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This article presents an educational overview of the latest mathematical developments in coupled cluster (CC) theory, beginning with Schneider's seminal work from 2009 that introduced the first local analysis of CC theory. We provide a tutorial review of second quantization and the CC ansatz, laying the groundwork for understanding the mathematical basis of the theory. This is followed by a detailed exploration of the most recent mathematical advancements in CC theory. Our review starts with an in‐depth look at the local analysis pioneered by Schneider which has since been applied to various CC methods. We subsequently discuss the graph‐based framework for CC methods developed by Csirik and Laestadius. This framework provides a comprehensive platform for comparing different CC methods, including multireference approaches. Next, we delve into the latest numerical analysis results analyzing the single reference CC method developed by Hassan, Maday, and Wang. This very general approach is based on the invertibility of the CC function's Fréchet derivative. We conclude the article with a discussion on the recent incorporation of algebraic geometry into CC theory, highlighting how this novel and fundamentally different mathematical perspective has furthered our understanding and provides exciting pathways to new computational approaches. [ABSTRACT FROM AUTHOR]

In this paper, a class of matrices, namely, Drazin-dagger ma- trices, in which the Drazin inverse and the Moore-Penrose inverse com- mute, is introduced. Also, some properties of the generalized inverses of these matrices, are investigated. Moreover, some results about the Moore- Penrose inverse, the Drazin inverse and the numerical range of some re- ciprocal matrices are obtained. In particular, the relations between re- ciprocal matrices, Drazin-Dagger matrices and star order are established. Also, some properties of the generalized inverses of the conjugate EP ma- trices are studied. To illustrate the results, some numerical examples are also given. [ABSTRACT FROM AUTHOR]

Abstract. In this paper, we study the interpolation properties of anisotropic net spaces Np,q(M), where p = (p1; ...., pn), q = (q1 ...., qn). It is shown that, with respect to the multidimensional interpolation method, the following equality holds [ABSTRACT FROM AUTHOR]

We analyze the SCARDO model in the case of the 3-element opinion space under specific constraints on the transition table parameters that allows to link the problem at stake to the case of the 2-element opinion space that has been thoroughly studied previously. We characterize the properties of fixed points and support our findings by numerical experiments. Further, we manage to find out those settings that ensure the system almost surely reaches a specific domain in the phase space, after which its behavior can be predicted analytically. [ABSTRACT FROM AUTHOR]

This paper discussed the synchronization of impulsive fuzzy neural networks (FNNs) with uncertainty of information exchange. Since the data of neural networks (NNs) cannot be completely measured in reality, we designed an observer-based impulsive controller on the basis of the partial measurement results and achieved the purpose of reducing the communication load and the controller load of FNNs. In terms of the Lyapunov stability theory, an impulsive augmented error system (IAES) was established and two sufficient criteria to guarantee the synchronization of our FNNs system were obtained. Finally, we demonstrated the validity of the results by a numerical example. [ABSTRACT FROM AUTHOR]

This paper mainly considered the finite-time lag synchronization for two-layer complex networks with impulsive effects. Different types of controllers were designed to achieve the lag synchronization of two-layer complex networks. Several sufficient conditions on lag synchronization in the sense of finite time were derived. The time for synchronization was also estimated. It is important to note that synchronization time was influenced by the initial value, as well as the impulses and impulse sequence. This implied that different impulse effects result in varying synchronization times. Additionally, desynchronizing impulses can extend the synchronization time, whereas synchronizing impulses have the opposite effect. Finally, a numerical example was presented to showcase the practicality and validity of the proposed theoretical criteria. [ABSTRACT FROM AUTHOR]

One of the most important things you can do to improve your mathematical application is to learn how to find numerical solutions. However, it was discovered that classrooms teaching methods that use numerical solutions are largely unable to provide students with the successful experience they should have in finding numerical solutions. Since conceptual and procedural knowledge, as well as the ability to perform computational mathematics, must be understood, simultaneously mastering all three can be difficult for most students. This study investigates combining GPT and Colab as learning tools for students to explore numerical solutions in the context of difference equations. The developed learning process works in tandem with the power of GPT and Colab to provide students with a successful experience in finding numerical solutions to difference equations. The survey results show that students have a high level of self-efficacy in finding numerical solutions to difference equations. This reflects today's power of innovation, which can be applied in classroom to improve student skills so that they can use the tools to solve problems. [ABSTRACT FROM AUTHOR]

