Definite Integrals Research Articles

The Third Romberg Extrapolate as a Numerical Integration

Modern techniques for quadrature include, but are not limited to, the Trapezium rule, the Midpoint rule, and Simpson’s rule. The accuracy of these methods can be improved by employing Romberg’s method. This is achieved by applying each method to a definite integral, subdividing it into multiple intervals, and then taking appropriate linear combinations of the resulting estimates to produce approximations with high-order accuracy. In this work, the third Romberg extrapolate is applied to a definite integral, and its solution is compared with the exact solution to demonstrate its accuracy.

A Comprehensive Survey on the Integrity of Localization Systems

This survey extends and refines the existing definitions of integrity and protection level in localization systems (localization as a broad term, i.e., not limited to GNSS-based localization). In our definition, we study integrity from two aspects: quality and quantity. Unlike existing reviews, this survey examines integrity methods covering various localization techniques and sensors. We classify localization techniques as optimization-based, fusion-based, and SLAM-based. A new classification of integrity methods is introduced, evaluating their applications, effectiveness, and limitations. Comparative tables summarize strengths and gaps across key criteria, such as algorithms, evaluation methods, sensor data, and more. The survey presents a general probabilistic model addressing diverse error types in localization systems. Findings reveal a significant research imbalance: 73.3% of surveyed papers focus on GNSS-based methods, while only 26.7% explore non-GNSS approaches like fusion, optimization, or SLAM, with few addressing protection level calculations. Robust modeling is highlighted as a promising integrity method, combining quantification and qualification to address critical gaps. This approach offers a unified framework for improving localization system reliability and safety. This survey provides key insights for developing more robust localization systems, contributing to safer and more efficient autonomous operations.

On Convolved Fibonacci Polynomials

This work delves deeply into convolved Fibonacci polynomials (CFPs) that are considered generalizations of the standard Fibonacci polynomials. We present new formulas for these polynomials. An expression for the repeated integrals of the CFPs in terms of their original polynomials is given. A new approach is followed to obtain the higher-order derivatives of these polynomials from the repeated integrals formula. The inversion and moment formulas for these polynomials, which we find, are the keys to developing further formulas for these polynomials. The derivatives of the moments of the CFPs in terms of their original polynomials and different symmetric and non-symmetric polynomials are also derived. New product formulas of these polynomials with some polynomials, including the linearization formulas of these polynomials, are also deduced. Some closed forms for definite and weighted definite integrals involving the CFPs are found as consequences of some of the introduced formulas.

An Improved Prediction Method for Failure Probability of Natural Gas Pipeline Based on Multi-Layer Bayesian Network

The failure probability of a pipeline is a quantification of the likelihood of an accident occurring in the pipeline, which is an indispensable part of the pipeline risk assessment process. To solve the problems of strong subjectivity, low feasibility, and low accuracy in the existing pipeline failure probability calculation methods, a three-layer Bayesian network topology model of “pipeline failure–failure cause–influencing factor” is proposed, with the pipeline failure as the subnode, the type of pipeline failure as the intermediate node, and the factors affecting the pipeline failure as the parent node of the network. Based on data fitting and fuzzy theory analysis methods, the functional relationship between the impact factor and the failure frequency of various pipelines is quantified. Using the mean value theorems for definite integrals and the analytic hierarchy process, the conditional probability of the directed edge in the network is calculated. The proposed function relationship provides a method to calculate the prior probability according to the parameters of the pipeline and its surroundings and a new idea to train the network model even without sufficient data.

Raising the level of integrity as a task of national-patriotic education of the Armed Forces of Ukraine

