Double Integral Research Articles

A simplified counterexample to the integral representation of the relaxation of double integrals

A simplified counterexample to the integral representation of the relaxation of double integrals

Model adaptive reference tracking control for uncertain robotic manipulators with input disturbance

AbstractThis article presents a simple approach for designing a linear adaptive controller, employing the model reference systems concept, to enable the tracking of uncertain robotic manipulators under the influence of input disturbances. The combined impact of model uncertainties and matched disturbances on the robot's behavior is considered as a total matched disturbance, attributed to a double integral system. Subsequently, a novel linear disturbance estimator, augmented by a feed‐forward correction term, is employed to estimate this lumped disturbance within the double integrator system. As a result of this procedure, the requisite adaptive law, based on the model reference system, is formulated for the nominal control parameter instead of arbitrary free‐parameter selections. The effectiveness of the proposed approach is theoretically justified and further supported by its application in the analysis of an active magnetic bearing system.

Near mass-shell double boxes

Two-loop multi-leg form factors in off-shell kinematics require knowledge of planar and nonplanar double box Feynman diagrams with massless internal propagators. These are complicated functions of Mandelstam variables and external particle virtualities. The latter serve as regulators of infrared divergences, thus making these observables finite in four space-time dimensions. In this paper, we use the method of canonical differential equations for the calculation of (non)planar double box integrals in the near mass-shell kinematical regime, i.e., where virtualities of external particles are much smaller than the Mandelstam variables involved. We deduce a basis of master integrals with uniform transcendental weight based on the analysis of leading singularities employing the Baikov representation as well as an array of complementary techniques. We dub the former asymptotically canonical since it is valid in the near mass-shell limit of interest. We iteratively solve resulting differential equations up to weight four in terms of multiple polylogarithms.

IMPLEMENTASI MODEL PEMBELAJARAN BERBASIS PROYEK UNTUK MENGIDENTIFIKASI KEMAMPUAN BERPIKIR KREATIF MAHASISWA MATEMATIKA

Advanced calculus is a mandatory subject in the Mathematics Study Program, Faculty of Science and Technology, Jambi University. Advanced Calculus studies functions of multiple variables, ordinary derivatives, partial derivatives, double integrals, fold integrals, applications of double integrals, triple integrals and applications of triple integrals. CPL from the advanced calculus course includes students being able to develop mathematical thinking starting from procedural/computational understanding to a broad understanding including exploration, logical reasoning, generalization, abstraction and formal proof. One learning model that facilitates students to fulfill the CPL for advanced Calculus courses is the project-based learning model (Project-based Learning). It is hoped that the Project-Based Learning model can be a learning alternative to help students think creatively in learning Advanced Calculus. It is hoped that the creative thinking abilities of Mathematics Study Program students will increase and support the fulfillment of CPL courses which will result in the formation of creative and systematic mathematics students who are skilled in solving problems and have the capabilities and competencies needed by the world of work (industry, services and government). The results of the analysis showed that the creative thinking abilities of students in class R001 were classified as creative and 1 group was very creative, while in R002 all groups were classified as very creative. The conclusions of this research show a tendency for students' creative thinking abilities to be in the creative category.

Sharp conditions for the validity of the Bourgain–Brezis–Mironescu formula

Following the seminal paper by Bourgain, Brezis, and Mironescu, we focus on the asymptotic behaviour of some nonlocal functionals that, for each $u\in L^2(\mathbb {R}^N)$ , are defined as the double integrals of weighted, squared difference quotients of $u$ . Given a family of weights $\{\rho _{\varepsilon} \}$ , $\varepsilon \in (0,\,1)$ , we devise sufficient and necessary conditions on $\{\rho _{\varepsilon} \}$ for the associated nonlocal functionals to converge as $\varepsilon \to 0$ to a variant of the Dirichlet integral. Finally, some comparison between our result and the existing literature is provided.

A mathematical model of the gravitational potential of the planet taking into account tidal deformations

