Integration Rules Research Articles

Differentiating under q–integral sign

The Leibniz integral rule enables us to interchange the order of differentiation and integration under some differentiability of the functions. It can be very useful in the computing the exact value of certain integrals. In this paper we will present analogs of such rule for q-integrals with functional borders and their properties.

Gauss quadrature rules for integrals involving weight functions with variable exponents and an application to weakly singular Volterra integral equations

Abstract This paper presents a numerical integration approach that can be used to approximate on a finite interval, the integrals of functions that contain Jacobi weights with variable exponents. A modification of the integrand close to the singularities is needed, and a new modification is proposed. An application of such a rule to the numerical solution of variable-exponent weakly singular Volterra integral equations of the second kind is also explored. In the space of continuous functions, the stability and the error estimates are demonstrated, and numerical tests that validate these estimates are conducted.

Improving the MSMEs data quality assurance comprehensive framework with deep learning technique

In the year of 2022 the ministry of cooperatives and small and medium enterprises (SMEs) executed a complete data collection program for the cooperatives and micro small and medium enterprises (MSMEs) profile. As the complexity of the process and the uniqueness of the data characteristics, plenty of risks must be mitigated. The most challenging risk is the possibility of reduced data quality. This study is performed to validate the proposed comprehensive framework to ensure the quality data of cooperatives and MSME. The proposed framework aims to prevent, detect, repair, and recover dirty data to achieve the required data quality minimum standard. We investigated many techniques namely rule-based, selection-based, and deep learning-based. By applying the framework, 6,850,000 missing values are found and corrected, whereas the number of instant data containing attribute values that do not follow the domain constraints or integrity rule is 4,082,630. The first deep learning task applied in the framework is MSME activity image description (image captioning) generated by the convolutional neural network-recurrent neural network (CNN-RNN) model. By using 1000 MSME images as data training, the model’s performance is quite good, achieving the average BLEU score of Culinary 0,3149, Fashion 0,4868, and creative products 0,5086. So far, the proposed framework can contribute to supporting MSME one data as the Indonesian government program.

Two linearized difference schemes on graded meshes for the time-space fractional nonlinear diffusion-wave equation with an initial singularity

Abstract In this article, two linearized difference schemes are considered to solve a class of time-space fractional nonlinear diffusion-wave equations with weak singularity near the initial time based on its equivalent integro-differential equation. The Crank-Nicolson method combined with the trapezoidal product integration rule with graded meshes is implemented to approximate the Riemann-Liouville integral. The fractional central difference formula and a novel compact difference method are used to construct fully discrete difference schemes. Using the energy method, the stability and convergence of the proposed schemes are demonstrated in terms of the L 2 norm with the accuracy of O N − min { r σ , 2 } + M − 2 and O N − min { r σ , 2 } + M − 4 , respectively. The theoretical analysis is complemented by numerical results.

Novel notions of symmetric Hahn calculus and related inequalities

In this manuscript, we demonstrate a graphical comparison analysis of the classical, quantum, and symmetric quantum derivatives for any continuous function, evaluated at z=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak{z}=1$\\end{document} and q=0.5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathtt{q}=0.5$\\end{document}. We introduce the new notions of symmetric Hahn calculus and give the basic interpretations of derivative and integration of a function in Hahn symmetric quantum calculus. This enable us to to derive general Hölder’s inequality in symmetric Hahn calculus and as an application, we extract corresponding symmetric Hahn Minkowski’s inequality in general settings. Later, we construct an exciting Lemma in which we derive power rule of integration in symmetric Hahn calculus. An example is provided in which we give explicit analysis on four cases. Finally, these findings allow us to construct Hermite–Hadamard inequality and its related kinds in symmetric Hahn calculus. It is pertinent to mention that the particular case of our results recapture the results of symmetric quantum calculus.

A general correction for numerical integration rules over piece-wise continuous functions

This article presents a novel approach to enhance the accuracy of classical quadrature rules by incorporating correction terms. The proposed method is particularly effective when the position of an isolated discontinuity in the function and the jump in the function and its derivatives at that position are known. Traditional numerical integration rules are exact for polynomials of certain degree. However, they may not provide accurate results for piece-wise polynomials or functions with discontinuities without modifying the location and number of data points in the formula. Our proposed correction terms address this limitation, enabling the integration rule to conserve its accuracy even in the presence of a jump discontinuity. The numerical experiments that we present support the theoretical results obtained.

