introduce and study the notion of rough I₂-lacunary statistical convergence of double sequences in normed linear spaces. We also introduce the notion of rough I₂-lacunary statistical limit set of a double sequence and discuss about some properties of this set.

In the first part of this article, we discuss and generalize the complete convergence introduced by Hsu and Robbins (1947) to the r-complete convergence introduced by Tartakovsky (1998). We also establish its relation to the r-quick convergence first introduced by Strassen (1967) and extensively studied by Lai (1976). Our work is motivated by various statistical problems, mostly in sequential analysis. As we show in the second part, generalizing and studying these convergence modes is important not only in probability theory but also to solve challenging statistical problems in hypothesis testing and changepoint detection for general stochastic non-i.i.d. models.

Using the method of the beta function, Sun has recently evaluated some series of the type $\sum_{k=0}^{\infty}(ak+b)x^k/\binom{mk}{nk}$. By factorization of the polynomial $t^{m-n}(1-t)^n-x$, we will give a general method to find new $\pi$-series.

In this paper, we study a co-variational inequality problem involving Yosida approximation operators in real uniformly smooth Banach space and define an iterative method to obtain its solution. We prove some properties of Yosida approximation operator for our purpose, that is strongly accretivity and Lipschitz continuity. An existence as well as convergence result is proved for co-variational inequality problem involving Yosida approximation operators using the concept of nonexpansive sunny retraction.

The authors intend to establish new oscillation criteria for a class of generalized third order functional difference equation of the form \begin{equation}{\label{eq01}} \Delta_{\ell}\left(a_2(n)\left[\Delta_{\ell}\left(a_1(n)\left[\Delta_{\ell}z(n)\right]^{\beta_1}\right)\right]^{\beta_2}\right)+q(n)f(x(g(n)))=0, ~~n\geq n_0, \end{equation} where $z(n)=x(n)+p(n)x(\tau(n))$. We also present sufficient conditions for the solutions to converges to zero. Suitable examples are presented to validate our main results.

In this paper, we introduce a new iteration scheme, named as the S**-iteration scheme, for approximation of fixed point of the nonexpansive mappings. This scheme is faster than Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, Thakur, and Ullah iteration schemes. We show the stability of our instigated scheme and give a numerical example to vindicate our claim. We also put forward some weak and strong convergence theorems for Suzuki's generalized nonexpansive mappings in the setting of uniformly convex Banach spaces. Our results comprehend, improve, and consolidate many results in the existing literature.

We construct a novel family of summation-integral type hybrid operators in terms of shape parameter $\alpha\in \lbrack 0,1]$ in this paper. Basic estimates, rate of convergence, and order of approximation are also studied using the Korovkin theorem and the modulus of smoothness. We investigate the local approximation findings for these sequences of positive linear operators utilising Peetre's K-functional, Lipschitz class, and second-order modulus of smoothness.

In this paper we consider constructions of first derivative approximations using the generating function. The weights of the approximations contain the powers of a parameter whose modulus is less than one. The values of the initial weights are determined, and the convergence and order of the approximations are proved. The paper discusses applications of approximations of first derivative for numerical solution of ordinary and partial differential equations and proposes an algorithm for fast computation of the numerical solution. Proofs of the convergence and accuracy of the numerical solutions are presented and the performance of the numerical methods considered is compared with the Euler method. The main goal of constructing approximations for integer-order derivatives of this type is their application in deriving high-order approximations for fractional derivatives, whose weights have specific properties. The paper proposes the construction of an approximation for the fractional derivative and its application for numerically solving fractional differential equations. The theoretical results for the accuracy and order of the numerical methods are confirmed by the experimental results presented in the paper.

This study compares the Melbourne Decision-Making Questionnaire (MDMQ) and the General Decision-Making Style (GDMS), two of the most widely used decision-making questionnaires in the literature, in a large age- and sex-weighted general population sample of 714 men (45.7%) and 848 women (54.3%) between 18 and 90 years old. The objective was to evaluate the convergent and construct validity between several aspects of decision-making styles questionnaires. The results indicate that the two questionnaires replicate the factorial structure of four and five factors reported in the original studies respectively, through exploratory and confirmatory procedures in our cross-cultural context. The domains of both questionnaires that represent a strong or large correlation are Vigilance with Rational (.50), Hypervigilance, Buck-passing, and Procrastination with Avoidant (.45, .52 and .60). A Structural Equations Model (SEM) between both questionnaires indicates that both latent factors formed by the domains of the MDMQ and the GDMS obtain a correlation of .96. It is concluded that the two questionnaires measure similar aspects of the decision-making construct.

This study compares the Melbourne Decision-Making Questionnaire (MDMQ) and the General Decision-Making Style (GDMS), two of the most widely used decision-making questionnaires in the literature, in a large age- and sex-weighted general population sample of 714 men (45.7%) and 848 women (54.3%) between 18 and 90 years old. The objective was to evaluate the convergent and construct validity between several aspects of decision-making styles questionnaires. The results indicate that the two questionnaires replicate the factorial structure of four and five factors reported in the original studies respectively, through exploratory and confirmatory procedures in our cross-cultural context. The domains of both questionnaires that represent a strong or large correlation are Vigilance with Rational (.50), Hypervigilance, Buck-passing, and Procrastination with Avoidant (.45, .52 and .60). A Structural Equations Model (SEM) between both questionnaires indicates that both latent factors formed by the domains of the MDMQ and the GDMS obtain a correlation of .96. It is concluded that the two questionnaires measure similar aspects of the decision-making construct.

It has been argued that the historical nature of evolution makes it a highly path-dependent process. Under this view, the outcome of evolutionary dynamics could have resulted in organisms with different forms and functions. At the same time, there is ample evidence that convergence and constraints strongly limit the domain of the potential design principles that evolution can achieve. Are these limitations relevant in shaping the fabric of the possible? Here, we argue that fundamental constraints are associated with the logic of living matter. We illustrate this idea by considering the thermodynamic properties of living systems, the linear nature of molecular information, the cellular nature of the building blocks of life, multicellularity and development, the threshold nature of computations in cognitive systems, and the discrete nature of the architecture of ecosystems. In all these examples, we present available evidence and suggest potential avenues towards a well-defined theoretical formulation.

Due to the broadening of services from industrial enterprises to customers, the industrial network has gained more significance than what is really thought. As a part of this revolution, this paper tried to introduce the best suitable routing option for the healthcare, banking and internet services industries to optimize the routing department of their enterprises. RIP is not associated with this study, being out of favor in the industries due to hop count limitation. The idea behind doing such a thing is to avoid improper implantation of routing inside the enterprises which causes lack of optimization in routing department. The method which is selected for this paper is qualitative case study of banking, healthcare and internet services industries with unstructured interviews of IT experts of the respected industries in an open-ended questions fashion. The results were illustrated based on the stories and the discussion of the stories is the main contributor to the result of the study. The evaluation results of the stories have illustrated, that EIGRP is a suitable routing option for health care and banking, while OSPF can be suitable for the internet services industry. The implication of the study can be optimized network of industries such as healthcare, banking and internet services and it introduces the best suitable routing option for the demanded industries with the case study method and narrative analysis of the stories, which start from load balancing type and ends to demanded routing protocol, while reaching to demanded routing protocol, it introduces the best suitable routing option of that industry

The fractional derivatives are a generalization the derivatives of integer order and find applications in studying memory processes in various scientific fields. Numerical methods are used to solve and analyze fractional models of real world problems. In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic formula, whose generating function is the polylogarithmic function. In the paper, we prove the convergence of the approximation and derive an estimate for the error and order. We consider an application of the approximation for the construction of finite difference schemes for numerical solution of the two-term ordinary fractional differential equation and the time-fractional Black-Scholes equation for option pricing. The properties of the approximation are used to prove the convergence of the methods used for numerical solution of the fractional differential equations. The theoretical results for the error and order of the methods are illustrated by the results of numerical experiments.

In this paper, we generalize the concept of I-statistical convergence, which is a recently introduced summability method, with an infinite matrix of complex numbers and a modulus function. Lacunary sequences will also be included in our definitions. The name of our new method will be A^{I}-lacunary statistical convergence with respect to a sequence of modulus functions. We also define strongly A^{I}-lacunary convergence and we give some inclusion relations between these methods.

We consider a stationary inclusion in a real Hilbert space X, governed by a set of constraints K, a nonlinear operator A and an element f∈X. Under appropriate assumptions on the data the inclusion has a unique solution, denoted by u. We state and prove a covergence criterion, i.e., we provide necessary and sufficient conditions on a sequence {un}⊂X which guarantee its convergence to the solution u. We then present several applications which provide the continuous dependence of the solution with respect to the data K, A and f, on one hand, and the convergence of an associate penalty problem, on the other hand. We use these abstract results in the study of a frictional contact problem with elastic materials which, in a weak formulation, leads to a stationary inclusion for the deformation field. Finally, we apply the abstract penalty method in the anlysis of two nonlinear elastic constitutive laws.

Balanced and coordinated economic development across regions is a critical goal of regional economic development and new-type urbanization in China. However, few studies have examined economic growth convergence clubs at the county level. To extend the research on convergence clubs, this research applies a log t convergence test and a dynamic spatial ordered Probit model (DSOP) to endogenously identify economic growth convergence clubs in counties and to examine the influence of initial states and structures on club convergence probability. The study sample covers 2286 counties of China from 1992 to 2010. The results show significant convergence club patterns at the county levels, resulting in the gradual formation of six convergence clubs. The DSOP estimation results show that per capita fixed assets, population density and industrialization have promoted convergence club formation to varying degrees.

