Integral Function Research Articles

Generalized integrals and point interactions

First we recall a method of computing scalar products of eigenfunctions of a Sturm-Liouville operator. This method is then applied to Macdonald and Gegenbauer functions, which are eigenfunctions of the Bessel, resp. Gegenbauer operators. The computed scalar products are well defined only for a limited range of parameters. To extend the obtained formulas to a much larger range of parameters, we introduce the concept of a generalized integral. The (standard as well as generalized) integrals of Macdonald and Gegenbauer functions have important applications to operator theory. Macdonald functions can be used to express the integral kernels of the resolvent (Green functions) of the Laplacian on the Euclidean space in any dimension. Similarly, Gegenbauer functions appear in Green functions of the Laplacian on the sphere and the hyperbolic space. In dimensions 1,2,3 one can perturb these Laplacians with a point potential, obtaining a well defined self-adjoint operator. Standard integrals of Macdonald and Gegenbauer functions appear in the formulas for the corresponding Green functions. In higher dimensions the Laplacian perturbed by point potentials does not exist. However, the corresponding Green function can be generalized to any dimension by using generalized integrals.

Adaptive Decentralized Control for Interconnected Time-Delay Uncertain Nonlinear Systems With Different Unknown Control Directions and Deferred Full-State Constraints.

In this article, the problem of adaptive decentralized control is investigated for a class of interconnected time-delay uncertain nonlinear systems with different unknown control directions and deferred asymmetric time-varying (DTV) full-state constraints. By constructing the novel time-varying asymmetric integral barrier Lyapunov function (TVAIBLF), the conservative limitation of constant integral barrier Lyapunov function (IBLF) or symmetric IBLF is reduced and the need on the prior knowledge of control gains is also avoided, while the deferred constraints directly imposed on the states of system are achieved by introducing the shifting function into the controller design. Furthermore, based on the Nussbaum-type functions, a new adaptive decentralized control strategy for interconnected time-delay nonlinear systems with subsystems having different control directions is proposed via backstepping method. And it is proven that the proposed control method can guarantee that all signals in closed-loop system are bounded and the transform errors asymptotically converge to zero. Finally, the effectiveness of the proposed control strategy is illustrated through the simulation results.

Adaptive Learning Control Algorithms for Infinite-Duration Tracking.

Learning control is applicable to systems that operate periodically or over finite time intervals. Currently, there is a lack of research results about learning control approaches to infinite-duration tracking, without requiring periodicity or repeatability. This article addresses the problem of adaptive learning control (ALC) for systems performing infinite-duration tasks. Instead of using integral adaptation, incremental adaptive mechanisms are exploited, by which the numerical integration for implementation can be avoided. The comparison with the conventional integral adaptive mechanisms indicates that the suggested methodology can be an alternative to the adaptive system designs. Using an error-tracking approach, the approximation-based backstepping design is carried out for systems in the strict-feedback form, where a novel integral Lyapunov function is shown to be efficient in the treatment of state-dependent control gain. Theoretical results for the performance analysis are presented in detail. In particular, the robust convergence of the tracking error is established, while the boundedness of the variables of the closed-loop system is characterized, with the aid of a key technical lemma. It is shown that the proposed control method can provide satisfactory tracking performance and simplify the controller designs. Numerical results are presented to demonstrate effectiveness of the learning control schemes.

Determination of the Reliability of Urban Electric Transport Running Autonomously through Diagnostic Parameters

The urban transport network involves complex processes, operating 24 h a day and 365 days a year. The sustainable development of the urban transport network using electric buses and trolleybuses that run autonomously is an urgent task since the transport network performs integral social functions and is the transport artery of any urban center. The social and economic life of a city as a whole depends on the reliability of the transportation network. A theory is proposed for the technical and economic evaluation of reliability improvement in electric buses and trolleybuses running autonomously, which enables the determination of the reliability parameters of electric buses and forecasts for the future from the point of view of optimal economic costs for the operation of electric equipment in electric buses. As a result of the application of the proposed theory, it was found that increasing the reliability of the transportation fleet can lead to a decrease in both specific operating costs and capital investments in the development of the fleet. This is achieved as a result of increasing the annual productivity of vehicles by reducing the time they are out of service to eliminate the consequences of failures and carry out maintenance and repair. The conducted experiments confirmed that the theory and methodology of optimal reliability level selection not only enable the rational use of the material resources of the urban transport network but also the release of funds for its scientific and technical development by reducing the number of failures in the electrical equipment of transport systems by 14%.