The purpose of this article is to characterize the enhancement of integrity as a task of national­patriotic education of the Armed Forces of Ukraine’s military personnel. It is found that the Law of Ukraine «On Prevention of Corruption» does not explicitly define the category of integrity, despite its frequent mention. This oversight, coupled with the brevity of other relevant legal acts, has led to varying interpretations of the concept. As a result, numerous integrity codes present inconsistent and often arbitrary definitions of integrity. Given the Armed Forces of Ukraine’s primary constitutional function of defending the country’s sovereignty and territorial integrity, the integrity of its personnel should be viewed as a matter of public interest rather than merely an internal military concern. This necessitates a unified understanding of integrity to prevent arbitrary interpretations by law enforcement agencies. Elevating integrity as a key component of national-patriotic education for Ukrainian military personnel requires a comprehensive approach. This involves implementing a range of organizational, legal, pedagogical, psychological, and other measures aimed at instilling in personnel a strong sense of duty to the nation, society, and the individual. This task is particularly challenging under martial law, as military units often consist of individuals with diverse values, motivations, and behavioral patterns. To address the issue of inconsistent interpretations, it is advisable to codify the concept of integrity in the Law of Ukraine «On Prevention of Corruption.» Additionally, the Code of Good Conduct and Professional Ethics for Military Officials, Civil Servants and Other Persons Authorized to Perform State Functions in the Ministry of Defense of Ukraine and the Armed Forces of Ukraine should be amended to emphasize the societal dimension of integrity. Furthermore, the Law of Ukraine «On the Basic Principles of State Policy in the Sphere of Strengthening Ukrainian National and Civil Identity» should explicitly include integrity education as a crucial component of national-patriotic education.

Improper Integrals of Type II—A New Look and Some Surprising Facts

The goal of this work is to present a new theoretical and simultaneously practical approach to the conception of improper integrals of type II and, consequently, to improper integrals of type I. Seven distinct definitions of improper integrals of type II are presented, each illustrated by suitable examples. Among them, particularly important are those defined on nets composed of sets symmetric with respect to points belonging to a given finite family of points. Connections between improper integrals of type II defined here are described in two main theorems. It should be emphasized that all definitions of improper integrals presented here are special cases of a very general definition of improper integrals on non-compact sets, which is presented at the end of this work. This definition is inspired by a concept introduced by Maurin.

A socio-epistemological approach articulated with problem-solving in higher education: Teaching of integral calculus

The training of mathematics teachers in universities in Colombia has as a transversal axis the resolution of problems based on their social and cognitive mission pillars. In this sense, this study relates to the re-signification and construction of the concept of definite integral (DI) (integral calculus) through socio-epistemological studies, action researches, and the typology of didactic situations. The results are obtained through content analysis, didactic sequences (GeoGebra), and a discussion group. The above allows us to conclude that the validation of meanings, historical contexts, and associated social practices leads to the construction of the concept of DI as a model of mathematical analysis. This structuring of knowledge from its epistemological framework enables the exploration of mathematical objects from the basic notions that emerge in the history of humanity and didactic processes that reconstruct the evolution of the concept in society.

Integrais definidas na determinação de características em estruturas metálicas na construção de um galpão

The objective proposed in this study will be to present a practical application of the use of the content of defined integrals in determining certain geometric characteristics in fundamental metallic structures in civil construction, knowing that there is a concern on the part of teachers and pedagogical coordinators regarding the teaching and learning of specific subjects, leading students to a lack of stimulation and, consequently, the much-feared dropout. As an application of the use of integrals, we can consider a civil engineer who has planned the construction of an agricultural shed and knows that the roof of the structure is described by a quadratic function of the type . Even though this structure is subject to a transverse shape resulting from its own weight and the weight of other structures that are supported on it. In cases where geometry proves to be inefficient, we will introduce the concept of the definite integral. Let

Numerical solution of fractional integro-differential equations using ultraspherical polynomials

In this work we will discuss the fractional integro-differential (FIDE) equations of the Fredholm and Volterra type, where we begin by presenting some special concepts about the definition of the fractional integral and the fractional derivative of Riemann-Liouville and Caputo, while addressing the Gegenbauer (Ultraspherical) polynomials and its special differential equation, mentioning the primary polynomials, in addition to presenting some theories such as the Krasnoselskii fixed point theorem, which guarantees the existence and uniqueness of solutions to fractional integro-differential equations (FIDE). To find the approximate solution of fractional integro-differential equations, we relied on orthogonal polynomials of the ultraspherical type (Gegenbauer), which have a great advantage in solving integral equations and integro-differential equations. These polynomials, with the application of fractional Caputo derivatives with the series method, in addition to fixing the values and changing the degree of the polynomials used with fixing the values of , lead to the equation becoming a linear algebraic system of equations. By solving this linear system, the approximate solution is obtained in the form of a Gegenbauer (ultraspherical) series. The change in the degree of the polynomials used will inevitably show the effectiveness and speed of the method through the examples that will be presented, where the approximate solutions obtained are compared with the exact solutions.

Recreational sports in prison: inmates' perspectives on coaching effectiveness.