Objectives. This paper investigates the gravitational potential of a viscoelastic planet moving in the gravitational field of a massive attracting center (star), a satellite and one or more other planets moving in Keplerian elliptical orbits relative to the attracting center. Celestial bodies other than a viscoelastic planet are modeled by material points. Within the framework of the linear model of the theory of viscoelasticity, the problem of finding the vector of elastic displacement has been resolved. Traditionally, a solid body model is used to determine the Earth’s gravitational field, while tidal deformations are taken into account in the form of small corrections to the coefficients of the geopotential model. In this work, the viscoelastic ball model is used to take into account tidal effects. The relevance of the research topic is associated with high-precision forecasting of the movement of artificial satellites of the Earth, high-precision measurement of the Earth’s gravitational field.Methods. In this study the asymptotic and analytical methods developed by V.G. Vilke are used for mechanical systems containing viscoelastic elements of high rigidity, as well as methods of classical mechanics, mathematical analysis. The graphs were plotted using the Octave mathematical package.Results. After resolving the quasi-static problem of elasticity theory by calculating triple integrals over a spherical area, a formula for the gravitational potential of a deformable planet was obtained. In addition, the gravitational potential of the Earth was also calculated taking into account solid-state tidal effects from the Moon, Sun, and Venus at an external point. Graphs were constructed to show the dependence of the Earth’s gravitational potential on time.Conclusions. The theoretical and numerical results established herein show that the main contribution to the gravitational potential of the Earth is made by the Moon and the Sun. The influence of other planets in the solar system is small. The value of the gravitational potential at the outer point of the Earth, taking into account tidal effects, depends both on the position of the point in the moving coordinate system and on the relative position of celestial bodies.

Impact of changes in collagen construction and molecular state on integrin – binding properties

The interaction between the integrin and collagen is important in cell adhesion and signaling. Collagen, as the main component of extracellular matrix, is a base material for tissue engineering constructs. In tissue engineering, the collagen structure and molecule state may be altered to varying degrees in the process of processing and utilizing, thereby affecting its biological properties. In this work, the impact of changes in collagen structure and molecular state on the binding properties of collagen to integrin α2β1 and integrin specific cell adhesion were explored. The results showed that the molecular structure of collagen is destroyed under the influence of heating, freeze-grinding and irradiation, the triple helix integrity is reduced and molecular breaking degree is increased. The binding ability of collagen to integrin α2β1 is increased with the increase of triple helix integrity and decays exponentially with the increase of molecular breaking degree. The collagen molecular state can also influences the binding ability of collagen to cellular receptor. The collagen fibrils binding to integrin α2β1 and HT1080 cells is stronger than to collagen monomolecule. Meanwhile, the hybrid fibril exhibits a different cellular receptor binding performance from corresponding single species collagen fibril. These findings provide ideas for the design and development of new collagen-based biomaterials and tissue engineering research.

TWO-DIMENSIONAL GENERALIZED GAMMA FUNCTION AND ITS APPLICATIONS

In this article, we present a new two-dimensional generalization of the gamma function based on the product of the one-dimensional generalized beta function and the one-dimensional generalized gamma function. As will become clear later, one of the properties of this generalization is a formula that generalizes the famous formula establishing the connection between the classical gamma and beta functions. Next, we present the properties of this generalization, some series for the generalized beta function and the generalized two-dimensional gamma function. As a practical application of the two-dimensional generalized gamma function, we will show how it can be used to represent a fairly wide class of double integrals in the form of functional series, or in the form of the product of two one-dimensional integrals, which makes it quite easy to find their values. That is, it will be seen how, with the help of simple transformations, many types of double integrals can be reduced to a generalized two-dimensional gamma function (23), which greatly simplifies work with them, thanks to formula (24). At the end of the article, a representation of the classical and generalized hypergeometric function in the form of a two-dimensional generalized gamma function is presented.

Full harmonic spectrum calculation of magnetic pressures in synchronous permanent magnet machines based on a new realistic partial analytical model: application to SVPWM and DPWM2

PurposeThe purpose of this paper is to propose a new method that allows to compare the magnetic pressures of different pulse width modulation (PWM) strategies in a fast and efficient way.Design/methodology/approachThe voltage harmonics are determined using the double Fourier integral. As for current harmonics and waveforms, a new generic model based on the Park transformation and a dq model of the machine was established taking saturation into consideration. The obtained analytical waveforms are then injected into a finite element software to compute magnetic pressures using nodal forces.FindingsThe overall proposed method allows to accelerate the calculations and the comparison of different PWM strategies and operating points as an analytical model is used to generate current waveforms.Originality/valueWhile the analytical expressions of voltage harmonics are already provided in the literature for the space vector pulse width modulation, they had to be calculated for the discontinuous pulse width modulation. In this paper, the obtained expressions are provided. For current harmonics, different models based on a linear and a nonlinear model of the machine are presented in the referenced papers; however, these models are not generic and are limited to the second range of harmonics (two times the switching frequency). A new generic model is then established and used in this paper after being validated experimentally. And finally, the direct injection of analytical current waveforms in a finite element software to perform any magnetic computation is very efficient.