In this study, we propose a conservative and compact finite difference scheme designed to preserve both the mass change rate and energy for solving the sixth-order Boussinesq equation with the surface tension. Theoretical analysis confirms that the proposed scheme achieves second-order accuracy in temporal discretization and fourth-order accuracy in spatial discretization. The solvability, convergence, and stability of the difference scheme are rigorously established through the application of the discrete energy method. Additionally, a series of numerical experiments are conducted to illustrate the effectiveness and reliability of the conservative scheme for long-time simulations.

This article introduces a novel two-step fifth-order Jacobian-free iterative method aimed at efficiently solving systems of nonlinear equations. The method leverages the benefits of Jacobian-free approaches, utilizing divided differences to circumvent the computationally intensive calculation of Jacobian matrices. This adaptation significantly reduces computational overhead and simplifies the implementation process while maintaining high convergence rates. We demonstrate that this method achieves fifth-order convergence under specific parameter settings, with a broad applicability across various types of nonlinear systems. The effectiveness of the proposed method is validated through a series of numerical experiments which confirm its superior performance in terms of accuracy and computational efficiency compared to existing methods.

In order to master the structure state of subway shield tunnel, reduce the diseases and avoid the potential safety hazards, the lateral convergence deformation of subway shield tunnels should be predicted. For the non-stationarity small sample size of subway shield tunnel lateral convergence deformation value, the existing prediction models have poor performance in terms of accuracy and stability. In this paper, a lateral convergence model of subway shield tunnel based on Kalman algorithm is constructed based on Kalman filtering theory. The model is efficient, adaptive and robust, and can accurately predict the lateral convergence deformation of subway shield tunnel. Taking the horizontal diameter of 200 rings shield segment in the interval section of a subway tunnel as an example, it is proved that the residuals of the Kalman prediction model are small, the residual distribution conforms to the normal distribution, and the prediction effect is great. The model is suitable for the prediction of more than 5 periods of data, and the prediction accuracy of the model is improved with the increase of the number of data periods. In addition, the Kalman model is compared with GM(1,1) model and GM-Markov model, and RMSE and MAPE are introduced as evaluation indexes. The results show that the Kalman model has higher prediction accuracy and is more suitable for predicting the lateral convergence deformation of subway shield tunnel.

Infant obesity has become one of the most serious global health challenges of our time. The combined prevalence of overweight and obesity has rapidly increased worldwide during the last two decades, especially in some developing countries where obesity is reaching levels on a par with some industrialized countries, or even higher. This fast growth has occurred especially in countries in the midst of rapid social-economic transitions. Most international comparisons focus on the adult population while analyses focusing on the child population are more limited. Using the methodology developed by Phillips and Sul [1], this paper studies the worldwide evolution of infant body mass index (BMI), overweight and obesity prevalence for a large sample of countries during the period 1975-2016. Our results indicate that the figures for BMI or the prevalence of obesity in different countries do not converge, while the opposite is the case for overweight prevalence in children. Furthermore, there is a non-linear relationship between obesity and income or human capital, indicating that low and middle-income countries require a strong initiative for health policies targeting obesity prevention.

Machine learning offers advanced tools for efficient management of radio resources in modern wireless networks. In this study, we leverage a multi-agent deep reinforcement learning (DRL) approach, specifically the Parameterized Deep Q-Network (DQN), to address the challenging problem of power allocation and user association in massive multiple-input multiple-output (M-MIMO) communication networks. Our approach tackles a multi-objective optimization problem aiming to maximize network utility while meeting stringent quality of service requirements in M-MIMO networks. To address the non-convex and nonlinear nature of this problem, we introduce a novel multi-agent DQN framework. This framework defines a large action space, state space, and reward functions, enabling us to learn a near-optimal policy. Simulation results demonstrate the superiority of our Parameterized Deep DQN (PD-DQN) approach when compared to traditional DQN and RL methods. Specifically, we show that our approach outperforms traditional DQN methods in terms of convergence speed and final performance. Additionally, our approach shows 72.2 % and 108.5 % improvement over DQN methods and RL method respectively in handling large-scale multi-agent problems in M-MIMO networks.

In this article, an iterative method of two step is proposed for finding all roots simultaneously of polynomial equations. The order of convergence of the proposed algorithm is 2m, by using any iterative scheme of order m. Numerical tests are performed to confirm the theoretical results and to compare the proposed scheme with existing methods for finding all roots simultaneously.

We carry out in a thin heterogeneous porous layer, the multiscale analysis of Smoluchowski's discrete diffusion-coagulation equations describing the evolution density of diffusing particles that are subject to coagulate in pairs. Assuming that the thin heterogeneous layer is made of microstructures that are uniformly distributed inside, we obtain in the limit an upscaled model in lower space dimension. We also prove a corrector-type result very useful in numerical computations. In view of the thin structure of the domain, we appeal to a concept of two-scale convergence adapted to thin heterogeneous media to achieve our goal.

This paper presents a numerical algorithm for solving high-order BVPs. We introduce the construction method of multiscale orthonormal basis in W^m[0; 1] by multiscale orthonormal basis in W¹[0; 1]. We define approximate solution, and obtain the approximate solution of high-order BVPs by using the approximate theory. Moreover, the convergence and stability of the algorithm are improved. At last, several numerical experiments show the feasibility of the proposed method.

Effective communication in automated chat systems hinges on the ability to understand and respond to context. Traditional models often struggle with determining when additional context is necessary for generating appropriate responses. This paper introduces Context-Aware BERT (CA-BERT), a transformer-based model specifically fine-tuned to address this challenge. CA-BERT innovatively applies deep learning techniques to discern context necessity in multi-turn chat interactions, enhancing both the relevance and accuracy of responses. We describe the development of CA-BERT, which adapts the robust architecture of BERT with a novel training regimen focused on a specialized dataset of chat dialogues. The model is evaluated on its ability to classify context necessity, demonstrating superior performance over baseline BERT models in terms of accuracy and efficiency. Furthermore, CA-BERT's implementation showcases significant reductions in training time and resource usage, making it feasible for real-time applications. The results indicate that CA-BERT can effectively enhance the functionality of chatbots by providing a nuanced understanding of context, thereby improving user experience and interaction quality in automated systems. This study not only advances the field of NLP in chat applications but also provides a framework for future research into context-sensitive AI developments.

Promoting the sustainable development of resource-based cities is of great significance to the ecological protection and high-quality development of the Yellow River Basin. Combining the environmental status and economic development, we objectively evaluate the ecological safety level of resource-based cities in the Yellow River Basin, compare and analysis the main factors affecting the ecological safety of different resource-based cities, and formulate precise and differentiated ecological safety protection measures in river basins to achieve high-quality development. Taking the Yellow River Basin as an example, this paper analyses the ecological level of the Yellow River Basin based on the theory of sustainable development, calculates the ecological safety level of 30 resource-based cities in the Yellow River Basin from 2006 to 2020 by TOPSIS model, and makes convergence analysis by classification. Through the DID empirical test of sustainable development planning policy on ecological safety factors, distinguish the development characteristics of different resource-based cities. The results show that: (1) the ecological safety level of resource-based cities in the Yellow River Basin is generally developing well, and there are differences among different types of resource-based cities; (2) the kinetic energy of sustainable development of resource-based cities in the Yellow River Basin is mainly industrial transformation, and the power of science and technology is insufficient; (3) local governments pay limited attention to environmental protection, and the sustainable development planning of resource-based cities in 2013 mainly promotes the adjustment of industrial structure of resource-based cities.

Quantum computing is an emerging field that have given a significant impact on optimization. Among the diverse quantum algorithm, the quantum gradient descent has become a prominent technique for solving unconstrained optimization problems. In this paper, we propose a quantum-spectral Polak-Ribi{\'e}re-Polyak (PRP) conjugate gradient approach. The technique is considered as a generalization of the spectral PRP method which employs a $q$-gradient that approximate the classical gradient with quadratically better dependence on the quantum variable $q$. Additionally, the proposed method reduces to the classical variant as quantum variable $q$ gets closer to $1$. The quantum search direction always satisfies the sufficient descent condition and does not depend on any line search. This approach is globally convergent with the standard Wolfe line search without any convexity assumption. Numerical experiments are conducted and compared with the existing approach to demonstrate the improvement of the proposed strategy.

In this paper, we present an innovative technique that improves the convergence order of iterative schemes that do not require the evaluation of Jacobian matrices. Using this procedure, we achieve a remarkable increase in the order of convergence, raising it from p to p + 3 units, which results in a remarkable improvement in the overall performance. We have conducted comprehensive numerical tests in various scenarios to validate the theoretical results, showing the efficiency and effectiveness of the new Jacobian-free schemes.

Generally, stochastic functional differential equations (SFDEs) pose a challenge as they often lack explicit exact solutions. Consequently, it becomes necessary to seek certain favorable conditions under which numerical solutions can converge towards the exact solutions. This article aims to delve into the convergence analysis of solutions for stochastic functional differential equations by employing the framework of G-Brownian motion. To establish the goal, we find a set of useful monotone type conditions and work within the space Cr((−∞,0];Rn). The investigation conducted in this article confirms the mean square boundedness of solutions. Furthermore, this study enables us to compute both LG2 and exponential estimates.