I examine the classical idea of 'algorithm' as a sequential, step-by-step, deterministic procedure (i.e. , the idea of 'algorithm' that was already in use by the 1930s), with respect to three themes, its relation to the notion of an 'effective procedure', its different roles and uses in logic, computer science, and mathematics (focused on numerical analysis), and its different formal definitions proposed by practitioners in these areas. I argue that 'algorithm' has been conceptualized and used in contrasting ways in the above areas, and discuss challenges and prospects for adopting a final foundational theory of (classical) 'algorithms'. [ABSTRACT FROM AUTHOR]

This work is concerned with solving parabolic Volterra partial integro-differential equations (PIDE) considering differentiable and singular kernels. The implicit finite difference scheme is implemented to approximate the differential operator, and the nonlocal term is discretized based on an open-type formula with two distinct time step sizes related to the nature of the time level to guarantee to avoid the singular terms at the endpoints and denominators. The properties of the plied scheme are investigated, more precisely, its stability and consistency. Four detailed examples are implemented to demonstrate the efficiency and reliability of the applied finite difference scheme.

We are concerned with the analysis of the neural networks of worms in wireless sensor networks (WSN). The concerned process is considered in the form of a mathematical system in the context of fractal fractional differential operators. In addition, the Banach contraction technique is utilized to achieve the existence and unique outcomes of the given model. Further, the stability of the proposed model is analyzed through functional analysis and the Ulam-Hyers (UH) stability technique. In the last, a numerical scheme is established to check the dynamical behavior of the fractional fractal order WSN model.

Fingering is the foundation of keyboard instrument performance and an important part of keyboard music, but few people pay attention to its development. The evolution from the early variety of playing fingerings to the thumb-centered five-fingerings is even less mentioned. At present, steganographic analysis mainly focuses on the diversity and high dimensionality of features. Faced with the emerging new steganography, it is difficult for a single feature to cover and express the influence of steganography process on the multi-dimensional distribution of images. Therefore, it is necessary to combine various features through certain methods to analyze the changes of image properties before and after steganography embedding in a larger range and more types. The experimental results show that the experimental comparison diagram of Comb-RichModel based on diversity features and other steganographic analysis algorithms can be seen by replacing filtering and adding SPAM features. Compared with numerical analysis, the detection accuracy of the Comb-RichModel with diversified features has been improved to some extent, and its detection performance has improved stably from low embedding rate to high embedding rate. It is proved that the numerical analysis can effectively analyze the diversity characteristics of keyboard instrument playing fingerings.

The relationship between convexity and symmetry is widely recognized. In fuzzy theory, both concepts exhibit similar behavior. It is crucial to remember that real and interval-valued mappings are special instances of fuzzy-number-valued mappings (F-N-V-Ms), as fuzzy theory relies on the unit interval, which is crucial to resolving problems with interval analysis and fuzzy number theory. In this paper, a new harmonic convexities class of fuzzy numbers has been introduced via up and down relation. We show several Hermite–Hadamard (H⋅H) and Fejér-type inequalities by the implementation of fuzzy Aumann integrals using the newly defined class of convexities. Some nontrivial examples are also presented to validate the main outcomes.

This paper investigates the dynamics of the SIR infectious disease model, with a specific emphasis on utilizing a harmonic mean-type incidence rate. It thoroughly analyzes the model’s equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. A sophisticated stability theory, primarily focusing on the characteristics of the Volterra–Lyapunov (V-L) matrices, is developed to examine the overall trajectory of the model globally. In addition to that, we describe the transmission of infectious disease through a mathematical model using fractal-fractional differential operators. We prove the existence and uniqueness of solutions in the SIR model framework with a harmonic mean-type incidence rate by using the Banach contraction approach. Functional analysis is used together with the Ulam–Hyers (UH) stability approach to perform stability analysis. We simulate the numerical results by using a computational scheme with the help of MATLAB. This study advances our knowledge of the dynamics of epidemic dissemination and facilitates the development of disease prevention and mitigation tactics.