Adaptive Event-Triggered Fuzzy Control of State-Constrained Stochastic Nonlinear Systems Using IBLFs

In this article, an adaptive tracking control problem is addressed for nonstrict-feedback stochastic nonlinear systems subject to state constraints. Fuzzy logic systems (FLSs) are used to model unknown nonlinearities and avoid the algebraic loop arising from the system structure. Appropriate integral Barrier Lyapunov functions (IBLFs) are chosen so that time-varying full state constraints can be guaranteed directly rather than by transforming the constraint object. In the framework of backstepping technology, the relative threshold strategy is introduced to modify the adaptive control scheme so that the controller can be updated only after the trigger condition has been met, which reduces the update frequency of the controller and the loss of the actuator. Combined with Lyapunov stability theory, it is shown that all closed-loop signals are bounded in probability, in which the states remain within the specified constraints, and there is no Zeno behavior. A series of simulation results are given to reveal the effectiveness of the constructed control scheme.

Novel Definitions of α-Fractional Integral and Derivative of the Functions

An α-fractional integral and derivative of real function have been introduced in new definitions and then, they compared with the existing definitions. According to the properties of these definitions, the formulas demonstrate that they are most significant and suitable in fractional integrals and derivatives. The definitions of α-fractional derivative and integral coincide with the existing definitions for the polynomials for 0 ≤ α < 1. Furthermore, if α = 1, the proposed definitions and the usual definition of integer derivative and integral are identical. Some of the properties of the new definitions are discussed and proved, as well, we have introduced some applications in the α- fractional derivatives and integrals. Moreover, α-power series and α–rule of integration by parts have been proposed and implemented in this study.

Drone audition: improved Gaussian mixture model Wiener filtering approach for audio signal enhancement

Audio signal enhancement is an integral function of drone audition systems which emerges intelligent audio applications for military and civil environments. Signal enhancement using a drone is challenging due to its emission of significant noise from drone motors and propellers resulting in an extremely low signal-to-drone noise ratio level. A method for signal enhancement from an on-board microphone array on a drone for static flight conditions in adverse noisy environments is proposed. In this paper, we present an improved Gaussian mixture model Wiener filter approach for a noisy drone platform in the presence of interfering noise sources such as wind, and traffic. The key to this method is to exploit the spatial properties of the noise sources by using a set of Gaussian components to derive an accurate estimate of the power spectral densities of both residual drone noise and interfering noise sources to obtain the enhanced target sound source signal. We demonstrate the method's general applicability for both speech, and bird calls using an extensive experimental study, including an experimental recording from a drone acoustics dataset using measured impulse responses, and an outdoor measurement from a hovering drone for a bioacoustic application, respectively.

Short-packet communication for MIMO NOMA relay networks: BLER and AoI analysis

This work considers a multiple-input multiple-output (MIMO) non-orthogonal multiple access (NOMA) dual-hop relay system in which the size of signal packets is finite and the propagation environment follows the Nakagami-m distribution. Based on an approximation of the Gaussian Q-function, Riemann integral, and incomplete Gamma function, we derive the closed-form expressions of the block error rate (BLER), age of information (AoI), and latency of the proposed system in both exact and asymptotic cases. On the other hand, the automatic repeat request (ARQ) protocol and acknowledgment (ACK) are applied to confirm the accuracy of the receiving data packet process. Moreover, the maximal-ratio transmission (MRT), and maximal-ratio combining (MRC) are used to improve the BLER, AoI, and latency of the proposed system. The accuracy of mathematical analysis frameworks is confirmed by simulation results. The calculation results show that the considered system satisfies the requirement of ultra-reliable and low-latency communications (URLLC). The results also represent that the MIMO-NOMA system outperforms the MIMO-OMA system in terms of BLER performance. The results also provide insight into the optimizations of the power allocation coefficient and the number of transmission bits that minimize the BLER and AoI of the considered system.

Alley greening: a tool for enhancing community resilience?

ABSTRACT In many cities across the world alleys are transitioning from residual spaces to hybrid places providing the foundation for new functions, uses, and identities to take root and coincide through a process of “alley greening”. Such manifestations are transforming the relationship between people (local residents) and place (alleyway – local area), most notably during the COVID-19 pandemic when a new urgency for the provision, or repurposing, of safe, social spaces emerged. Yet, the potential of alley greening to affect people-place relationships and engender community resilience has been relatively unexplored. Adopting a mixed-methods approach, including questionnaires, interviews, and case study analyses, this paper critically investigates the experience and perspectives of green alleys from various place-based actors in Belfast, Northern Ireland. The findings reveal that, even in the absence of institutional and policy support, green alley projects have the potential to stimulate positive people-place relationships in various ways and enhance wider community resilience to shocks and stresses. However, barriers prevail, curtailing the reach and purpose of such projects both in Belfast and elsewhere. The paper considers how governance arrangements might best overcome such hurdles and strengthen pro-environmental and pro-social behaviours that are fundamental to community resilience. Key policy highlights Despite their integral form and function in the city, alleyways, nor their potential, are rarely recognised in the policy. The COVID-19 pandemic exposed a policy-implementation gap in the provision of locally accessible greenspace. Policy inertia exacerbated this gap preventing the fulfilment of changing community needs. Alley greening emerged as a tactical urban response. A lack of place-based approaches within policy has polarised institutions from communities. People-place relationships, essential to resilience-building and green alley longevity, are subsequently inadequately engaged with and fostered. An opportunity exists for alley greening to be a place-based policy instrument to stimulate pro-social and pro-environmental behaviours for building community resilience.