Influenced by the quasi-stationary Mei-yu front and the development of mesoscale low vortices on the Mei-yu front, a torrential rain event occurred in Suzhou on July 6, 2020. The 24-hour rainfall recorded by the automatic station exceeded 200 millimeters, with the highest hourly intensity surpassing 70 millimeters, accompanied by thunderstorms and strong winds. This heavy precipitation process was characterized by suddenness, extremity, long duration, high precipitation efficiency, and backward propagation. Using various data sources such as the European ERA5 reanalysis data, ground-based automatic stations, sounding, Doppler weather radar, and wind profile radar, this paper analyzed in detail the circulation characteristics and radar echo structures of this torrential rain event, and further studied the triggering, feature evolution, and maintenance mechanisms of the heavy precipitation.The results show: (1) The continuously strengthening low-level jet stream propagates downward forming an ultra-low-level jet stream. Heavy precipitation is located on the left front side of the jet exit area, where strong convergence and lifting in the boundary layer overlap with the ascending branch of the secondary circulation above the shear line, resulting in heavy precipitation. (2) A supercell storm with intense precipitation produced this heavy rainfall event, characterized by a low centroid of strong echo; echo-free vault, bounded weak echo region, and other supercell structural features. During this event, the high reflectivity factor areas corresponded to high KDP value areas, indicating a large amount of liquid water content. Vertically, the KDP column can track the rainwater mixing ratio well, and its location corresponds to the vicinity of the short-term heavy precipitation occurrence. The high KDP value area mainly consists of numerous precipitation particles, and the vertical vortex wind field structure causes rapid growth of precipitation particles through continuous rolling. Furthermore, it can be found through correlation coefficients (CC) that there are always high-value areas for CC since they correspond to areas of intense precipitation. (3) Continuous triggering of mesoscale convective cloud clusters on the east-west ground convergence line leads to new development and forms a "train effect." Strong frontogenesis in both horizontal and vertical directions triggers and sustains this heavy precipitation event. The significant increase in liquid water and water vapor density has indicative significance for the occurrence of heavy precipitation.

The latest World Atlas of Desertification represents a turning point in the diagnosis of desertification. While it forgoes desertification mapping due to the intrinsic complexity of the phenomenon and the impossibility of measuring it using a single indicator, it introduces the Convergence of Evidence paradigm, which identifies socioeconomic and biophysical variables whose behavior allows pointing out those areas prone to desertification. The Spanish National Action Program Against Desertification (PAND), back to 2008, already implemented a similar approach to identify five “desertification landscapes” within Spain using both socio-economic and climatic information. The PAND was not only pioneering but also, unfortunately, accurate. Desertification in Spain has continued to worsen and the first two decades of the 21st century have consolidated an agri-food model whose dynamics have exacerbated the desertification processes identified in the PAND. Despite its scientific value, the PAND lacked a proper action plan and was completely detached from the diagnosis. As a result the diagnosis it provided was not followed by effective actions to halt desertification in Spain. The Spanish Government's recent declaration of climate and environmental emergency requires a new strategy to combat desertification. This commitment is an excellent opportunity to update the diagnosis of the situation and, more crucially, to unify the different desertification sectoral policies and actions under a single front. We provide here elements for a roadmap to be designed around the pressures, impacts, and drivers that define today's Spanish desertification landscapes to manage them effectively and avoid their further degradation.

The theory of the origin of carcinomas suggests that neoplasias from epithelial tissues are the consequence of the reactivation of developmental programs. It proposes a model in which the epithelial cells undergo cellular transitions due to replicative senescence and inflammation towards a mesenchymal-undifferentiated phenotype with cancerous behavior. The conserved pattern of histological progression and the molecular biology of carcinomas is congruent with the view of cancer as a developmental disease. In support of the theory, the experimental literature regarding the molecular, cellular, and histopathological mechanisms associated with epithelial carcinogenesis were aligned according to the premises of the hypothesis. Thereby identified a generic process in the carcinogenesis of the breast, endometrium, prostate, colon, lung, pancreas, bladder, liver, and cervix. Then, is provided a methodology overview of modeling in systems biology derived from previous research testing the hypothesis. The results illustrate the value of the complex systems approach to recover behavior that cannot be inferred only by traditional methods. Specifically, the model suggests that the consistency in the cell types and the directionality of the observed cellular transitions during epithelial carcinogenesis arise from structural constraints in the molecular networks associated with the carcinomas. Overall, the results of the dynamical analysis agree with the premises of the hypothesis and provide an insightful perspective of the potential mechanisms underlying the cellular plasticity displayed during epithelial carcinogenesis. In an era of big data and yet few advances in the underlying causes of chronic diseases, the manuscript also aims to inspire molecular biologists to integrate existing empirical evidence with systems biology modeling in the pursuit of understanding.

Evolutionary convergence provides natural opportunities to investigate how, when, and why novel traits evolve. Many convergent traits are complex, highlighting the importance of explicitly considering convergence at different levels of biological organization, or ‘multi‐level convergent evolution’. To investigate multi‐level convergent evolution, we propose a holistic and hierarchical framework that emphasizes breaking down traits into several functional modules. We begin by identifying long‐standing questions on the origins of complexity and the diverse evolutionary processes underlying phenotypic convergence to discuss how they can be addressed by examining convergent systems. We argue that bioluminescence, a complex trait that evolved dozens of times through either novel mechanisms or conserved toolkits, is particularly well suited for these studies. We present an updated estimate of at least 94 independent origins of bioluminescence across the tree of life, which we calculated by reviewing and summarizing all estimates of independent origins. Then, we use our framework to review the biology, chemistry, and evolution of bioluminescence, and for each biological level identify questions that arise from our systematic review. We focus on luminous organisms that use the shared luciferin substrates coelenterazine or vargulin to produce light because these organisms convergently evolved bioluminescent proteins that use the same luciferins to produce bioluminescence. Evolutionary convergence does not necessarily extend across biological levels, as exemplified by cases of conservation and disparity in biological functions, organs, cells, and molecules associated with bioluminescence systems. Investigating differences across bioluminescent organisms will address fundamental questions on predictability and contingency in convergent evolution. Lastly, we highlight unexplored areas of bioluminescence research and advances in sequencing and chemical techniques useful for developing bioluminescence as a model system for studying multi‐level convergent evolution.

Maximum likelihood estimation (MLE) is a popular method for parameter estimation in both applied probability and statistics but MLE cannot solve the problem of incomplete data or hidden data because it is impossible to maximize likelihood function from hidden data. Expectation maximum (EM) algorithm is a powerful mathematical tool for solving this problem if there is a relationship between hidden data and observed data. Such hinting relationship is specified by a mapping from hidden data to observed data or by a joint probability between hidden data and observed data. In other words, the relationship helps us know hidden data by surveying observed data. The essential ideology of EM is to maximize the expectation of likelihood function over observed data based on the hinting relationship instead of maximizing directly the likelihood function of hidden data. Pioneers in EM algorithm proved its convergence. As a result, EM algorithm produces parameter estimators as well as MLE does. This tutorial aims to provide explanations of EM algorithm in order to help researchers comprehend it. Moreover some improvements of EM algorithm are also proposed in the tutorial such as combination of EM and third-order convergence Newton-Raphson process, combination of EM and gradient descent method, and combination of EM and particle swarm optimization (PSO) algorithm.

In this paper, we investigate the dynamics of following higher order difference equation x_{n+1}=A+B((x_{n})/(x_{n-m}²)) with A,B and initial conditions are positive numbers, and m∈{2,3,⋯}. Especially we study the boundedness, periodicity, semi-cycles, global asymptotically stability and rate of convergence of solutions of related higher order difference equations.

We identify the source of the Mw = 5.6 earthquake that hit west-central Epirus on March 21, 2020 00:49:52 UTC. We use synthetic aperture radar interferograms tied to one permanent Global Navigation Satellite System (GNSS) station (GARD). We model the source by inverting the INSAR displacement data. The inversion model suggests a shallow source on a low-angle fault (39°) dipping towards east with a centroid depth of 8.5 km. The seismic moment deduced from our model agrees with those of the published seismic moment tensors. This geometry is compatible with the Margariti thrust fault within the collision zone between Apulia and Eurasia. We also processed new GNSS data and estimate a total convergence rate between Apulia and Eurasia of 8.9 mm yr-1 , of which shortening of the crust between the Epirus coastal GNSS stations and station PAXO in the Ionian Sea is equivalent to ~ 50% of it or 4.6 mm yr−1. A 60-km wide deformation zone takes up nearly most of the convergence between Apulia-Eurasia, trending N318°E. Its central axis runs along the southwest coast of Corfu, along the northeast coast of Paxos, heading toward the northern extremity of the Lefkada island.

Relying on the USA, Canada and Mexico extract from the cross-national data sample on the environmental affection and cognition of adolescent students (Niankara, 2019), along with seemingly unrelated bivariate weighted ordered probit regression modeling (Niankara and Zoungrana, 2018), this study reports on the convergence of technological awareness and expectations within the context of international trade. We achieve this by adopting a regional perspective in investigating the effects of affective, cognitive and situational factors on youth's awareness and expectations about genetically modified organisms (GMOs) and nuclear power technology (NPT) within the North American free trade block. Identification of model parameters is achieved using maximum simulated likelihood methods. The findings show that although it has been over 20 years as of 2015 that USA, Canada, and Mexico ratified the north American free trade agreement (NAFTA), the diffusion of technology and information within the trade block has not succeeded in homogenizing awareness and expectations about GMOs and Nuclear power technology, as observed in the youth population across the three countries. Indeed, with regards to technological awareness, compared to youth from the USA, those from Canada show 15% (GMOs) and 7.1% (NPT) more awareness respectively; while those in Mexico are respectively 34.4% and 19.5% less aware about GMOs and NPT. With respect to technological expectations, compared to youth from the USA, those from Canada and Mexico are respectively 34.4% and 39.9% more optimistic about GMOs, while 15% and 49.7% more optimistic about NPT. Overall, youth within NAFTA country members are respectively 2.5% and 6.7% more optimistic about GMOs and NPT for every level increase in their awareness about the two technologies.