This study examines inmates' perspectives on the effectiveness of sports coaches in prison. According to the integrative definition of effective coaching, coaches require professional, interpersonal, and intrapersonal knowledge to enhance athletes' outcomes within a specific context. Five male inmates of a German prison were interviewed after they participated in sports programs. The data were analyzed using directed content analysis. The inmates recognized professional knowledge in coaches who demonstrated clear training structures and spoke in an educated manner. Interpersonal knowledge was highly valued, with inmates expressing a preference for coaches who showed closeness, truthfulness, and responsiveness, while intrapersonal knowledge was attributed through coaches' adaptability. Inmates identified all four possible outcomes and emphasized that sensitivity, authenticity, energetic demeanor, physical appearance, and athletic skills are crucial qualities for an effective coach in a prison. The balance between an authentic and energetic demeanor, including a certain degree of strictness, emerges as a key factor in effective coaching within the prison context.

Analisis Kemampuan Pemahaman Matematis Mahasiswa dalam Menyelesaikan Soal Integral Tentu Berdasarkan Teori APOS

This study aims to describe the mathematical understanding ability of students regarding the concept of definite integral based on APOS Theory, and describe the achievement of each indikator of students' mathematical understanding ability regarding the concept of definite integral of algebraic functions based on APOS Theory. The research method used is descriptive qualitative. The data collection techniques used in this study were written tests and interviews. The results showed that the achievement in question number one on the K indikator 88.89%, L 83.33% and S 69.44%, most of the students who were able to solve the problem meant that they had been able to organise the actions, processes, and objects they had so that based on APOS theory, the student was at the level of understanding the scheme. Achievement in question number two on indikator K 80.55%, L 58.33% and S 50%. Most students are only able to identify properties and conditions, meaning that students only have the ability to understand at the action stage. Achievement in question number two on indikator K 86.11%, L 66.67% and S 63.88%. Most students are only able to identify properties and conditions and work on problems according to the steps.

Generalized Taylor Series and Peano Kernel Theorem

ABSTRACTAs in the polynomial case, non‐polynomial divided differences can be viewed as a discrete analog of derivatives. This link between non‐polynomial divided differences and derivatives is defined by a generalization of the derivative operator. In this study, we obtain a generalization of Taylor series using the link between non‐polynomial divided differences and derivatives, and state generalized Taylor theorem. With the definition of a definite integral, the relation between the non‐polynomial divided difference and non‐polynomial B‐spline functions is given in terms of integration. Also, we derive a general form of the Peano kernel theorem based on a generalized Taylor expansion with the integral remainder. As in the polynomial case, it is shown that the non‐polynomial B‐splines are in fact the Peano kernels of non‐polynomial divided differences.MSC2020 Classification: 65D05, 65D07

Object-Oriented Zero-Information-Optimised Product of Monte Carlo Integrations

Abstract We propose an object-oriented Monte Carlo framework for computation of definite integrals with a primary focus on generality and simplicity, improving in find points existing software libraries. We introduce an optimised sampling method for a product of Monte Carlo integrations. The method does not require any a-priori knowledge of the integrands or the integration domains, and can be combined with other optimisation methods. Using our framework, we implement a naive uniform sampling method and the optimised one, and empirically compare them on the Monte Carlo computation of volumes of N-dimensional balls. The experiments confirm our theoretical expectations.

Urban Heritage Resilience: An Integrated and Operationable Definition from the SHELTER and ARCH Projects

Resilience, initially a concept rooted in psychology, has traversed disciplinary boundaries, finding application in fields such as urban planning and development since the 2010s. Despite its broad application, most definitions remain too abstract to allow their practical integration into urban planning and development contexts. Addressing this challenge, the European research projects SHELTER and ARCH offer a practicable integration of resilience with planning and development practices surrounding urban heritage. Following a systemic approach to resilience, both projects integrate perspectives from urban development, climate change adaptation, disaster risk management, and heritage management, supported with tools and guidance to anchor resilience in existing practices. This paper presents the results from both projects, including similarities and differences.

Solution of Well-known Physical Problems with Fractional Conformable Derivative

This paper presents a new definition of fractional derivatives and integrals through the conformable derivative approach. This innovative framework offers a closer alignment with classical derivative concepts while providing a more practical and intuitive basis for fractional calculus. The new definition is applicable in two primary ranges: 0 ≤ α < 1 and n − 1 ≤ α < n, where n is a positive integer. It is shown that when α = 1, this definition corresponds precisely to the classical first-order derivative. Key benefits of this approach include its improved consistency with traditional calculus and greater computational ease, making it a useful tool for both theoretical research and practical applications. By integrating fractional calculus with conventional derivative ideas, this definition simplifies the analysis and interpretation of fractional differential equations and their solutions. Additionally, we examine the definition’s effects on stability and convergence in numerical methods and provide examples demonstrating its effectiveness and applicability.