Eğitsel Süreçlerde Mesleki Yeterlik ve Yeterlik Elementlerinin Entegrasyonu

The most important of the methods used in training and development is to shape the knowledge, skills and competencies of the employees of the organization. Education is essentially the internalization of knowledge, while skills and attitude/behavior are related to each other and successive roles. The development is that this trio quickly becomes an observable action in practice as competence. In modern business discourses, the concept of skill has turned into the concept of competence. Competence is a difference in performance that distinguishes effective managers from ineffective managers in a particular organization. since the 1970s, competencies have been the main starting point for evaluating personnel performance in human resources management; competence has been defined as the successful fulfillment of critical tasks and tasks, the potential to apply and use the knowledge, skills and behaviors necessary for a specific role or position. According to a different view, competencies are an output of continuous learning adaptation, the creation of new skills and the sustainability of achievements. Education is an activity of acquiring or producing competencies that are potentially produced in the process of professional development of a person. Since the earliest historical times, the integration of learning activities of knowledge, skills and procedures, the practice, rules and impact of education have been part of the sustainability of society. In social/cultural theories, the knowledge and skills of individuals are seen as inseparable from the social world in which they think and act. Knowledge and skills are the elements that make up competence. When attitude/behavior is developed after knowledge and skills are acquired, declarative and procedural knowledge turns into implicit knowledge, and functional competence emerges, and this triple integration process is completed, the entire process turns into professional competence. The professional competencies that they must acquire by completing the knowledge-skill-attitude integration process have been defined as basic and professional (professional) competencies as a functional competence. Key Words: Educational Processes, Professional Competence, Integration of Competence Elements

The strategic prize game Le Her

AbstractLe Her is a strategic two-person prize game the classic version of which uses a deck of cards. Despite its simple rules, the analysis of the game is surprisingly complex. I describe a modern variant of Le Her that is amenable to elementary analysis, yet retains and highlights all central aspects of the classical game. This elementary analysis combines naturally central concepts of calculus (partial derivatives, multidimensional optimization, double integrals, fixed and saddle points) with major probabilistic concepts (random variable, event, independence) and basic notions in decision theory (Nash equilibrium). Due to its stimulating context as a strategic prize game Le Her offers an attractive illustration and an appealing application of learning goals and curricular contents which are otherwise often perceived as remote and abstract.

Analysis of the Operational Efficiency of Public Hospitals in Tianjin City based on a Three-Stage Data Envelopment Analysis Model

The purpose is to analyze the operational efficiency of 132 public hospitals in Tianjin in 2021, providing reference for them to fully play the role of "leading institutions" and optimize resource allocation. We constructed an evaluation index system with multiple inputs, outputs, and environmental variables and utilized a three-stage DEA model to assess efficiency by removing the influence of random errors and environmental factors. The average comprehensive technical efficiency, pure technical efficiency, and scale efficiency of public hospitals were 0.865, 0.989, and 0.875, respectively. Among them, seven districts, including Heping, Hedong, and Hexi, had a comprehensive technical efficiency of 1. Among the remaining nine districts, except for Baodi, the rest showed an increasing trend in scale efficiency. Public hospitals in Tianjin City exhibit high operational efficiency, but significant variations exist among different districts, leading to uneven development. The main reason for this disparity is the presence of scale inefficiency. It is recommended to prioritize resource optimization and address the shortcomings that hinder efficiency improvement in a targeted manner. Emphasizing the "triple integration" of service capacity, efficiency, and quality in public hospitals is essential to maximize both social and economic benefits."

Numerical Approximation for a Stochastic Fractional Differential Equation Driven by Integrated Multiplicative Noise

We consider a numerical approximation for stochastic fractional differential equations driven by integrated multiplicative noise. The fractional derivative is in the Caputo sense with the fractional order α∈(0,1), and the non-linear terms satisfy the global Lipschitz conditions. We first approximate the noise with the piecewise constant function to obtain the regularized stochastic fractional differential equation. By applying Minkowski’s inequality for double integrals, we establish that the error between the exact solution and the solution of the regularized problem has an order of O(Δtα) in the mean square norm, where Δt denotes the step size. To validate our theoretical conclusions, numerical examples are presented, demonstrating the consistency of the numerical results with the established theory.

Gradient higher integrability for double phase problems on metric measure spaces

We study local and global higher integrability properties for quasiminimizers of a class of double phase integrals characterized by nonstandard growth conditions. We work purely on a variational level in the setting of a metric measure space with a doubling measure and a Poincaré inequality. The main novelty is an intrinsic approach to double phase Sobolev-Poincaré inequalities.