This contribution proposes a variational symplectic integrator aimed at linear systems issued from the least action principle. An internal quadratic finite-element interpolation of the state is performed at each time step. Then, the action is approximated by Simpson’s quadrature formula. The implemented scheme is implicit, symplectic, and conditionally stable. It is applied to the time integration of systems with quadratic Lagrangians. The example of the linearized double pendulum is treated. Our method is compared with Newmark’s variational integrator. The exact solution of the linearized double pendulum example is used for benchmarking. Simulation results illustrate the precision and convergence of the proposed integrator.

Author(s): Sergei Grudsky [sup.1], Helmuth R. Malonek [sup.2], Vladimir Rabinovich [sup.3], Ioannis Stratis [sup.4] Author Affiliations: (1) grid.512574.0, Departamento de Matemáticas, CINVESTAV del I.P.N., , Apartado Postal 14-740, C.P. 07360, [...]

In this article, we analyze tensor approximation schemes for continuous functions. We assume that the function to be approximated lies in an isotropic Sobolev space and discuss the cost when approximating this function in the continuous analogue of the Tucker tensor format or of the tensor train format. We especially show that the cost of both approximations are dimension-robust when the Sobolev space under consideration provides appropriate dimension weights., Author(s): Michael Griebel [sup.1] [sup.2], Helmut Harbrecht [sup.3] Author Affiliations: (1) grid.10388.32, 0000 0001 2240 3300, Institut für Numerische Simulation, Universität Bonn, , Friedrich-Hirzebruch-Allee 7, 53115, Bonn, Germany (2) grid.418688.b, [...]

In this paper, we study the dynamics of the diphtheria outbreak among the immunocompromised group of people, the Rohingya ethnic group. Approximately 800,000 Rohingya refugees are living in the Balukhali refugee camp in Cox’s Bazar. The camp is densely populated with the scarcity of proper food, healthcare, and sanitation. Subsequently, in November 2017 a diphtheria epidemic occurred in this camp. To keep up with the pace of the disease spread, medical demands, and disaster planning, we set out to predict diphtheria outbreaks among Bangladeshi Rohingya immigrants. We adopted a modified Susceptible-Latent-Infectious-Recovered (SLIR) transmission model to forecast the possible implications of the diphtheria outbreak in the Rohingya camps of Bangladesh. We discussed two distinct situations: the daily confirmed cases and cumulative data with unique consequences of diphtheria. Data for statistical and numerical simulations were obtained from \cite{Matsuyama}. We used the fourth-order Runge-Kutta method to obtain numerical simulations for varying parameters of the model which would demonstrate conclusive estimates. Daily and cumulative data predictions were explored for alternative values of the parameters i.e., disease transmission rate $(\beta)$ and recovery rate $(\gamma)$. Additionally, the average basic reproduction number for the parameters $\beta$ and $\gamma$ was calculated and displayed graphically. Our analysis demonstrated that the diphtheria outbreak would be under control if the maintenance could perform properly. The results of this research can be utilized by the Bangladeshi government and other humanitarian organizations to forecast disease outbreaks. Furthermore, it might help them to make detailed and practical planning to avoid the worst scenario.

Significance: The study of non-transient heat transport mechanism in mono nano as well as ternary nanofluids attracts the researchers because of their promising heat transport characteristics. Applications of these fluids spread in industrial and various engineering disciplines more specifically in chemical and applied thermal engineering. Due of huge significance of nanofluids, the study is organized for latest class termed as ternary nanofluids along with induced magnetic field. Methodology: The model development done via similarity equations and the properties of ternary nanoparticles, resulting in a nonlinear mathematical model. To analyze the physical results with parametric values performed via RKF-45 scheme. Study findings: The physical results of the model reveal that the velocity $ F{'}\left(\eta \right) $ increased with increasing $ m = 0.1, 0.2, 0.3 $ and $ {\lambda }_{1} = 1.0, 1.2, 1.3 $. However, velocity decreased with increasing $ {\delta }_{1} $. Tangential velocity $ G{'}\left(\eta \right) $ reduces rapidly near the wedge surface and increased with increasing $ {M}_{1} = 1.0, 1.2, 1.3 $. Further, the heat transport in ternary nanofluid was greater than in the hybrid and mono nanofluids. Shear drag and the local thermal gradient increased with increasing $ {\lambda }_{1} $ and these quantities were greatest in the ternary nanofluid.

We aim to give the numerical method for solving the fractional Volterra integral equations of first and second kinds. We here use the techniques based upon rational Chebyshev functions and Riemann-Liouville fractional integrals. Some illustrative experiments with a view of estimating error and graphics are given in order to show the validity and applicability of the technique. Our experiments show that the new technique has high accuracy and is very efficient when compare to the other approaches existing in literature. [ABSTRACT FROM AUTHOR]

Despite the variety of methods available to solve nonlinear optimal control problems, numerical methods are still evolving to solve these problems. This paper deals with the numerical solution of nonlinear optimal control affine problems by the interpolated variational iteration method, which was introduced in 2016 to improve the variational iteration method. For this purpose, the optimality conditions are first derived as a two-point boundary value problem and then converted to an initial value problem with the unknown initial values for costates. The speed and convergence of the method are compared with the existing methods in the form of three examples, and the initial values of the costates are obtained by an efficient technique in each iteration. [ABSTRACT FROM AUTHOR]

The quadratic Riccati equations are first-order nonlinear differential equations with numerous applications in various applied science and engineering areas. Therefore, several numerical approaches have been derived to find their numerical solutions. This paper provided the approximate solution of the quadratic Riccati equation via the cubic b-spline method. The convergence analysis of the method is discussed. The efficiency and applicability of the proposed approach are verified through three numerical test problems. The obtained results are in good settlement with the exact solutions. Moreover, the numerical results indicate that the proposed cubic b-spline method attains a superior performance compared with some existing methods.

The loss and degradation of habitat, Allee effects, climate change, deforestation, hunting-overfishing and human disturbances are alarming and significant threats to the extinction of many species in ecology. When populations compete for natural resources, food supply and habitat, survival to extinction and various other issues are visible. This paper investigates the competition of two species in a heterogeneous environment that are subject to the effect of harvesting. The most realistic harvesting case is connected with the intrinsic growth rate, and the harvesting functions are developed based on this clause instead of random choice. We prove the existence and uniqueness of the solution to the model. Theoretically, we state that, when species coexist, one may drive the other to die out, so both species become extinct, considering all possible rational values of parameters. These results highlight a worthy-of attention study between two populations based on harvesting coefficients. Finally, we solve the model for two spatial dimensions by using a backward Euler, decoupled and linearized time-stepping fully discrete algorithm in a series of examples and observe a match between the theoretical and numerical findings.

Currently, designing path-planning concepts for autonomous robot systems remains a topic of high interest. This work applies computational analysis through a numerical approach to deal with the path-planning problem with obstacle avoidance over a robot simulation. Based on the potential field produced by Laplace's equation, the formation of a potential function throughout the simulation configuration regions is obtained. This potential field is typically employed as a guide in the global approach of robot path-planning. An extended variant of the over-relaxation technique, namely the skewed block two-parameter over relaxation (SBTOR), otherwise known as the explicit decoupled group two-parameter over relaxation method, is presented to obtain the potential field that will be used for solving the path-planning problem. Experimental results with a robot simulator are presented to demonstrate the performance of the proposed approach on computing the harmonic potential for solving the path-planning problem. In addition to successfully validating pathways generated from various locations, it is also demonstrated that SBTOR outperforms existing over-relaxation algorithms in terms of the number of iterations, as well as the execution time.

In this work, a set of nonlinear equations capable of describing the transit of the membrane potential's spiking-bursting process which is shown in experiments with a single neuron was taken into consideration. It is well known that this system, which is built on dynamical dimensionless variables, can reproduce chaos. We arrived at the chaotic number after first deriving the equilibrium point. We added different nonlocal operators to the classical model's foundation. We gave some helpful existence and uniqueness requirements for each scenario using well-known theorems like Lipchitz and linear growth. Before using the numerical solution on the model, we analyzed a general Cauchy issue for several situations, solved it numerically and then demonstrated the numerical solution's convergence. The results of numerical simulations are given.

This is the first English translation of the paper originally published in the first issue of 'Problems in Mathematical Analysis' in 1966 and reproduced to the 85th birthday of V. Maz'ya. Bibliography: 15 titles. Illustrations: 3 figures., 1. In this paper, we consider the Dirichlet problem with zero boundary condition for the elliptic equation [Please download the PDF to view the mathematical expression] (1) in a finite [...]