The Application of Definite Integral In Different Aspects

A definite integral is a form of a calculation of area beneath of function, using infinitesimal silver or stripes of the region. Different from indefinite integral which representing a function, definite integral is a specific value. The value of a definite integral only depends on the interval of this integral, which means for the same function and interval, the result won’t change with different variables. Integrals may represent the area of a region, the accumulated value of a function changing over time, or the quantity of an item given its density. This method can be used to solve many problems in different fields. This paper will explore some applications of definite integral. This article first introduces the applications of definite integrals in geometry, and then studies their applications in physics, such as variable force work and pumping work. Finally, how to use the properties of definite integrals to solve limit problems will be further studied.

The First Mean Value Theorem of integral in Perplex Plane

Perplex numbers are an extension of the real numbers, consisting of pairs of real numbers that form a commutative ring. This article considers the partial order structure of perplex numbers and discusses the concept of perplex interval. Based on the definition of integration of perplex functions of perplex variable, this article we investigate the first mean value theorem of integral in perplex plane which can significantly enrich the mathematical toolbox by providing essential methods for estimating and approximating perplex number functions. It also simplifies the solution of complex integral problems involving perplex numbers and fosters theoretical extensions by exploring the behavior of perplex number functions under various conditions. This, in turn, promotes the development of perplex calculus, enhancing the overall theoretical and practical framework of perplex number analysis. In physics, particularly in relativity, studying the first mean value theorem for perplex numbers can provide more precise methods for describing and estimating physical quantities, enhancing the application of hyperbolic numbers in these fields.

Evaluation of the Second Virial Coefficient for the Mie Potential Using the Method of Brackets

The second virial coefficient for the Mie potential is evaluated using the method of brackets. This method converts a definite integral into a series in the parameters of the problem, in this case this is the temperature T. The results obtained here are consistent with some known special cases, such as the Lenard-Jones potential. The asymptotic properties of the second virial coefficient in molecular thermodynamic systems and complex fluid modeling are described in the limiting cases of T → 0 and T → ∞ .

Performance Analysis and Parallel Scalability of Numerical Methods for Fractional-in-Space Diffusion Problems with Adaptive Time Stepping

We study numerical methods and algorithms for time-dependent fractional-in-space diffusion problems. The considered anomalous diffusion is modelled by the fractional Laplacian (−Δ)α, 0<α<1, following the integral definition. Fractional diffusion is non-local, and the finite element method (FEM) discretization in space leads to a dense stiffness matrix. It is well known that numerically solving such non-local boundary value problems is expensive. Difficulties increase significantly when the problem is time-dependent. The aim of the article is to develop computationally efficient methods and algorithms. There are two main features of our approach. Hierarchical semi-separable (HSS) compression is applied for an approximate solution of the arising linear systems. For time discretization, we use an adaptive forward–backward Euler scheme. The properties of the composite algorithm thus obtained are investigated. In particular, the block representation of HSS compression allowed us to upgrade the HSS solver to efficiently handle varying diagonally perturbed transition matrices corresponding to changing time steps. The contribution of the paper is threefold. The methods are completely constructive, which allows for a clearly structured description of the algorithms. A theoretical estimate of the computational complexity is presented. It shows the advantages of the adaptive time stepping in combination with the HSS solver. Theoretical results are supported by representative numerical experiments. Both sequential and parallel scalability and efficiency are analyzed. The presented results provide convincing proof of the concept of the proposed methods and algorithms.

Z22-graded supersymmetry via superfield on minimal Z22-superspace

AbstractA superfield formalism for the minimalZ22-graded version of supersymmetry is developed. This is done by using the recently introduced definition of integration on the minimalZ22-superspace. It is shown that one may constructZ22-supersymmetric action by the procedure similar to the standard supersymmetry. However, the Lagrangian obtained has very general interaction terms, which give rise to aZ22-graded extension of many known theories defined in two-dimensional spacetime. As an illustration, we will give aZ22-extension of the sine-Gordon model different from the one already discussed in the literature.