Centralized/Cloud Radio Access Networks (C-RAN) are deployed in converged fixed-mobile networks to exploit the flexibility coming from joint application of Network Function Virtualization (NFV) and Software Defined Networking (SDN). In this context, optical links connecting C-RAN nodes, possibly based on the Ethernet standards, may carry traffic with different requirements in terms of latency and throughput. This paper considers the problem of traffic aggregation on C-RAN optical Ethernet links with latency control for fronthaul traffic and throughput capability for backhaul traffic. Integrated hybrid nework technique is applied to show how time transparency can be enforced for Ethernet encapsulated Common Public Radio Interface (CPRI) traffic while allowing statistical multiplexing of backhaul traffic. Simulation results show the effectiveness of segmentation of backhaul traffic to allow exploitation of the available bandwidth even with high capacity CPRI options.

This article is aimed at studying new finite difference methods for one-dimensional (1D) and two-dimensional (2D) space Riesz variable-order (VO) nonlinear fractional diffusion equations (SRVONFDEs). In the presented model, fractional derivatives are defined in the Riemann-Liouville type. Based on 4-point weighted-shifted-Gr\"unwald-difference (4WSGD) operators for Riemann-Liouville constant-order (CO) fractional derivatives, which have a free parameter and have at least third order accuracy, we derive 4WSGD operators for space Riesz VO fractional derivatives. In order that the fully discrete schemes have good stability and can handle the nonlinear term efficiently, we apply the implicit Euler (IE) method to discretize the time derivative, which leads to IE-4WSGD schemes for SRVONFDEs. The stability and convergence of the IE-4WSGD schemes are analysed theoretically. In addiction, a parameter selection strategy is derived for 4WSGD schemes and banded preconditioners are put forward to accelerate the GMRES methods for solving the discretization linear systems. Numerical resutls demonstrate the effectiveness of the proposed schemes and preconditioners.

In conventional Federated Learning (FL), clients work together to train a model managed by a central server, intending to speed up the learning process. However, this approach imposes significant computational and communication burdens on clients, particularly with complex models. Additionally, while FL strives to protect client privacy, the server's access to local and global models raises security concerns. To address these challenges, Split Learning (SL) separates the model into parts handled by the client and the server, though it suffers from inefficiencies due to sequential client participation. To overcome these issues, SplitFed Learning (SFL) was proposed, which combines the parallelism of FL with the model-splitting strategy of SL, enabling simultaneous training by multiple clients. Our main contribution is the theoretical analysis of SFL, which, for the first time, includes non-i.i.d. datasets, non-convex loss functions, and both full and partial client participation. We provide convergence proofs for a state-of-the-art SFL algorithm based on conventional convergence analysis assumptions for FL. Our results prove that we can recover the linear convergence rate of conventional FL for the SFL algorithm with the distinction that increasing the number of local steps or clients may not speed up the convergence in SFL.

We overview the main equations of the Rayleigh-Ritz variational method and discuss their connection with the problem of simultaneous diagonalization of two Hermitian matrices.

Randomized block Kaczmarz Method and randomized extended block Kaczmarz Method are proposed for solving matrix equation AXB=C, where the matrix A and B may be full rank or rank deficient. These methods are iterative methods without matrix multiplication, and are especially suitable for solving large-scale matrix equations. It is theoretically proved these methods converge to the solution or least square solution of the matrix equation. The numerical results show that these methods are more efficient than the existing algorithms for high-dimensional matrix equations.

Recently, the triple-network convergence system (TNCS) has emerged from the deep integration of the power grid, transportation network, and information network. Fault recovery research in the TNCS is important since this system's complexity and interactivity can expand the faults scale and increase faults impact. Currently, fault recovery focuses primarily on single power grids and cyber-physical systems, but there are certain shortcomings, such as ignoring uncertainties including generator start-up failures and the occurrence of new faults during recovery, energy supply-demand imbalances leading to system security issues and communication delay caused by network attacks. In this study, we propose a recovery method based on the improved TD3 algorithm, factoring in shortcomings of the existing research. Specifically, we establish a TNCS model to analyze interaction mechanisms and design a state matrix to represent the uncertainty changes in the TNCS, a negative reward to reflect the impact of unit start-up failures, a special reward to reflect the impact of communication delay and an improved Actor network update mechanism. Experimental results show that our method obtains the optimal recovery decisions, maximizes restoration benefit in power grid failure scenarios and demonstrates a strong resilience against communication delay caused by DoS attacks.

In this paper, we analyze the convergence of a finite difference method with the implicit forward time central space (FTCS) scheme for solving the three-dimensional advection-diffusion-reaction equation (ADRE). It is proved that the method is unconditionally convergent. We apply the scheme to obtain numerical solutions for the transport of pollutants in street tunnel problems with various reaction coefficients and various rates of change of concentrations of sources or sinks of pollution.

In this paper we investigate the global asymptotic stability of following system ofhigher order difference equations with quadratic terms:xn+1=A+Byn/yn−m^2, yn+1=A+Bxn/xn−m^2, where A and B are positive numbers and the initial values are positive numbers.We also study the boundedness, rate of convergence and oscillation behaviour of thesolutions of related system.

This paper aims to investigate the global asymptotic stability of following system of higher order difference equations with quadratic terms: x_{n+1}=A+B((y_{n})/(y_{n-m}²)),y_{n+1}=A+B((x_{n})/(x_{n-m}²)) where A and B are positive numbers and the initial values are positive numbers. We also study the rate of convergence and oscillation behaviour of the solutions of related system.

This research work deals with the formulation of numerical scheme with algorithm to compute singularly perturbed partial differential equations(SP-PDEs) involving a significant negative shift. The problem involves small parameter which causes a rapidly changing boundary layers in the vicinity of a body, and the negative shift term causes interior layer. Such appearance of an abruptly varying layers causes difficulties to find the exact solution and it is not adequate to employ classical numerical methods. The technique presented in this research work simplifies these challenges and yields accurate numerical solution.The stability and convergence analyses of the methods are examined and proven. To test the developed technique, numerical experiments are carried out and confirmed with theoretical analysis.

The present work concerns with presenting an explicit formula for new shifted wavelet (NSW) functions. Wavelet functions have many applications and advantages in both applied and theoretical fields. They are formulated with different orthogonal polynomials to construct new techniques for treating some problems in sciences, and engineering. A new important differentiation property of NSW in terms of NSW themselves is obtained and proved in this paper. Then it is utilized together with the state parameterization technique to find solution of optimal control problem (OCP) approximately. The suggested method converts the OCP into a quadratic programming problem, which can be easily determined on computer. As a result, the approximate solution closes with the exact solution even with a small number of NSW utilized in estimation. The error bound estimation for the proposed method is also discussed. Some test numerical examples are solved to demonstrate the applicability of the suggested method. For comparison, the exact known solutions against the obtained approximate results are listed in Tables.

This paper proposed a time two-mesh (TT-M) finite difference numerical scheme to improve the efficiency of solving the symmetric regularized long wave (SRLW) equation. The TT-M Crank-Nicolson discretization and finite difference method are employed in time and space approximation respectively. The scheme involves three main steps: firstly, the time interval is divided into coarse and fine time meshes, then the nonlinear system is solved on the coarse time mesh; secondly, coarse numerical solutions on the fine time mesh are computed using an interpolation formula based on the solutions derived in the step one; lastly, the TT-M finite difference numerical solutions can be obtained through constructing the linearized fine time mesh system using Taylor’s formula. Compared to the currently existing TT-M numerical methods, the novelty of this study is that the nonlinear term including derivatives is linearized by Taylor’s formula for a function with three variables, whose error analysis is more complex. Finally, some numerical examples, including computational time and accuracy, preservation of conservation laws, are given to verify the efficiency of the scheme. By comparing it with the standard nonlinear finite difference scheme, this method can reduce CPU time without sacrificing accuracy.

In this paper, we study the global asymptotically stability of following system of difference equations with quadratic terms: x_{n+1}=A+B((y_{n})/(y_{n-1}²)),y_{n+1}=A+B((x_{n})/(x_{n-1}²)) where A and B are positive numbers and the initial values are positive numbers. We also investigate the rate of convergence and oscillation behaviour of the solutions of related system.

This paper examines whether the Great Recession has altered the disparities of the US regional health care expenditures. We test the null hypothesis of convergence for the US real per capita health expenditure for the period 1980-2014. Our results indicate that the null hypothesis of convergence is clearly rejected for the total sample as well as for the pre-Great Recession period. Thus, no changes are found in this regard. However, we find that the Great Recession has modified the composition of the estimated convergence clubs, offering a much more concentrated picture in 2014 than in 2008, with most of the states included in a big club, and only 5 (Nevada, Utah, Arizona, Colorado and Georgia) exhibiting a different pattern of behavior. These two estimated clubs diverge and, consequently, the disparities in the regional health sector have increased.

The aim of this article is to extend the local as well as the semi-local convergence analysis of multi-point iterative methods using center Lipschitz conditions in combination with our idea, of the restricted convergence region. It turns out that this way a finer convergence analysis for these methods is obtained than in earlier works and without additional hypotheses. Numerical examples favoring our technique over earlier ones completes this article.