The semiclassical approximation of multiple functional integrals

In this paper, we study the semiclassical approximation of multiple functional integrals. The integrals are defined through the Lagrangian and the action. Of all possible trajectories, the greatest contribution to the integral is given by the classical trajectory x̅cl for which the action S takes an extremal value. The classical trajectory is found as a solution of the multidimensional Euler – Lagrange equation. To calculate the functional integrals, the expansion of the action with respect to the classical trajectory is used, which can be interpreted as an expansion in powers of Planck’s constant. The numerical results for the semiclassical approximation of double functional integrals are given.

Asymptotic analysis for confluent hypergeometric function in two variables given by double integral

We study an integrable connection with irregular singularities along a normally crossing divisor. The connection is obtained from an integrable connection of regular singular type by a confluence, and has irregular singularities along \(x=\infty\) and \(y=\infty\). Solutions are expressed by a double integral of Euler type with resonances among the exponents in the integrand. We specify twisted cycles that give main asymptotic behaviors in sectorial domains around \((\infty,\infty)\). Then we obtain linear relations among the twisted cycles, and get an explicit expression of the Stokes multiplier. The methods to derive the asymptotic behaviors for double integrals and to get linear relations among twisted cycles in resonant case, which we developed in this paper, seem to be new.

Efficient α-Dense Curve Strategies for Multiple Integrals over Hyper-rectangle Regions

Abstract In this paper, we propose an approximation technique to compute multiple integrals of a non-negative real continuous function over a hyper-rectangle Ω of ℝ n . The main idea is to use a reducing transformation procedure obtained by using α-dense curves. First, the region Ω f whose measure represents the value of the integral, is densified by a specific curve ℓ α(t) of finite length. Therefore, the multiple integral can be approached by a simple integral corresponding to ℓ α (t). Some numerical examples are given.

Generalized Gibbons-Werner method for stationary spacetimes

The Gibbons-Werner (GW) method is a powerful approach in studying the gravitational deflection of particles moving in curved spacetimes. The application of the Gauss-Bonnet theorem (GBT) to integral regions constructed in a two-dimensional manifold enables the deflection angle to be expressed and calculated from the perspective of geometry. However, different techniques are required for different scenarios in the practical implementation which leads to different GW methods. For the GW method for stationary axially symmetric (SAS) spacetimes, we identify two problems: (a) the integral region is generally infinite, which is ill-defined for some asymptotically nonflat spacetimes whose metric possesses singular behavior, and (b) the intricate double and single integrals bring about complicated calculation, especially for highly accurate results and complex spacetimes. To address these issues, a generalized GW method is proposed in which the infinite region is replaced by a flexible region to avoid the singularity, and a simplified formula involving only a single integral of a simple integrand is derived by discovering a significant relationship between the integrals in conventional methods. Our method provides a comprehensive framework for describing the GW method for various scenarios. Additionally, the generalized GW method and simplified calculation formula are applied to three different kinds of spacetimes — Kerr spacetime, Kerr-like black hole in bumblebee gravity, and rotating solution in conformal Weyl gravity. The first two cases have been previously computed by other researchers, affirming the effectiveness and superiority of our approach. Remarkably, the third case is newly examined, yielding a innovative result for the first time.

Relaxed Stability Criteria for Delayed Memristor-Based Neural Network Systems via a Novel Matrix-Separation Legendre Inequality.

This article studies the issue of stability in memristor-based neural network (MNN) systems with time-varying delays. First, a novel matrix-separation Legendre inequality is proposed to achieve a tight hierarchical bound on augmented-type integral terms. To derive implementable inequality conditions, several delay-dependent matrices are introduced to eliminate the reciprocal terms associated with time-varying delay. Furthermore, a new Lyapunov-Krasovskii (L-K) functional is proposed by incorporating augmented-type double integrals and delay-product terms. A series of free-weighting matrices are introduced into the L-K functional, leveraging the zero-sum equations and the S-procedure pertaining to both the delay and its derivative. Based on the proposed matrix-separation Legendre inequality and L-K functional, the derived stability conditions exhibit reduced conservatism, as validated by three numerical cases and simulation results.

Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets

We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of logarithmic or algebraic type. In a recent paper (Gibbs et al. Numer. Algorithms 92, 2071–2124 2023), it was shown that when the fractal set is “disjoint” in a certain sense (an example being the Cantor set), the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular integrals, which can be readily approximated numerically. In this paper, we present a methodology for extending these results to cases where the fractal is non-disjoint but non-overlapping (in the sense that the open set condition holds). Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake.