The Henstock–Kurzweil integral, which is also known as the generalized Riemann integral, arose from a slight modification of the classical Riemann integral more than 50 years ago. This relatively new ...integral is known to be equivalent to the classical Perron integral; in particular, it includes the powerful Lebesgue integral. This book presents an introduction of the multiple Henstock–Kurzweil integral. Along with the classical results, this book contains some recent developments connected with measures, multiple integration by parts, and multiple Fourier series. The book can be understood with a prerequisite of advanced calculus.
•We make a revision of recent generalizations of the Choquet integral that appear in the literature.•We show some of the most relevant theoretical features of these extensions.•We also discuss some ...applications where these extensions have provided good results.
In 2013, Barrenechea et al. used the Choquet integral as an aggregation function in the fuzzy reasoning method (FRM) of fuzzy rule-based classification systems. After that, starting from 2016, new aggregation-like functions generalizing the Choquet integral have appeared in the literature, in particular in the works by Lucca et al. Those generalizations of the Choquet integral, namely CT-integrals (by t-norm T), CF-integrals (by a fusion function F satisfying some specific properties), CC-integrals (by a copula C), CF1F2-integrals (by a pair of fusion functions (F1, F2) under some specific constraints) and their generalization gCF1F2-integrals, achieved excellent results in classification problems. The works by Lucca et al. showed that the aggregation task in a FRM may be performed by either aggregation, pre-aggregation or just ordered directional monotonic functions satisfying some boundary conditions, that is, it is not necessary to have an aggregation function to obtain competitive results in classification. The aim of this paper is to present and discuss such generalizations of the Choquet integral, offering a general panorama of the state of the art, showing the relations and intersections among such five classes of generalizations. First, we present them from a theoretical point of view. Then, we also summarize some applications found in the literature.
To derive a less conservative stability criterion via Lyapunov-Krasovskii functional (LKF) method, in previous literature, multiple integral terms are usually introduced into the construction of ...LKFs. This article generalizes the results of previous literature by proposing a polynomial-type LKF, which contains the LKFs with multiple integral terms as special cases. In addition, a Jacobi-Bessel inequality is presented to bound the derivative of such LKF. As a result, an improved stability criterion of time-delay systems is established. Finally, two numerical examples are given to illustrate the effectiveness, and advantages of our method.
In this paper, an improvement is made to an efficient direct method for numerical evaluation of high order singular curved boundary integrals. Then this improved singular integral evaluation method ...is employed to solve the strongly and hypersingular integrals involved in the dual boundary element method (Dual BEM), which combine the use of displacement and traction boundary integral equations to solve crack problems in a single domain formulation. The singular integral evaluation method is carried out based on a parameter plane expansion and radial integral approach, this paper proposed a new strategy for treating the singular radial integral, which plays a vital role in this method. In isoparametric coordinate system, the singular curved boundary integral is mapped into a singular square plane integral in intrinsic coordinates, then the radial integration method (RIM) is employed to transform the singular square plane integral into a regular line integral over the contour of intrinsic square plane and a singular radial integral over the path from source point to the contour of intrinsic square plane. A singularity isolation technique is utilized to divide the singular radial integral into two parts, the regular radial integral can be evaluated normally using Gauss quadrature and the singular radial that can be evaluated analytically by expanding the non-singular part of the integrand function into a power series. Compared with conventional local interpolation approach to deal with the singular radial integral, the newly proposed method has a more rigorous mathematical derivation, and can achieve more stable and precise results. Based on the successful implementation of direct evaluation of singular boundary integrals, Dual BEM is successfully applied to solve two- and three-dimensional elastic crack problems including straight and curved crack paths with continuous or discontinuous elements. Two different approaches, geometrical extrapolation method and J-integral method are used in the evaluation of stress intensity factors. Several numerical examples are given to validate effectiveness of the presented method.
•Based on singularity isolation, an efficient 2D and 3D singular curved boundary integral evaluation method is proposed.•Continuous elements are used in the discretization of straight or curved crack surfaces.
In the paper, the authors establish some new inequalities of the Grüss type for conformable fractional integrals. These inequalities generalize some known results.
In 2015, Abdeljawad 1 has put an open problem, which is stated as: “Is it hard to fractionalize the conformable fractional calculus, either by iterating the conformable fractional derivative ...(Grunwald–Letnikov approach) or by iterating the conformable fractional integral of order 0<α≤1 (Riemann approach)?. Notice that when α=0 we obtain Hadamard type fractional integrals”. In this article we claim that yes it is possible to iterate the conformable fractional integral of order 0<α≤1 (Riemann approach), such that when α=0 we obtain Hadamard fractional integrals. First of all we prove Cauchy integral formula for repeated conformable fractional integral and proceed to define new generalized conformable fractional integral and derivative operators (left and right sided). We also prove some basic properties which are satisfied by these operators. These operators (integral and derivative) are the generalizations of Katugampola operators, Riemann–Liouville fractional operators, Hadamard fractional operators. We apply our results to a simple function. Also we consider a nonlinear fractional differential equation using this new formulation. We show that this equation is equivalent to a Volterra integral equation and demonstrate the existence and uniqueness of solution to the nonlinear problem. At the end, we give conclusion and point out an open problem.
The control of a nuclear power plant is important due to its nonlinear dynamics. An optimized proportional–integral–derivative (PID) controller is a popular tool to this end. PID is tuned by various ...methods, including a genetic algorithm (GA). The quality and speed of GA optimization depend on the objective function as the performance index. In this paper, integral of absolute error (IAE), integral of the square error (ISE), integral of time-absolute error (ITAE), and integral of time-square error (ITSE) objective functions are evaluated. The results comparison is done based on overshoot/undershoot and the number of function evaluations (NFE) criteria. The PID gains are tuned to control of a typical pressurized water reactor (PWR), based on the point-kinetics model. The results show the high control performance compared to an empirically tuned PID. Also, ISE and ITSE are better performance indexes than IAE and ITAE in steady-states as critical situations. Also, these performance indexes have signified a good robustness against dynamics parametric uncertainties compared to an empirically tuned PID in every power mode. This comparative approach provides a good idea for experts to choose an appropriate performance index.
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•PID controller is used to a PWR-type nuclear reactor power control, based on the point-kinetics model.•GA is used to tune PID gains, based on the different performance indexes.•IAE, ISE, ITAE, and ITSE objective functions are used as the performance index.•Performance index quality and speed are compared in the number of function evaluations (NFE).
Three dimensional (3-D) imaging and display have been subjects of much research due to their diverse benefits and applications. However, due to the necessity to capture, record, process, and display ...an enormous amount of optical data for producing high-quality 3-D images, the developed 3-D imaging techniques were forced to compromise their performances (e.g., gave up the continuous parallax, restricting to a fixed viewing point) or to use special devices and technology (such as coherent illuminations, special spectacles) which is inconvenient for most practical implementation. Today's rapid progress of digital capture and display technology opened the possibility to proceed toward noncompromising, easy-to-use 3-D imaging techniques. This technology progress prompted the revival of the integral imaging (II)technique based on a technique proposed almost one century ago. II is a type of multiview 3-D imaging system that uses an array of diffractive or refractive elements to capture the 3-D optical data. It has attracted great attention recently, since it produces autostereoscopic images without special illumination requirements. However, with a conventional II system it is not possible to produce 3-D images that have both high resolution, large depth-of-field, and large viewing angle. This paper provides an overview of the approaches and techniques developed during the last decade to overcome these limitations. By combining these techniques with upcoming technology it is to be expected that II-based 3-D imaging systems will reach practical applicability in various fields.
Some Hermite-Hadamard type inequalities for generalized
k
-fractional integrals (which are also named
(
k
,
s
)
-Riemann-Liouville fractional integrals) are obtained for a fractional integral, and an ...important identity is established. Also, by using the obtained identity, we get a Hermite-Hadamard type inequality.
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