To solve large scale linear equations involved in the Fast Multipole Boundary Element Method (FM-BEM) efficiently, an iterative method named the generalized minimal residual method (GMRES)(m)algorithm with Variable Restart Parameter (VRP-GMRES(m) algorithm) is proposed. By properly changing a variable restart parameter for the GMRES(m) algorithm, the iteration stagnation problem resulting from improper selection of the parameter is resolved efficiently. Based on the framework of the VRP-GMRES(m) algorithm and the relevant properties of generalized inverse matrix, the projection of the error vector r_m+1 on r_m is deduced. The result proves that the proposed algorithm is not only rapidly convergent but also highly accurate. Numerical experiments further show that the new algorithm can significantly improve the computational efficiency and accuracy. Its superiorities will be much more remarkable when it is used to solve larger scale problems. Therefore, it has extensive prospects in the FM-BEM field and other scientific and engineering computing.

Changes in hydrology from aridification in the U.S. Southwest have led to tension at the urban-rural divide about conservation and land management strategy. Strain on multi-generational agricultural livelihoods and a nearly 150-year-old Colorado River water adjudication system presents an opportunity for multidisciplinary solutions and symbiotic partnerships. In this study, producers in a small San Juan River headwater community in southwestern Colorado participated in a convergence research study developed collaboratively with community partners, prioritizing local objectives and policy needs. Acknowledging the historic and sometimes perceived role of academic institutions as representing urban interests, our goal was to highlight how research could support rural governance and identify rural interests. Respondents identified water availability and climate changes as the factor that most negatively affects operations. Respondents had interest in nature-based agroforestry methods like windbreak trees and drought-resistant crop species. Community members identified cost as the leading perceived barrier to implementing nature-based agricultural solutions, with over half of respondents indicating that programs to subsidize or eliminate cost would improve willingness to try nature-based methods. Analyses identified that satisfaction with community resources was positively correlated with scale of production, satisfaction with irrigation equipment, and familiarity with water rights. Further, our results outline barriers to perceptions about nature-based solutions on rural landscapes. We hope to contribute this model for wider applications in other regions to uncover place-based solutions to resource challenges and contribute to addressing stress at the urban-rural divide.

Homotopy perturbation and analysis methods have been widely used to obtain both approximate and exact so- lutions to nonlinear problems. In general, these two methods are based on the Taylor series with respect to an embedding parameter. Many researchers have compared the two methods and raised more concerns on the homo- topy perturbation method (HPM) because the homotopy analysis method (HAM) contains a convergence-control parameter ~: For this reason, in this article, a more general form of HPM is introduced as the -homotopy per- turbation method (-HPM), which contains a control parameter : The introduction of parameter  in this new modication gives a better way to adjust and control the convergence region and the rate of the series solution. We conrm through the given examples in this study that the HPM is a special case of the -HPM. The error and convergence analysis of this proposed method are also presented

In this paper, we present a three-step sixth-order iterative schemes to estimate the solutions of a nonlinear systems of equations, for predicting the shear strength of a reinforced concrete beam. This procedure is designed by means of a weight function technique. The values for the parameters of this system were randomly selected inside the prescribed ranges by technical standards for structural concrete; moreover, some of this parameters were fixed taking into consideration the solvability region of the adopted steel constitutive model. The efficiency of the new class is compared with other current schemes, with very good results.

This study used a two-stage network data envelopment analysis model to measure the water use efficiency of 108 cities in the Yangtze River Economic Belt in the initial water use and wastewater treatment phases from 2009 to 2019. We used the Dagum Gini coefficient to decompose the urban water use efficiency of six major urban clusters in the Yangtze River Economic Belt. We also used σ convergence and β convergence types to test the convergence characteristics of urban water use efficiency of six major urban clusters in the Yangtze River Economic Belt. This study found that the overall low level of water use efficiency in cities in the Yangtze River Economic Zone mainly stems from the low level of water use efficiency in the wastewater treatment stage. The 108 cities in the Yangtze River Economic Zone are divided into four types based on the average values of water use efficiency in the initial use and wastewater treatment phases; the highest number of cities are in the double-low category, with low average values of water use efficiency in the initial use and wastewater treatment phases. During the study period, spatial differences in urban water use efficiency in the Yangtze River Economic Zone narrowed, with the differences stemming mainly from hyperdensity, followed by intra- and inter-regional differences. Meanwhile, there is convergence in urban water use efficiency in the Yangtze River Economic Belt; significant β convergence in the urban agglomerations of the Yangtze River Delta, Jianghuai, middle reaches of the Yangtze River, Chengdu–Chongqing, and Central Yunnan; and insignificant β convergence in the Central Qian urban agglomeration. After considering control factors, such as industrial structure, financial development level, environmental regulation, economic development level, and science and education development level, the water use efficiency of the six major urban clusters in the Yangtze River Economic Belt converges faster, but the influence of these control factors on the water use efficiency of each urban cluster is heterogeneous. Research results have reference value for the development of improvement strategies on differentiated urban water use efficiency in the Yangtze River Economic Belt.

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

Optimization techniques are pivotal across various scientific domains, typically involving solving problems among feasible alternatives based on specific goals, alternatives, or constraints. In mathematical optimization, traditional methods like Simulated Annealing (SA) do not guarantee finding global minima/optima, especially in cases that contain high dimensions in objective functions. This paper introduces a novel approach where a free-energy driven self-adaptive SA algorithm is designed to handle such cases by incorporating free-energy costs within the Metropolis-Hastings framework. This algorithm not only dynamically adjusts to the changing dimensions of the objective function, but it also enhances a faster optimization process. Furthermore, as examples we demonstrate its capability on a convex unimodal function and the non-convex Rastrigin function, revealing faster convergence to search global minima compared to standard SA algorithm. At last, parameter estimation of noisy exponential data has been executed by our Simulated Annealing driven by the Free-Energy (SAFE) algorithm. Our results suggest that this novel approach may significantly improve optimization speed and accuracy, providing a robust tool for complex multidimensional optimization problems.

Pulsating high frequency voltage injection can be used for sensorless control of SPMSM at zero and low speed ranges. However, the sensorless capability still faces challenges to the requirements of industrial application, especially at heavy load status. Aiming at this issue, this article proposes a sensorless capability expansion method for SPMSM based on inductance parameter identification. First, incremental inductances at d-q-axis and cross-coupling inductance are identified by three steps combining the rotating high frequency voltage injection and pulsating high frequency voltage injection, then, using polynomial curve fitting algorithm, apparent inductances are calculated. Second, positive DC current injection at d-axis is proposed to enhance the saliency ratio based on the analysis of parameter identification results. Compared with the conventional =0 or < 0 method, when positive DC current is injected at d-axis, saliency ratio is enhanced. Third, convergence region is expanded at heavy load status and accuracy of rotor position estimation is improved using the proposed method. Finally, the experiment results validate that the sensorless capability of SPMSM is expanded.

In this paper, we present a new parametric class of optimal fourth-order iterative methods to estimate the solutions of nonlinear equations. After the convergence analysis, we study the stability of this class by using the tools of complex discrete dynamics. This allows us to select those elements of the class with lower dependence on initial estimations. We verify by means of numerical tests that the stable members on quadratic polynomials perform better than the unstable ones, when applied to other non-polynomial functions. We also compare the performance of the best elements of the family with known methods, such as the schemes by Newton or Chun, among others, showing robust and stable behaviour.

In this paper, an iterative procedure to find the solution of a nonlinear structural model is introduced. The model presents different multiplicities where parameters are randomly selected within a solvability region. To achieve this aim, a class of multipoint fixed-point iterative schemes for single roots is modified to find multiple roots, reaching the fourth order of convergence. Complex discrete dynamics techniques are employed to select the members with the most stable performance. The structural problem, as well as some academic problems involving multiple roots, are solved numerically to verify the theoretical analysis, robustness and applicability of the proposed scheme.

The Chebyshev collocation method (ChC) implemented as Chebfun is used in order to solve a class of second order one-dimensional singular and genuinely nonlinear boundary value problems. Efforts to solve these problems with conventional ChC have generally failed, and the outcomes obtained by finite differences or finite elements are seldom satisfactory. We try to fix this situation using the new Chebfun programming environment. However, for toughest problems we have to loosen the default Chebfun tolerance in Newton's solver as the ChC runs into trouble with ill-conditioning of the spectral differentiation matrices. Although in such cases the convergence is not quadratic the Newton updates decrease monotonically. This fact, along with the decreasing behaviour of Chebyshev coefficients of solutions, suggests that the outcomes are trustworthy, i.e., the collocation method has exponential (geometric) rate of convergence or at least an algebraic rate. We consider first a set of problems that have exact solutions or prime integrals and then another set of benchmark problems that do not possess these properties. Actually, for each test problem carried out we have determined how Chebfun solution converges, its length, the accuracy of the Newton method and especially how well the numerical results overlap with the analytical ones (existence and uniqueness).

Arguments of Mises and Hayek, who opposed the planned economy (PE), are used in the paper as a starting point for establishing the objective area of effective application of the PE. The abstract model of PE, based on the definitions of Mises and Hayek, leads to the conclusion that for the effective use of PE, it must be a part of mixed economy and it must produce a limited amount of essential goods of irreducible demand. These goods must be distributed among all members of society free of charge, evenly, without competition. Examples of a mixed economy are given that meet this requirement. Calculations of the personal benefit in the transition to a mixed model of the economy have been carried out. The positive and negative qualities of the planned and market methods of organization are considered. Mixed economy model combines these qualities in optimal construction. An analytical framework has been introduced for the construction of product characterization curves. Such curves provide criteria for determining the efficiency of manufacturing of this product in a planned economy. The general economic prerequisites for the usability of the PE are clarified. The applicability and advantages of the PE for the organization of the universal basic income (UBI) system are demonstrated. The possibility of using PE to solve the problem of guaranteed employment is mentioned.