A novel approach to nonlinear fractional volterra integral equations

Nonlinear Fractional Volterra integral equations (FVIEs) of the first kind present challenges due to their intricate nature, combining fractional calculus and integral equations. In this research paper, we introduce a novel method for solving such equations using Leibniz integral rules. Our study focuses on a thorough analysis and application of the proposed algorithm to solve fractional Volterra integral equations. By using Leibniz integral rules, we offer a fresh perspective on handling these equations, shedding light on their fundamental properties and behaviours. As a result of this study, we anticipate contributing distinctively to the broader development of analytical tools and techniques. By bridging the gap between fractional calculus and integral equations, our approach not only offers a valuable computational methodology but also paves the way for new insights into the application domains in which such equations arise.

A new linearized ADI compact difference method on graded meshes for a nonlinear 2D and 3D PIDE with a WSK

In this work, a new linearized alternating direction implicit (ADI) compact difference method (CDM) is proposed for solving nonlinear two-dimensional (2D) and three-dimensional (3D) partial integrodifferential equation (PIDE) with a weakly singular kernel (WSK). The time derivative is treated by Crank-Nicolson (CN) method and the Riemann-Liouville (R-L) integral by product integration (PI) rule on graded meshes. The linear interpolation combining with Taylor formula is applied in time to deal with nonlinear term v∇v in interval (t0,t1), and linear interpolation concerning two previous time points is employed to deal with v∇v in intervals (tn−1,tn),n≥2. A linearized semi-discrete scheme is obtained, which can achieve second-order convergence in time. Then via introducing two kinds of compact difference operators to discretize the spatial derivatives. To improve the computing efficiency, we construct a ADI compact difference method. It is the first time that the ADI compact difference method is applied for the nonlinear 2D and 3D PIDE with a WSK. The advantage of our proposed scheme is that it not only has second-order accuracy in time and fourth-order accuracy in space, but also fast computational speed, just by solving the linear coupled equations for tridiagonal matrices. In addition, we prove the existence, uniqueness and convergence of 2D scheme. Four numerical examples in 2D and 3D are present to demonstrate our proposed method.

LAWFARE: COMO INSTRUMENTOS DA DEMOCRACIA SÃOUSADOS CONTRA O PRÓPRIO ESTADO DEMOCRÁTICO DEDIREITO

This article explores the concept of Lawfare and its application as a strategy to weaken democraciesthrough the manipulation of the legal system for political purposes. We analyze how, by substitutingmilitary force with judicial measures, Lawfare aims to delegitimize opponents and neutralize politicalleaders, creating a crisis of confidence in democratic institutions. The research adopts a qualitativeapproach, based on a literature review and case analysis, highlighting the threat that Lawfare poses tothe Rule of Law and democratic integrity. The conclusion points out that the use of Lawfare not onlycompromises justice but also generates a cycle of distrust and social polarization.

An equivalent average shear stress yield criterion based on Simpson's numerical integration rule and its application in burst pressure analysis of thin-walled pipelines

Pipelines are a crucial transportation infrastructure for the long-distance transport of natural gas, oil, and other hydrocarbons, typically conducted through thin-walled pipes (D/t≥20). Accurate prediction of the burst pressure of thin-walled pipelines is essential for their safe and reliable operation. To enhance the prediction accuracy of the burst pressure for defect-free, straight, thin-walled pipes, a new numerical integration yield criterion was proposed between the Tresca and Mises criteria, using the mean value of Simpson's numerical integration (MSI) as the equivalent shear stress. A new burst pressure prediction formula was then constructed using the MSI yield criterion. The burst pressure predictions based on different yield criteria were compared with experimental data. Additionally, the burst pressure curves of the new prediction formula were plotted, and the relative errors between the new prediction formula and four existing prediction formulas were analyzed and compared. The results indicate that the new burst pressure prediction formula using the MSI yield criterion closely matches the experimental data, with the best relative error result within a confidence interval of ±1.3 %. The Lord parameter curve and plane stress curve of the MSI yield criterion exhibit good consistency with the experimental data of various ductile metal materials. The new burst pressure formula established by incorporating the MSI criterion provides a safe, economical, and reliable theoretical foundation for the design, manufacture, inspection, and safety assessment of thin-walled pipelines in production practice.

