Taking Limits under the Integral Sign

1967 ◽  
Vol 40 (4) ◽  
pp. 179 ◽  
Author(s):  
F. Cunningham,
Keyword(s):  
Integral Sign

LIMITS UNDER THE INTEGRAL SIGN

1989 ◽  
Vol 15 (1) ◽  
pp. 62
Author(s):  
Henstock
Keyword(s):  
Integral Sign

Differentiation under the integral sign

2007 ◽  
pp. 157-158
Keyword(s):  
Integral Sign

RISE-backstepping-based robust control design for induction motor drives

2017 ◽  
Vol 36 (4) ◽  
pp. 906-937 ◽  
Author(s):  
Yosra Rkhissi-Kammoun ◽  
Jawhar Ghommam ◽  
Moussa Boukhnifer ◽  
Faiçal Mnif
Keyword(s):  
Induction Motor ◽  
Control Design ◽  
Good Control ◽  
Motor Drives ◽  
Continuous Control ◽  
Backstepping Design ◽  
Integral Sign ◽  
Reference Tracking ◽  
Content Type ◽  
Robust Control Design

Purpose This paper aims to address the speed and flux tracking problem of an induction motor (IM) drive that propels an electric vehicle (EV). A new continuous control law is developed for an IM drive by using the backstepping design associated with the Robust Integral Sign of the Error (RISE) technique. Design/methodology/approach First, the rotor field-oriented IM dynamic model is derived. Then, a RISE-backstepping approach is proposed to compensate for the load torque disturbance under the assumptions that the disturbances are C2 class functions with bounded time derivatives. Findings The numerical validation results have presented good control performances in terms of speed and flux reference tracking. It is also robust against load disturbances rejection and IM parameters variation compared to the conventional Field-Oriented Control design. Besides, the asymptotic stability and the boundedness of the closed-loop signals is guaranteed in the context of Lyapunov. Originality/value A very relevant strategy based on a conjunction of the backstepping design with the RISE technique is proposed for an IM drive. The approach remains simple and can be scaled to different applications.

scholarly journals An Integral Sign Conductor in the Frontal Himalaya Region and Its Tectonic Interpretation

1997 ◽  
Vol 49 (11) ◽  
pp. 1649-1658 ◽  
Author(s):  
C. D. Reddy
Keyword(s):  
Integral Sign ◽  
Tectonic Interpretation

scholarly journals Using Mathematical Equations to Communicate and Think About Karma

2021 ◽  
Vol 11 (1) ◽  
pp. 300-317
Author(s):  
Kien Lim ◽  
Christopher Yakes
Keyword(s):  
External Reality ◽  
English Sentence ◽  
Integral Sign ◽  
Literal Translation ◽  
Mathematical Symbols ◽  
Mathematical Equations

Two equations are presented in this article to communicate a particular understanding of karma. The first equation relates future experiences to past and present actions. Although the equation uses variables and mathematical symbols such as the integral sign and summation symbol, it reads more like a literal translation of an English sentence. Based on the key idea in the first equation, a second equation is then created to highlight the viability of using math to communicate concepts that are not readily quantifiable. Analyzing such equations can stimulate thinking, enhance understanding of spiritual concepts, raise issues, and uncover tensions between our ordinary conceptions of external reality and transcendental aspects of spirituality.

scholarly journals Gauss-Simpson Quadrature Algorithm for Calcaulting Additional Stress in Foundation Soils

2021 ◽  
Author(s):  
Shuai WANG ◽  
Chao WANG ◽  
Chenliang XING
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Elasticity Theory ◽  
Quadrature Formula ◽  
Gaussian Quadrature ◽  
Additional Stress ◽  
The Finite Element Method ◽  
Integral Sign ◽  
Gaussian Quadrature Formula ◽  
Element Method

Abstract The additional pressure at the bottom of a building’s foundation produces an additional stress in the foundation soils under the building’s foundation. In order to overcome the limitations of traditional elastic theory methods and the finite element method when calculating the additional stress in foundation soils, we use the Gauss-Simpson formula to derive the Gauss-Simpson Quadrature Algorithm based on the elasticity theory. The Gauss-Simpson Quadrature Algorithm is a method designed to calculate the additional stress in foundation soils under an irregularly shaped foundation and an irregular load distribution. This new method is based on the fact that the Gaussian quadrature formula and the Simpson formula are independent of the specific type of integrand. The finite element method with n interpolation points can only achieve an algebraic accuracy of n. The interpolation points of the Gaussian quadrature formula are n zeros of orthogonal polynomials, which can achieve an algebraic accuracy of 2n + 1. Moreover, the weights of the nodes in the quadrature formula are all positive, and thus, it has a high numerical stability. In the proposed method, the Simpson formula is necessary. The Simpson formula is used to transform the implicit additional stress formula with the integral sign into an explicit cumulative integral, which can be considered similar to the rectangular domain case to obtain an explicit analytical algebraic formula for solving the additional stress approximation. In engineering applications, we only need to provide the field engineers with the locations of the interpolation points of the Gauss-Legendre formula, the interpolated weight coefficients, and the specific type of Simpson's formula, and then, the results of the additional stress can be calculated manually, which is nearly impossible using the traditional methods and finite element methods. From the point of view of academic rigor and theoretical completeness, it is possible to use the compound Gauss-Simpson Quadrature Algorithm in conjunction with the looping function in computer programs. Under standard conditions, the proposed Gauss-Simpson Quadrature Algorithm is in good agreement with the results of the traditional elasticity theory.

scholarly journals Difference methods for solving nonlocal boundary value problems for fractional-order differential convection-diffusion equations with memory effect

2021 ◽  
Vol 21 (1) ◽  
pp. 3-25
Author(s):  
Murat Beshtokov ◽  
◽  
M. Z. KHudalov ◽  
Keyword(s):  
Boundary Value Problems ◽  
Memory Effect ◽  
Fractional Order ◽  
A Priori Estimates ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Uniform Grid ◽  
Integral Sign ◽  
Nonlocal Boundary Value Problems ◽  
Convection Diffusion

In the present paper, in a rectangular domain, we study nonlocal boundary value problems for one-dimensional in space differential equations of convection-diffusion of fractional order with a memory effect, in which the unknown function appears in the differential expression and at the same time appears under the integral sign. The emergence of the integral term in the equation is associated with the need to take into account the dependence of the instantaneous values of the characteristics of the described object on their respective previous values, i.e. the effect of its prehistory on the current state of the system. For the numerical solution of nonlocal boundary value problems, two-layer monotone difference schemes are constructed that approximate these problems on a uniform grid. Estimates of solutions of problems in differential and difference interpretations are derived by the method of energy inequalities. The obtained a priori estimates imply the uniqueness, as well as the continuous and uniform dependence of the solution on the input data of the problems under consideration and, due to the linearity of the problem under consideration, the convergence of the solution of the difference problem to the solution of the corresponding differential problem with the rate $O(h^2+\tau^2)$.

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