Decision tree algorithms have been among the most popular algorithms for interpretable (transparent) machine learning since the early 1980s. On the other hand, deep learning methods have boosted the capacity of machine learning algorithms and are now being used for non-trivial applications in various applied domains. But training a fully-connected deep feed-forward network by gradient-descent backpropagation is slow and requires arbitrary choices regarding the number of hidden units and layers. In this paper, we propose near-optimal neural regression trees, intending to make it much faster than deep feed-forward networks and for which it is not essential to specify the number of hidden units in the hidden layers of the neural network in advance. The key idea is to construct a decision tree and then simulate the decision tree with a neural network. This work aims to build a mathematical formulation of neural trees and gain the complementary benefits of both sparse optimal decision trees and neural trees. We propose near-optimal sparse neural trees (NSNT) that is shown to be asymptotically consistent and robust in nature. Additionally, the proposed NSNT model obtain a fast rate of convergence which is near-optimal up to some logarithmic factor. We comprehensively benchmark the proposed method on a sample of 80 datasets (40 classification datasets and 40 regression datasets) from the UCI machine learning repository. We establish that the proposed method is likely to outperform the current state-of-the-art methods (random forest, XGBoost, optimal classification tree, and near-optimal nonlinear trees) for the majority of the datasets.

Not only does this paper present a novel type-2 fuzzy system for identification and behavior prognostication of an experimental solar cell set and a wind turbine, but also it brings forward an exquisite technique to acquire an optimal number of membership functions and the corresponding rules. It proposes a seven-layered NCPRT2FS. For fuzzification in the first two layers, Gaussian type-2 fuzzy membership functions with uncertainty in the mean, are exploited. The third layer comprises rule definition and the forth one embeds fulfillment of type reduction. The three last remained layers are the ones in which resultant left–right firing points, two end-points and output all get assessed correspondingly. It should not be neglected off the nutshell that recurrent feedback at the fifth layer exerts delayed outputs ameliorating efficiency of the suggested NCPRT2FS. Later in the paper, a modern structural learning, established on type-2 fuzzy clustering, is held forth. An adaptively rated learning back-propagation algorithm is extended to adjust the parameters ensuring the convergence as well. Eventually, solar cell photo-voltaic and wind turbine are deemed as case studies. The experimental data are exploited and the consequent yields emerge so persuasive.

Fourier expansions by the polyharmonic-Neumann eigenfunctions showed improved convergence compared to the Fourier expansions by the classical trigonometric system due to the rapid decay of the corresponding Fourier coefficients. Based on this evidence, we investigate interpolations on a finite interval that are exact for the polyharmonic-Neumann eigenfunctions and show the same benefits. Further, we improve the convergence of the interpolations by applying the idea of quasi-periodicity, where the basis functions are periodic on a slightly extended interval. We show that those interpolations have much better convergence rates away from the endpoints of the approximation interval. Moreover, the interpolations are more accurate on the entire interval. We prove those properties for a specific case of the polyharmonic-Neumann eigenfunctions known as the modified Fourier system. For other basis functions, we provide evidence based on the results of numerical experiments.

We consider a Cauchy problem for differential equations in a Hilbert space $X$. The problem is stated in a time interval $I$, which can be finite or infinite. Weuse a fixed point argument for history-dependent operators to prove the unique solvability of the problem. Then, we state and prove convergence criteria for both a general fixed point problem and thecorresponding Cauchy problem. These criteria provide necessary and sufficient conditions on a sequence $\{u_n\}$ which guarantee its convergence to the solution of the corresponding problem, in the space of both continuous and continuously differentiable functions. We then specify our results in the study of a particular differential equation governed by two nonlinear operators. Finally, we provide an application in viscoelasticity and give a mechanical interpretation of the corresponding convergence result.

This study identified, analysed, and prioritised significant factors for standardising the Korean smart city project evaluation system. We analysed the efficiency and productivity of companies currently providing smart city services to consider both policy and practical aspects. The prioritisation of smart city planning reflects the latest trends in South Korea, where urban planning is moving towards smart city planning. Furthermore, the capacity building of public-private/private-private partnerships indicate the importance of business scalability. This indicates that smart city services are only stabilised when the private sector is involved and leads the project, rather than focusing on public development. The feasibility of building intelligent facilities indicates that smart city projects should be implemented after securing cost-benefit feasibility. The results were used as the basis for building an evaluation system, showing that in smart city convergence alliances, small- and medium-sized enterprises achieved the highest efficiency by reducing inputs to 81% and 86%, under the assumptions of constant and variable returns to scales, respectively. As the operational aspect is more problematic than the technical aspect, policy alternatives are necessary for smooth business progress, including increased flexibility of laws and institutions and the activation of policies that temporarily relieve regulations to demonstrate smart city projects.

Because of the rapid advancement in the use of photovoltaic (PV) energy systems, it has become critical to look for ways to improve the energy generated by them. The extracted power from the PV modules is proportional to the output voltage. The relationship between output power and array voltage has only one peak under uniform irradiance, whereas it has multiple peaks under partial shade circumstances (PSC). There is only one global peak (GP) and many local peaks (LPs), where the typical maximum power point trackers (MPPT) may become locked in one of the LPs, significantly reducing the PV system's generated power and efficiency. The metaheuristic optimization algorithms (MOAs) solved this problem, albeit at the expense of the convergence time, which is one of these algorithms' key shortcomings. Most MOAs attempt to lower the convergence time at the cost of the failure rate and the accuracy of the findings because these two factors are interdependent. To address these issues, this work introduces the dandelion optimization algorithm (DOA), a novel optimization algorithm. The DOA's convergence time and failure rate are compared to other modern MOAs in critical scenarios of partial shade PV systems to demonstrate the DOA's superiority. The results obtained from this study showed substantial performance improvement compared to other MOAs, where the convergence time is reduced to 0.4 s with zero failure rate compared to 0.9 s, 1.25 s, and 0.43 s for other MOAs under study. The optimal number of search agents in the swarm, optimal initialization of search agents, and optimal design of the dc-dc converter is introduced for optimal MPPT performance.

The influence of neutrosophy in the previous period is constantly growing in many areas of science and technology. Moreover, various applications of the neutrosophic approach have become more common in recent years. Our goal in this research is to utilize the neutrosophy to improve the performance of the Dai-Liao conjugate gradient (CG) method. Specifically, in this research, we propose and investigate a new neutrosophic logic system to calculate the key parameter t involved in the Dai–Liao CG iterations. Theoretical analysis and numerical experience indicate that the efficiency and robustness of the new rule for determining t. Combining the neutrosophy and the Dai-Liao conjugate gradient method, we propose and explore a new Dai-Liao CG iterations for solving large-scale unconstrained optimization models. The global convergence is established under common assumptions and the backtracking line search. Finally, by conducting numerical experiments, computational evidence demonstrates that the new fuzzy neutrosophic Dai-Liao conjugate gradient method is computationally effective and robust.

DGJ (Daftardar-Gejii-Jafaris) method is used to obtain numerical solution of the third order fractional differential equation. Providing the DGJ method converges, the approximate solution is a good and effective numerical result which is close to the exact solution or the exact solution. For this,the examples of the explaning the method are presented. The proposed method is implemented for the approximation solution of the third order nonlinear fractional partial differential equations. The method was shown to be unsuitable and inconsistent for an example of a nonlinear fractional partial differential equation depend on initial-boundary value conditions. The fact that these numerical results are not consistent can be explained by the fact that the method is not convergent.

Neural network regularized learning has garnered significant attention in recent years. We give a systematic investigation on the performance of regularized regression associated zonal translation networks. We propose the concept of Marcinkiewicz-Zygmund inequality Setting (MZIS) for the scattered nodes collected from the unit sphere. We show that, under the MZIS, the corresponding convolutional zonal translation network has reproducing property. Based on these facts,we propose a kind of kernel regularized regression learning framework and provide upper bound estimate for the learning rate with the kernel approach. We also give proof for the density of the zonal translation network with spherical Fourier analysis.We provide the approximation error with a K-functional.

The fast decoding of low-density parity-check (LDPC) is crucial for its future applications. In this paper, we propose to use Early Termination (ET) and Force Convergence (FC) techniques to accelerate the decoding of two new local/global two-phases decoding algorithms to reduce the Globally Coupled (GC) LDPC codes. Specifically, ET is applied in the local decoding phase of two- phases decoding of GC-LDPC codes to skip the local decoding which is highly likely to fail and avoid the unnecessary iterations. FC is applied in global decoding phase where converged variable nodes are identified and removed from Tanner graph and the decoding can be speed up shrinking parity check matrices. Two variants are proposed for the ET operation in local decoding phase. ET-1 merely stops iteration if it meets the stopping criterion while in ET-1, the iteration number saved in local decoding is added up in global decoding to improve the decoding performance. The simulation results show that the ET-1-FC scheme can save up to 42% decoding time complexity in the low SNR region while keeping the error rate performance nearly unchanged, the ET-2-FC can improve the error rate performance by 0.18dB at bit error rate 0.001 or 0.23dB at frame error rate 0.1. In the waterfalling region of BER/FER curves, both schemes can save about 25% decoding time complexity.

In the present study, a mathematical model of non-steady partial differential equation from the process of oxygen mass transport in the human pulmonary circulation is proposed. Mathematical modelling of this kind of problems lead to a non-steady partial differential equation and for its numerical simulation, we have used finite differences. The aim of the process is the exact numerical analysis of the study, wherein consistency, stability and convergence is proposed. The necessity of doing the process is that, we would like to increase the order of numerical solution to a higher order scheme. An increment in the order of numerical solution makes the numerical simulation more accurate, also makes the numerical simulation being more complicated. In addition, the process of numerical analysis of the study in this order of solution needs more research work.

The purpose of this study is to apply the convergence-confinement method (CCM) based on the assumption of hydrostatic stress to the surrounding rock excavated in a circular tunnel whose rock mass characteristics meet the requirements of the non-linear Hoek-Brown failure criterion. Based on a consistent and rigorous method of theoretical analysis, a closed-form analytical solution for the stress/displacement of the ground reaction induced by the advancing excavation of the tunnel, especially in the elastic and plastic regions, is derived and investigated under the condition of an isotropic stress state. By taking the value of a confinement loss as an explicit analysis process in an incremental manner, not only is the computationally executable analytical solution achieved, but it can also be performed using a simple spreadsheet. The validity of the analytical solution implemented by the incremental procedure was examined with published data to study the effect of nonlinear failure criteria on the ground reaction curve; in particular the mechanical behavior at the intrados of the tunnel, and the distribution of stress/displacement on the cross-section of the tunnel. A comparison between the published results and the solution proposed in this study, which interprets the closed-form by the incremental procedure, shows consistent and favorable trends.

In this work, we offer a novel and accurate method in order to find the solution of the linear differential equations over the intervals [0, 1) based on the generalization of Legendre wavelets. The mechanism is still upon workable implementation of the operational matrix of integration and its derivatives. This method reduces the problems into algebraic equations via the properties of generalized Legendre wavelet (GLW) together with the operational matrix of integration. The function approximation has been picked out in such a way so as to enumerate the connection coefficients in an facile manner. The proposed numerical technique, based on the GLW, has been examined on three linear problems as a consequence of this investigation. The outcomes have shown that this method, as opposed to some other existing numerical and analytical methods, is a very useful and advantageous for tackling such problems.

The search algorithm based on symbiotic organisms’ interactions is a relatively recent bio-inspired algorithm of the swarm intelligence field for solving numerical optimization problems. It is meant to optimize applications based on the simulation of the symbiotic relationship among the distinct species in the ecosystem. The modified SOS algorithm is developed to solve independent task scheduling problems. This paper proposes a modified symbiotic organisms search based scheduling algorithm for efficient mapping of heterogeneous tasks to access cloud resources of different capacities. The significant contribution of this technique is the simplified representation of the algorithm's mutualism process, which uses equity as a measure of relationship characteristics or efficiency of species in the current ecosystem to move to the next generation. These relational characteristics are achieved by replacing the original mutual vector, which uses an arithmetic mean to measure the mutual characteristics with a geometric mean that enhances the survival advantage of two distinct species. The modified symbiotic organisms search algorithm (G_SOS) aimed to minimize the task execution time (Makespan), response, degree of imbalance and cost and improve the convergence speed for an optimal solution in an IaaS cloud. The performances of the proposed technique have been evaluated using a Cladism toolkit simulator, and the solutions are found to be better than the existing standard (SOS) technique and PSO.

Trajectory planning determines whether a UAV can successfully complete its mission, and the reasonable planning of UAV trajectory in 3D environment is a complex global optimization problem, which needs to take into account many constraints such as urban environment, mountain terrain, obstacles, no-fly zones, flight boundaries, flight distances, and trajectory change rates. In view of the shortcomings of the whale optimization algorithm in 3D trajectory planning, such as slow convergence speed, low accuracy and easy to fall into the local optimum, this paper increases the diversity of the initial population through the introduction of the reverse learning mechanism, and optimizes the coordination of the global and local search ability by integrating the nonlinear convergence factor and the random number generating mechanism, so as to realize the improved whale optimization algorithm that can cope with the higher degree of freedom and the more complex constraints. The simulation results show that the optimization algorithm has improved the convergence accuracy by 22.1% and reduced the standard deviation by 74.1%, which can effectively deal with the 3D UAV path planning problem.

The study aims to utilize the convergence-confinement method (CCM) by considering non-hydrostatic stress assumptions in the analysis of the surrounding rock in a circular tunnel. The rock mass properties should adhere to the criteria of the non-linear Hoek-Brown failure criterion. Through a thorough theoretical analysis approach, an analytical solution is derived to determine the stress and displacement induced by tunnel excavation, particularly in the elastic and plastic zones. This solution, applicable under anisotropic stress conditions, involves accounting for confinement loss incrementally for computational feasibility. The implementation of this analytical solution, facilitated by a straightforward spreadsheet, was validated against existing data to evaluate the impact of non-linear failure criteria on ground reaction behavior. The study scrutinizes the mechanical response at the tunnel's inner curve and assesses stress-displacement distribution across the tunnel cross-section. A comparison between the proposed solution and published results demonstrates a consistent and promising correlation.

This paper represents the first investigation into the consensus problem of linear time-varying multi-agent systems utilizing an event-triggered communication scheme. First, a general event-triggered consensus control scheme is proposed for a general category of linear time-varying multi-agent systems. Under some suitable assumptions, it is demonstrated that all agents’ states will converge exponentially, with Zeno behaviour being ruled out. Second, the consensus problem in a network of linear time-varying multi-agent systems with a spanning tree is investigated using the proposed control strategy. It demonstrates that the consensus issue for the specified system can be reformulated as a stabilization problem for an error system through a time-varying linear transformation. Then, the event-triggered consensus problem is just a special instance of the general event-triggered consensus problem mentioned above. Finally, to illustrate the efficacy of the event-triggered method proposed in this study, simulation results are shown.

This article presents a new methodology for modeling and analyzing discrete dynamic systems through the construction of inverse algebraic models, giving rise to the Theory of Inverse Discrete Dynamical Systems. Key concepts such as inverse modeling, structural analysis in inverse algebraic trees, and the establishment of topological equivalences for the analytical transfer of properties between the canonical system and its inverted counterpart are introduced. Central theorems on homeomorphic invariance and topological transport are demonstrated, consolidating the validity of transferring cardinal attributes between both equivalent dynamic representations. As a concrete application, an alternative proof of the historic Collatz Conjecture is presented through the rigorous construction of the associated inverse model and the analytical transfer of properties exhibited in the inverted tree structure. Furthermore, evidence has been presented on Conway's Conjecture in the Game of Life through meticulous inverse modeling in bounded cases, overcoming initial limitations of the theory in combinatorial explosions. Although a complete proof of the conjecture is not achieved, this positively expands the modeling and analysis capacity of the methodology. The impact of properly applying this novel theory to expand understanding and solve open problems in discrete dynamical systems seems vast and profound.

That birds are maniraptoran theropod dinosaurs is now considered an evolutionary consensus. An “open” (i.e., completely, or substantially perforate) acetabulum is considered an important synapomorphy verifying the bird-dinosaur nexus. Here I present anatomical evidence from morphological aspects of the acetabulum and its important appurtenances, the supracetabular crest and the antitrochanter, that the hip anatomy differs substantially between dinosaurs and birds and for the current hypothesis of origins to be correct unlikely anatomical reversals would have been required. Given the thin bone of the acetabular walls and the varied tissue, both hard and soft, in the acetabular region and especially the lower part of the basin, it is apparent that many avian skeletons exhibit some anatomical loss of soft tissue and thin bone, some perhaps related to changes in gait, but also in part related to the dramatic trend in bone reduction associated with flight, especially in more advanced crown taxa. Many basal birds and early diverging neornithines tend to have a nearly closed or various levels of partially closed acetabula, thus rendering the current terms “open” or “closed” acetabula inaccurate; they should be modified or replaced.

Purpose of this article is to introduce a modification of Phillips operators on the interval $\left[ \frac{1}{2},\infty \right) $ via Dunkl generalization. This type of modification enables a better error estimation on the interval $\left[ \frac{1}{2},\infty \right) $ rather than the classical Dunkl Phillips operators on $\left[ 0,\infty \right) $. We discuss the convergence results and obtain the degrees of approximations. Furthermore, we calculate the rate of convergence by means of modulus of continuity, Lipschitz type maximal functions, Peetre's $K$-functional and second order modulus of continuity.

Purpose of this article is to introduce a modification of Phillips operators on the interval $\left[ \frac{1}{2}% ,\infty \right) $ via Dunkl generalization. This type of modification enables a better error estimation on the interval $\left[ \frac{1}{2},\infty \right) $ rather than the classical Dunkl Phillips operators on $\left[ 0,\infty \right) $. We discuss the convergence results and obtain the degrees of approximations. Furthermore, we calculate the rate of convergence by means of modulus of continuity, Lipschitz type maximal functions, Peetre's $K$-functional and second order modulus of continuity.

This paper studies an effective finite difference scheme for solving two-dimensional Heston stochastic volatility option pricing model problems. A dynamically balanced up-downwind strategy for approximating the cross-derivative is implemented and analyzed. Semi-discretized and spatially nonuniform platforms are utilized. The numerical method comprised is simple, straightforward with reliable first order overall approximations. The spectral norm is used throughout the investigation and numerical stability is proven. Simulation experiments are given to illustrate our results.

In the paper, we prove a joint limit theorem in terms of weak convergence of probability measures on $\mathbb{C}^2$ defined by means of the Epstein ζ(s;Q) and Hurwitz ζ(s,α) zeta-functions. The limit measure in the theorem is explicitly given. For this, some restrictions on the matrix Q and parameter α are required. The theorem obtained extends and generalizes Bohr-Jessen’s results characterising asymptotic behaviour of the Riemann zeta-function.

In this paper, we obtain the existence and uniqueness theorem for solutions of Caputo type fractional stochastic delay differential systems (FSDDSs) with Poisson jumps by utilizing delayed perturbation of Mittag-Leffler function. Moreover, by using Burkholder-Davis-Gundy's inequality, Doob's martingale inequality and Holder inequality, we prove that the solution of the averaged FSDDSs converges to that of the standard FSDDSs in the sense of Lp. Some known results in the literature are extended.

A system of simultaneous multi-variable nonlinear equations can be solved by the Newton’s method with local q-quadratic convergence if the Jacobian is analytically available. If this is not the case, then quasi-Newton methods with local q-superlinear convergence give solutions by approximating the Jacobian in some way. Unfortunately, the quasi-Newton condition (secant equation) doesn’t completely specify the Jacobian approximate in multi-dimensional case, so its full-rank update is not possible with classic variants of methods. The suggested new iteration strategy (“T-Secant”) allows full-rank update of the Jacobian approximate in each iteration by determining two independent approximates for the solution. They are used to generate a set of new independent trial approximates, then the Jacobian approximate can fully be updated. It is shown, that the T-Secant approximate is in the vicinity of the classic quasi-Newton approximate, providing that the solution is evenly surrounded by the new trial approximates. The suggested procedure increases the super-linear convergence of the secant method (φS = 1.618...) to super-quadratic (φT = φS + 1 = 2.618...) and the quadratic convergence of the Newton-method (φN = 2) to cubic (φT = φN + 1 = 3) in one dimensional case. The Broyden-type efficiency (mean convergence rate) of the suggested method in multi-dimensional case is an order higher than the efficiency of other classic low-rank update quasi-Newton methods as shown by numerical examples on a Rosenbrock-type test-function with up to 1000 variables. The geometrical representation (hyperbolic approximation) in single variable case helps explaining the basic operations and a vector-space description is also given in multi-variable case.

In this paper, we study a three step iterative scheme to approximate fixed points of Suzuki's generalized non-expansive mappings. We establish some weak and strong convergence results for such mappings in uniformly convex Banach spaces. Further, we show numerically that iterative scheme (1.8) converges faster than some other known iterations for Suzuki's generalized non-expansive mappings. To support our claim, we give an illustrative example and approximate fixed points of such mappings using Matlab program. Our results are new and generalize several relevant results in the literature.

In this paper, we present a rigorous proof demonstrating the stabilization of percolation probability in supercritical regimes for percolation models. We analyze a sequence of expanding compact balls in ℝ௡, incorporating correction terms that vanish asymptotically, and show that the percolation probability within these regions converges to a finite, non‐zero value as the balls expand to cover the entire space. Our approach combines key concepts from percolation theory with measure‐theoretic tools, such as the Monotone Convergence Theorem and Fatouʹs Lemma, to rigorously establish the existence and uniqueness of the limiting percolation probability. The result extends classical results on percolation probability in lattice models and provides a new framework for understanding convergence in infinite systems. The non‐triviality of the limit is demonstrated in the supercritical regime, where percolation occurs with positive probability. The framework introduced here could serve as a bridge between discrete and continuous models in statistical physics

This paper presents a novel approach to demonstrate the Collatz Conjecture, an unsolved problem in mathematics stating that all positive integers will eventually converge to 1 when subjected to a recursive sequence of operations. We introduce Algebraic Inverse Trees (AITs), innovative data structures that characterize relationships within the Collatz sequence by tracing inverse transformations back to the origin. Leveraging properties unveiled through this inverted representation, including absence of non-trivial cycles and guaranteed convergence of paths, we construct a topological framework to formally prove that all trajectories invariably lead to 1 under the Collatz iterative process. This modern technique provides not only the machinery to manifest a rigorous proof, but also grants intuitive insights into the deep complexity of this notorious yet elementary conjecture.

Purpose: This research compared the dissociated phoria at near and distance fixation in free space using three techniques: the Howell test, alternate Cover test, and Thorington test. Methods: 220 healthy Mexican children (age 5.06-17.11) participated in this study. Phorias were quantified at both distances using each test, from the least to the most disruptive. The stereopsis degree and near point of convergence (break/recovery) were analyzed to understand their role on the visual system's sensorimotor balance. Results: For the exophoria/esophoria at near, the alternate Cover and Thorington tests showed similar values. The exophoria at distance, was similar with the alternate Cover and Howell tests, while esophoria, with the alternate Cover and Thorington tests. The break value and near exophoria relates to each other and affect the stereopsis degree, whereas age is associated with the stereopsis degree and break value. Conclusions: The analysis revealed that alternate Cover and Thorington tests can substitute one another, but not with the Howell test. The test's specific dissociation method may explain these results. Analyzing the binocular visual system provides information about its sensorimotor balance and fragility when faced to the technique’s degree of dissociation.

The Subtropical Front (STF) plays a key role in the ecology of Chilean jack mackerel Trachurus murphyi. Nonetheless, there are few remote sensing studies of the STF in the open Southeast Pacific Ocean, and almost all of them have been conducted by satellite oceanographers in Russia and Ukraine to support respective large-scale fisheries of jack mackerel in this region. We reviewed these studies that documented long-term seasonal and interannual variability of the STF from sea surface temperature (SST) and sea surface height (SSH) data. We also mapped the STF from satellite sea surface salinity (SSS) data of the SMOS mission (2012-2019). The Subtropical Front consists of two fronts -- North and South STF about 500 km apart -- that border the Subtropical Frontal Zone (STFZ) in-between. The STF is density-compensated, with spatially divergent manifestations in temperature and salinity. In the temperature field, the STF extends in the WNW to ESE direction in the Southeast Pacific. In the salinity field, the STFZ appears as a broad frontal zone, extending zonally between 30-35°S across the entire South Pacific. Three major types of satellite data – SST, SSH, and SSS – can be used to locate the STF. The SSH data is most advantageous with regard to the jack mackerel fisheries owing to the all-weather capability of satellite altimetry and the radical improvement of the spatial resolution of SSH data in the near future. Despite the dearth of dedicated in situ studies of the South Pacific STFZ, there is a broad consensus regarding the STFZ being the principal spawning and nursing ground of T. murphyi as well as a major migration corridor between Chile and New Zealand.

In the paper, by induction and recursively, the author proves that the generating function of multivariate logarithmic polynomials and its reciprocal are a Bernstein function and a completely monotonic function respectively, establishes a Lévy-Khintchine representation for the generating function of multivariate logarithmic polynomials, deduces an integral representation for multivariate logarithmic polynomials, presents an integral representation for the reciprocal of the generating function of multivariate logarithmic polynomials, computes real and imaginary parts for the generating function of multivariate logarithmic polynomials, derives two integral formulas, and denies the uniform convergence of a known integral representation for Bernstein functions.

The urgency to develop vaccines against Covid-19 is putting pressure on the long and expensive development timelines which are normally required for development of lifesaving vaccines. There is a unique opportunity to take advantage of new technologies, smart and flexible design of clinical trials, and evolving regulatory science to speed up vaccine development against Covid-19 and transform vaccine development altogether.

In this paper we propose and study in the setting of a Hilbert space an averaged Halpern type algorithm for approximating the solution of a common fixed point problem for a couple of nonexpansive and demicontractive mappings with a variational inequality constraint. The strong convergence of the sequence generated by the algorithm is established under feasible assumptions on the parameters involved. In particular, we also obtain the common solution of the fixed point problem for nonexpansive or demicontractive mappings and of a variational inequality problem. Our results extend and generalize various important related results in literature that were established for the pairs (nonexpansive, nonspreading) or (nonexpansive, strongly quasi-nonexpansive) mappings. Numerical tests to illustrate the superiority of our algorithm over the ones existing in literature are also reported.

This paper explores the Wilsonian renormalization group of probe-D-branes, focusing on the Dirac-Born-Infeld (DBI) action. The analysis involves deriving the constant of motion and formulating the Wilsonian renormalization group equation. The study extends to the infrared regime, considering both zero and non-zero temperatures. Additionally, thermodynamic aspects related to the functional form of the grand potential are discussed. The investigation provides insights into the renormalization of probe-D-branes and their thermodynamic properties.

The notion of Critical Point Theory has extensive application in the filed of Partial Differential Equations, one of which is studying the existence and commenting on the uniqueness and multiplicity (in case when the solution is not unique) of weak solutions to Elliptic PDEs under certain boundary conditions.\par This article provides a brief survey about specific concepts like : differentiation on Banach Spaces, maxima and minima and its applications to PDEs. Furthermore, we've discussed the notion of defining a weak topology on Banach Spaces, before introducing the Variational Principle and its applications.\par The epilogue of the article primarily deals with results which studies the existence of weak solutions to Dirichlet Boundary Value Problems under specific conditions. The most notable aspect of this article is the proof of \textit{Rabinowitz's Saddle Point Theorem} using the method of \textit{Brouwer Degree}. Readers who are highly motivated in pursuing research in any of the topics relevent to the contents of this paper will surely find the \texttt{References} section to be extremely resourceful.