Improved eight-node non-conforming hexahedral element for structures of various shapes

A previous 8-node 24-degree-of-freedom hexahedral element H8i9 is modified to improve its performance in structures of various shapes, especially in plates and shells. First, the complicated correction coefficients for non-conforming modes required by the patch test are reduced to constant values independent of the element. Second, the element strain field is enhanced with bilinear modes by the introduction of a trilinear non-conforming mode. Third, the iterative regularization used to address the ill-conditioned stiffness matrix is replaced by a special 9-point integration rule. Benchmark tests demonstrate that the new element H8ij10 outperforms the previous version, particularly with a notable improvement in coarse-mesh accuracy in plate and shell issues.

Course gamification: Applying the KANO model to gather student perceptions

Design for Six Sigma (DFSS) is a critical component of the Six Sigma methodology, focusing on product, process, and service design. DFSS is proactive, focusing on design rather than existing processes. Its wide application in various sectors, from manufacturing to service industries, has proven its effectiveness. Despite this, its adoption in education remains limited. This research explores the application of DFSS in education, particularly in redesigning a course through gamification to enhance student engagement. This study addresses this gap by illustrating how DFSS principles can enhance education, mainly through gamification. Gamification, incorporating game elements into learning, holds promise for engaging students and improving learning outcomes. Using the DMADV framework, the study defines project objectives, gathers the voice of the customer using the Kano model, analyzes survey data, designs gamification elements, and verifies their effectiveness through student feedback. Results indicate several attractive elements identified through the Kano model, such as smartphone integration and clear rules. Implementation challenges, particularly in a Jeopardy-style review session, are identified and addressed using DFSS tools such as failure mode and effects analysis. Subsequent surveys capture student feedback and suggest improvements, highlighting the iterative nature of DFSS. The study underscores the potential of DFSS in educational redesign, particularly through gamification, to enhance student engagement and learning outcomes. While focused on a specific course, the methodology's generalizability and applicability to diverse educational contexts warrant further exploration for broader implementation.

Using maple software to reduce problem solving errors in teaching mathematics to future computer science teachers

This article discusses some possible uses of Maple in mathematics. Mathematics is currently in a constant state of development in the application of computer technology in education and research. The use of algebraic systems in education is still relatively small, but growing research and interest in them point to their widespread use. There are complex and abstract ideas that present an insurmountable obstacle for many students in solving basic concepts and proving many mathematical concepts. One of the most powerful computer programs that helps in learning mathematics is Maple. Maple allows you to explain classic mathematical concepts in a practical and effective way. The effectiveness of Maple is due to its ability to visualize concepts and solve complex problems that are difficult to solve manually. Maple's features help students develop new methods and better understand, learn and teach complex concepts, advanced mathematical concepts and more. Ideal for students who are developing problem solving skills and advanced development skills. Allows students to solve mathematical problems step by step. Students can practice using various integration rules.

The Interaction Between Literature and Commercial Films: A Case Study of Hong Kong Cinema

This paper explores the interaction between literature and commercial films, focusing on Hong Kong cinema. In modern society, as the economy develops and people s spiritual needs increase, the integration of literature and commercial films has become a new cultural phenomenon. With its unique cultural background, Hong Kong cinema has successfully blended literary works with commercial films, creating a distinctive artistic style. The paper analyzes how literary works provide creative inspiration for films and how films, through commercialization, promote the dissemination of literature. Through case studies, the paper examines the impact of different types of literature on films, the reciprocal influence of films on literary creation, and how writers adapt to and cater to public demands in the face of commercialization. The paper aims to uncover the inherent rules of integration and development between literature and commercial films, encouraging the use of more excellent literary works in the film and television industry.

Evolutional behavior of viscoelastic materials: A formulation based on logarithmic approach

In this paper, a new formulation for large deformation of visco-elastometic materials is derived. The formulation is based on logarithmic strain, and the internal variable is defined as the viscous part of the logarithmic strain. A derivation of thermodynamically consistent formulation for visco-elastomeric materials is provided. The evolution equation which is related to the deviatoric part of the strain for viscoelastic formulation is solved by the implicit trapezoidal integration rule which leads to a recursive relation.

A semantic axiomatic design for integrity in IoT

AbstractIn the complex, critical, and rapidly evolving era of Industry 4.0 manufacturing, mature engineering disciplines can be developed to validate system designs at an abstract level, enabling effective fault‐free environments. However, designing integrated IoT platforms presents significant challenges due to their inherent complexity and interoperability issues. To address this challenge, the proposed framework integrates axiomatic design principles with considerations for data integrity and domain ontology. Furthermore, this paper introduces a multi‐level integrity rule repository for IoT application to provides a robust foundation for designing and maintaining IoT systems. To validate the effectiveness of this approach, we conducted a study across four domains of smart cities. The design's quality was assessed using model quality parameters, including completeness, consistency and validity. Our analysis shows that the semantic‐aware axiomatic design framework can effectively address the problem of inconsistencies and low‐quality design in scientific IoT environments, providing a more reliable and efficient solution.

A Filon-Clenshaw-Curtis-Smolyak rule for multi-dimensional oscillatory integrals with application to a UQ problem for the Helmholtz equation

A Filon-Clenshaw-Curtis-Smolyak rule for multi-dimensional oscillatory integrals with application to a UQ problem for the Helmholtz equation

Recurrence relations for the joint distribution of the sum and maximum of independent random variables

Abstract In this paper, the joint distribution of the sum and maximum of independent, not necessarily identically distributed, nonnegative random variables is studied for two cases: (i) continuous and (ii) discrete random variables. First, a recursive formula of the joint cumulative distribution function (CDF) is derived in both cases. Then recurrence relations of the joint probability density function (PDF) and the joint probability mass function (PMF) are given in the former and the latter case, respectively. Interestingly, there is a fundamental difference between the joint PDF and PMF. The proofs are simple and mainly based on the following tools from calculus and discrete mathematics: differentiation under the integral sign (also known as Leibniz’s integral rule), the law of total probability, and mathematical induction. In addition, this work generalizes previous results in the literature, and finally presents several extensions of the methodology.

Needlets liberated

Spherical needlets were introduced by Narcowich, Petrushev, and Ward to provide a multiresolution sequence of polynomial approximations to functions on the sphere. The needlet construction makes use of integration rules that are exact for polynomials up to a given degree. The aim of the present paper is to relax the exactness of the integration rules by replacing them with QMC designs as introduced by Brauchart, Saff, Sloan, and Womersley (2014). Such integration rules (generalised here by allowing non-equal cubature weights) provide the same asymptotic order of convergence as exact rules for Sobolev spaces Hs, but are easier to obtain numerically. With such rules we construct “generalised needlets”. The paper provides an error analysis that allows the replacement of the original needlets by generalised needlets, and more generally, analyses a hybrid scheme in which the needlets for the lower levels are of the traditional kind, whereas the new generalised needlets are used for some number of higher levels. Numerical experiments complete the paper.

Impacts of operating variables in forward roll coating process of viscous hybrid nanofluid

Abstract Roll coating plays a significant role in various coating industries such as magnetic records, wallpapers, wrapping, adhesive tapes, books and magazines, photographic and plastic films. The thin layer coating of a magnetohydrodynamic (MHD) viscous hybrid nanofluid by passing through the space between two co-rotating rolls has been studied in an isothermal and incompressible analysis. The governing equation of mass and momentum are obtained then dimensionless using lubrication approximation theory (LAT). The velocity, pressure gradient, and pressure distribution are determined by the exact solution. Using Simpson’s (3/8) rule for numerical integration, the complex integral is examined. Important engineering parameters including power and roll separating force delivered by the rolls to the fluid are also estimated numerically. Raising the volume fraction of nanoparticles raises the pressure distribution and pressure gradient while having little effect on the velocity profile. It seems that the magnetic field and hybrid nanofluid, both seem very advantageous for the efficient roll coating process, controlling the separation force, power input, and distance between the attachment and separation point.