The Primacy of National Policies in Navigating the Local-Global Nexus: A Case Study of the Plastic Crisis

Against the backdrop of the escalating plastic crisis, exacerbated by human activities and posing a significant public concern, this study seeks to evaluate the role of various policy-making levels in environmental intervention. The core subject of investigation is the integral function of national lawmakers in spearheading effective environmental policy strategies while contextualizing the coordination of local and international policymakers. Using practical policy examples as primary tools for analysis, the study delves into the dynamics between the three tiers of policy-making and gleans evidence underlining the centrality of national players. Results indicate that while local and international policies offer solutions, they are often impeded by inherent policy limitations. National policies, powered by vast resources and political clout, emerge as pivotal, capable of overhauling plastic production, consumption, and disposal mechanisms. The research concludes that to holistically address the plastic crisis spawned by market failures, a blend of governmental intervention, primarily driven by national policymakers, and complemented by local and international cooperation, is imperative. Additionally, the future necessitates not only a reduction in plastic use but also the pursuit of viable plastic alternatives to combat pollution comprehensively.

Midpoint-type inequalities via twice-differentiable functions on tempered fractional integrals

In this paper, we obtain an equality involving tempered fractional integrals for twice-differentiable functions. By using this equality, we establish several left Hermite–Hadamard-type inequalities for the case of tempered fractional integrals. Moreover, we derive our results by using special cases of obtained theorems.

Master integrals for four-loop massless form factors

We present analytical results for all master integrals for massless three-point functions, with one off-shell leg, at four loops. Our solutions were obtained using differential equations and direct integration techniques. We review the methods and provide additional details.

New definitions of fractional derivatives and integrals for complex analytic functions

In this paper, we introduce a ground-breaking approach to defining fractional calculus for a selected class of analytic functions. Our new definitions, based on a novel and intuitive understanding of fractional derivatives and integrals, offer improved mathematical tractability for a variety of applications, including physics, engineering and finance. Our approach significantly simplifies the complexity of mathematical functions compared to the traditional Riemann-Liouville approach, by using simple functions rather than special functions, while preserving the intrinsic sense of fractional calculus. This article not only presents our proposed definitions but also provides a thorough analysis of their properties and advantages. The conclusion of this paper discusses the potential for future research in the field of fractional calculus.

Model-free adaptive dynamic event-triggered robust control for unknown nonlinear systems using iterative neural dynamic programming

This paper proposes an adaptive dynamic event-triggered (ADET) robust control method for unknown nonlinear systems using iterative neural dynamic programming (INDP). Firstly, the ADET robust control problem is transformed into an ADET optimal control problem by introducing an infinite domain integral function. Then, dynamic variables and adaptive thresholds are designed for discrete-time systems with auxiliary variables. The proposed ADET method reduces computation and transmission costs compared to existing static and dynamic event triggering methods. INDP is utilized to learn the optimal solution of the ADET Hamilton-Jacobi-Bellman equation within the heuristic dynamic programming (HDP) framework, providing system stability and algorithm convergence. The INDP algorithm employs model, action, and critic neural networks and includes a method to directly minimize the iterative cost function in the back-propagation process. The neural network implementation of the INDP algorithm is detailed. Simulation results demonstrate that the proposed ADET method based on INDP achieves faster convergence with fewer transmitted data and control updates.