This paper establishes convergence rates and error estimates for the pseudo-polyharmonic div-curl and elastic interpolation. This type of interpolation is based on a combination of the divergence and the curl of a multivariate vector field and minimizing an appropriate functional energy related to the divergence and curl. Convergence rates and error estimates are established when the interpolated vector field is assumed to be in the classical fractional vectorial Sobolev space on an open bounded set with a Lipschitz-continuous boundary. The error estimates introduced in this work are sharp and the rate of convergence depends algebraically on the fill distance of the scattered data nodes. More precisely, the order of convergence depends, essentially, on the smoothness of the target vector field, on the dimension of the Euclidean space and on the null space of corresponding Sobolev semi-norm.

A set of nonlinear ordinary differential equations has been considered in this paper. The work tries to establish some theoretical and analytical insights when the usual time-deferential operator is replaced with the Caputo fractional derivative. Using the Caratheodory principle and other additional conditions, we established that the system has a unique system of solutions. A variety of well-known approaches were used to investigate the system. The stochastic version of this system was solved using a numerical approach based on Lagrange interpolation, and numerical simulation results were produced.

There is ongoing debate regarding what performance on the number line estimation task represents and its role in mathematics learning. The patterns followed by children’s estimates on the number line task could provide insight into this. This study investigates children’s estimation patterns on the number line task and assesses whether mathematics achievement is associated with these estimation patterns. Singaporean children (n = 324, Age M = 6.2 years, Age SD = 0.3 years) in their second year of kindergarten were assessed on the number line task (0-100) and their mathematical performance (Numerical Operations and Mathematical Reasoning subtests from WIAT II). The results show that most children’s number line estimation patterns can be explained by at least one mathematical model (i.e., linear, logarithmic, unbounded power model, one-cycle power model, two-cycle power model). But the findings also highlight the high percentage of participants for which more than one model shows similar support. Children’s mathematical achievement differed based on the models that best explained children’s estimation patterns. Children whose estimation patterns corresponded to a more advanced model tended to show higher mathematical achievement. Limitations of drawing conclusions regarding what performance on the number line task represents based on models that best explain the estimation patterns are discussed. [ABSTRACT FROM AUTHOR]

Mastery of mathematics depends on the people’s ability to manipulate and abstract values such as negative numbers. Knowledge of arithmetic principles does not necessarily generalize from positive number arithmetic to arithmetic involving negative numbers (Prather & Alibali, 2008, https://doi.org/10.1080/03640210701864147). In this study, we evaluate the relationship between participant’s knowledge of the Relation to Operands arithmetic principle in both positive and negative numbers and their spontaneous on numerical relations. Additionally, we tested if the feedback that directs attention to relations affects participants’ attention to relation and their arithmetic principle knowledge. This study contributes to our understanding of the specific skills and cognitive processes that are associated with understanding high-level mathematics. [ABSTRACT FROM AUTHOR]

Numerical literacy refers to the knowledge and ability to use various numbers and basic mathematical symbols to solve problems, while math self-concept means the assessment of students' skills, abilities, enjoyment, and interest in the subject. However, children with special needs and normal students in inclusive Elementary Schools are yet to sufficiently acquire learning that accommodates literacy and maths self-concept. This causes a need for the implementation of a children-friendly learning process. Therefore, this study aimed to identify the factors influencing the numeracy level and math self-concept, and also explore the obstacles in implementing children-friendly learning in order to facilitate students' abilities. A qualitative method was applied because of in-depth data exploration regarding children with special needs, while the utilized instruments include tests, questionnaires, and interviews. Both the data collected and the analysis are qualitative, which are obtained through excavation, identification, and description. Consequently, this paper was able to (a) describe the factors influencing the numeracy level and math self-concept in inclusive elementary schools; (b) explore the barriers to implementing children-friendly learning; and (c) identify the relationship between students' numeracy and math self-concept. [ABSTRACT FROM AUTHOR]

In this study, the suggested residual power series transform method is used to compute the numerical solution of the fractional-order nonlinear Gardner and Cahn-Hilliard equations and the result is discovered in a fast convergent series. The leverage and effcacy of the suggested technique are demonstrated by the test examples provided. The achieved results are proved graphically. The current method handles the series solution in a sizable admissible domain in a powerful way. It provides a simple means of modifying the solution's convergence zone. Results with graphs expressly demonstrate the effectiveness and abilities of the suggested method. [ABSTRACT FROM AUTHOR]

Currently, designing path-planning concepts for autonomous robot systems remains a topic of high interest. This work applies computational analysis through a numerical approach to deal with the path-planning problem with obstacle avoidance over a robot simulation. Based on the potential field produced by Laplace's equation, the formation of a potential function throughout the simulation configuration regions is obtained. This potential field is typically employed as a guide in the global approach of robot path-planning. An extended variant of the over-relaxation technique, namely the skewed block two-parameter over relaxation (SBTOR), otherwise known as the explicit decoupled group two-parameter over relaxation method, is presented to obtain the potential field that will be used for solving the path-planning problem. Experimental results with a robot simulator are presented to demonstrate the performance of the proposed approach on computing the harmonic potential for solving the path-planning problem. In addition to successfully validating pathways generated from various locations, it is also demonstrated that SBTOR outperforms existing over-relaxation algorithms in terms of the number of iterations, as well as the execution time. [ABSTRACT FROM AUTHOR]

Linear operators with off‐diagonal decay appear in many areas of mathematics including harmonic and numerical analysis, and their stability is one of the basic assumptions. In this paper, we consider a family of localized integral operators in the Beurling algebra with kernels having mild singularity near the diagonal and certain Hölder continuity property, and prove that their weighted stabilities for different exponents and Muckenhoupt weights are equivalent to each other on a space of homogeneous type with Ahlfors regular measure. [ABSTRACT FROM AUTHOR]

Memory has been well-established as a predictor of mathematics achievement in child development. Nevertheless, empirical evidence remains elusive on the unique role of the different forms of memory and their specific mechanisms as predictors of mathematics development. Therefore, in this study, the role of visuospatial short-term memory, visuospatial working memory, verbal short-term memory, and verbal longterm memory was investigated at three key stages of the development of mathematics (5-6 years, 6-7 years, 7-8 years), as well as their interactions across development. The relation between the different memory types and informal and formal mathematics was also studied. The findings of this study provide empirical support for a shift in the relation between different memory types and mathematics achievement over development with: 1) visuospatial short-term memory predicting informal mathematics achievement at the age of 5-6 years; 2) visuospatial working memory predicting informal and formal mathematics achievement at the age of 6-7 years; and 3) verbal short-term memory predicting formal mathematics achievement at the age of 7-8 years. These shifts clearly appear consistent with children's mathematics curriculum content over time and the requirements of mathematics acquisition at specific stages in development. With these findings, the unique role of various forms of memory in the development of mathematics and the timeframe in which they play a crucial part is highlighted, which should be taken into consideration for future research and possible intervention studies in children's mathematics achievement. [ABSTRACT FROM AUTHOR]

In order to improve the expertise and skills of students and compensate the growing lack of interest towards mathematics in Industrial Design Engineering, during the last years several innovative teaching experiences have been developed, coordinating academic activities between the areas of Mathematics and Design Projects. These coordinated experiences allowed the introduction of advanced essential contents in parametric representation, numerical analysis and manufacturing in the first course, contents which are barely included in the curriculum of Industrial Design Engineering university degrees. Despite the issues and complexity of their planning and practical implementation, results have been excellent; transforming the perception of mathematics by students from a simple working tool to a stimulating new way of creation and inspiration to explore. [ABSTRACT FROM AUTHOR]

This paper aims to present some computations related to numerical semigroups with embedding dimension 3. We study a simple problem regarding the packaging methods of some products connected with some invariants of the numerical semigroup. We tried to define the algorithm that produces simultaneously a presentation of the semigroup by three generators and three relations and show how to use it for visualizing the different ways of buying certain products in the market We use the GAP package to find the exact number of products we want to buy, no one above no one below, where amounts correspond with finite sums of integers in the set {𝑎, 𝑏, 𝑐} or numerical semigroup 〈𝑎, 𝑏, 𝑐〉. [ABSTRACT FROM AUTHOR]