Inequalities involving general fractional integrals of p-convex functions

Inequalities involving general fractional integrals of p-convex functions

A novel method for calculating Bose–Einstein correlation functions with Coulomb final-state interaction

Measurements of Bose–Einstein correlations played a crucial role in the discovery and the subsequent detailed exploration of the Quark-Gluon-Plasma (QGP) created in high-energy collisions of heavy nuclei. Such measurements gave rise to femtoscopy, a flourishing sub-field of high-energy physics, and there are important new directions to explore and discoveries to be made in the near future. One of these important current topics is the precise investigation of the shape of the correlation functions utilizing Lévy-stable sources. In this paper, we utilize Lebesgue’s dominated convergence theorem and Fubini’s theorem to present a novel method of calculating the shape of the two-particle correlation functions, including the Coulomb final-state interaction. This method relies on an exact calculation of a large part of the necessary integrals of the Coulomb wave function and can be utilized to calculate the correlation function for any source function with an easily accessible Fourier transform. In this way, it is eminently applicable to Lévy-stable source functions. In this particular case, we find that the new method is more robust and allows the investigation of a wider parameter range than the previously utilized techniques. We present an easily applicable software package that is ready to use in experimental studies.

Formulas for Computing Euler-Type Integrals and Their Application to the Problem of Constructing a Conformal Mapping of Polygons

This paper deals with Euler-type integrals and the closely related Lauricella function , which is a hypergeometric function of many complex variables. For new analytic continuation formulas are found that represent it in the form of Horn hypergeometric series exponentially converging in corresponding subdomains of, including near hyperplanes of the form, . The continuation formulas and identities for found in this paper make up an effective apparatus for computing this function and Euler-type integrals expressed in terms of it in the entire complex space , including complicated cases when the variables form one or several groups of closely spaced neighbors. The results are used to compute parameters of the Schwarz–Christoffel integral in the case of crowding and to construct conformal mappings of polygons.

Analytical Inverse Preisach Model and Its Comparison With Inverse Jiles–Atherton Model in Terms of Accuracy and Computational Speed

Inverse hysteresis model with magnetic flux density and magnetic field intensity as input and output respectively is more preferred, compared with the forward one, for resolving the electromagnetic issues of electrical equipment in terms of magnetic vector potential or with voltage sources. And an inverse hysteresis model with high accuracy and high computation speed both has broader range of applications. However, the well known classical Preisach hysteresis model is formulated with the forward form, and has the double integration of its distribution function, making it inconvenient and unpractical in the electrical engineering. In this paper, an analytical inverse Preisach model is proposed for the first time. Firstly, the analytical expressions of permeability of initial magnetization curve, descending and ascending branches of the hysteresis loop are derived using one closed form of Everett integral function. Subsequently the analytical inverse Preisach model is derived by the difference method, considering the reversible magnetization component that classical Preisach model cannot simulate with an odd function analytically. The experiment of one grain oriented silicon steel sample and one non-oriented silicon steel sample, under different magnetic excitation levels is conducted, and the accuracy and efficiency of the proposed analytical inverse Preisach model are confirmed by comparing its simulated hysteresis loops with the experimental ones and that computed with the widely used inverse Jiles-Atherton hysteresis model.

ИНКРЕМЕНТАЛЬНЫЕ МЕТОДЫ РЕШЕНИЯ ГЕОМЕТРИЧЕСКИ НЕЛИНЕЙНЫХ ЗАДАЧ

In the sequence of displacement – deformation – stress, two types of nonlinear dependence are possible: geometric and physical. Physically related, mainly with the plastic properties of the material. Geometric nonlinearity is due to a wider range of reasons. Both nonlinearities create a nonlinear displacement-load relationship, which, in most cases, can be expressed by some integral function. In this case, the solution can be obtained by incremental (step) methods. Two incremental methods are considered: the tangential stiffness method and the calculation method using a deformed scheme. The methods are applied to three problems of geometric nonlinearity: 1) small deformations with large displacements, 2) displacements and deformations occur in different directions, 3) large deformations with small displacements. The studies considered both uniaxial stress-strain bars and bars modeled by volumetric elements. Conclusions are drawn about the universality and accuracy of the methods considered.

Strong Differential Subordinations and Superordinations for Riemann–Liouville Fractional Integral of Extended q-Hypergeometric Function

The notions of strong differential subordination and its dual, strong differential superordination, have been introduced as extensions of the classical differential subordination and superordination concepts, respectively. The dual theories have developed nicely, and important results have been obtained involving different types of operators and certain hypergeometric functions. In this paper, quantum calculus and fractional calculus aspects are added to the study. The well-known q-hypergeometric function is given a form extended to fit the study concerning previously introduced classes of functions specific to strong differential subordination and superordination theories. Riemann–Liouville fractional integral of extended q-hypergeometric function is defined here, and it is involved in the investigation of strong differential subordinations and superordinations. The best dominants and the best subordinants are provided in the theorems that are proved for the strong differential subordinations and superordinations, respectively. For particular functions considered due to their remarkable geometric properties as best dominant or best subordinant, interesting corollaries are stated. The study is concluded by connecting the results obtained using the dual theories through sandwich-type theorems and corollaries.