This paper introduces a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs. The G-PDE is derived as a nonlocal reparametrization of the assignment flow approach that was introduced in [J. Math. Imaging Vision, 58 (2017), pp. 211-238]. Due to this parameterization, solving the G-PDE numerically is shown to be equivalent to computing the Riemannian gradient flow with respect to a nonconvex potential. We devise an entropy-regularized difference of convex (DC) functions decomposition of this potential and show that the basic geometric Euler scheme for integrating the assignment flow is equivalent to solving the G-PDE by an established DC programming scheme. Moreover, the viewpoint of geometric integration reveals a basic way to exploit higher-order information of the vector field that drives the assignment flow, in order to devise a novel accelerated DC programming scheme. A detailed convergence analysis of both numerical schemes is provided and illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

This paper contributes to the problem of chaos and hyperchaos localization in the fundamental structure of analog building blocks dedicated to single-tone harmonic signal generation. This time, the known Reinartz sinusoidal oscillator is addressed, considering its conventional topology, both via numerical analysis and experiments using a flow-equivalent lumped electronic circuit. It is shown that physically reasonable values of circuit parameters can result in robust dynamical behavior characterized by a pair of positive Lyapunov exponents. Mandatory numerical results prove that discovered strange attractors exhibit all necessary fingerprints of structurally stable chaos. The new “chaotic” parameters are closely related to the standard operation of the investigated analog functional block. A few interestingly shaped, strange attractors have been captured as oscilloscope screenshots.

Typhoid fever is an infectious disease that affects humanity worldwide; it is particularly dangerous in areas with communities of a lower socio-economic status, where many individuals are exposed to a dirty environment and unclean food. A mathematical model is formulated to analyze the impact of control measures such as vaccination of susceptible humans, treatment of infected humans and sanitation in different socio-economic communities. The model assumed that the population comprises of two socio-economic classes. The essential dynamical system analysis of our model was appropriately carried out. The impact of the control measures was analyzed, and the optimal control theory was applied on the control model to explore the impact of the different control measures. Numerical simulation of the models and the optimal controls were carried out and the obtained results indicate that the overall combination of the control measures eradicates typhoid fever in the population, but the controls are more optimal in higher socio-economic status communities.

During the past few years, freelancing has grown exponentially due to the pandemic and subsequent economical changes in the world. In fact, in the last ten years, a drastic increase in freelancing has been observed; people quit their jobs to be their own boss. There are various reasons for this: downsizing of employees, not having fun in their jobs, unemployment, part time work to earn more, etc. Observing this vast change, many individuals on Facebook/YouTube, NGOs, and government departments started teaching freelancing as a course; to date, thousands of youngsters have been trained to start their careers as freelancers. It has been observed that the ratio of informed freelancers is more successful than those who start their careers independently. We construct a compartmental model to explore the influence of information on the expansion of freelancing in this article, which was motivated by this surge in freelancing. Following that, the model is subjected to dynamical analysis utilizing dynamical systems and differential equation theory. To validate our analytical conclusions, we used numerical simulation.

We study the monotonicity method to analyse nabla positivity for discrete fractional operators of Riemann-Liouville type based on exponential kernels, where $ \left({}_{{c_0}}^{C{F_R}}\nabla^{\theta} \mathtt{F}\right)(t) > -\epsilon\, \Lambda(\theta-1)\, \bigl(\nabla \mathtt{F}\bigr)(c_{0}+1) $ such that $ \bigl(\nabla \mathtt{F}\bigr)(c_{0}+1)\geq 0 $ and $ \epsilon > 0 $. Next, the positivity of the fully discrete fractional operator is analyzed, and the region of the solution is presented. Further, we consider numerical simulations to validate our theory. Finally, the region of the solution and the cardinality of the region are discussed via standard plots and heat map plots. The figures confirm the region of solutions for specific values of $ \epsilon $ and $ \theta $.

Harmonic potential fields are commonly used as guidance in a global approach for self-directed robot pathfinding. These harmonic potentials are generated using Laplace's equation solutions. The computation of these harmonic potentials often requires the use of immense amounts of computing resources. This study introduces a numerical technique called Rotated Block Accelerated Over-Relaxation (AOR), also known as Explicit Decoupled Group AOR (EDGAOR), to deal with pathfinding problem. Several robot navigation simulations were performed in a static, structured, known indoor environment to validate the efficiency of the suggested approach. The paths generated by the simulations are shown using several different starting and target positions. The performance of the proposed approach in computing harmonic potentials for solving pathfinding problems is also discussed.