Integral Function Research Articles

Different Variants of Bernstein Kantorovich Operators and Their Applications in Sciences and Engineering Field

In this article, we investigate various Bernstein-Kantorovich variants together with their approximation properties. Nowadays, these variants of Bernstein-Kantorovich operators have been a source of inspiration for researchers as it helps to approximate integral functions also which is not feasible in the case of discrete operators. Chaos theory has also been referred to as complexity theory. Using chaos theory complexity is also reduced as in approximation theory. Thus in order to reduce complexity and to have better understanding of images in sciences and engineering field, sampling Kantorovich operators of approximation theory are widely used in this regard for enhancement of images. Thus, we discuss the important applications of Kantorovich operators depicting pragmatic and theoretical aspects of approximation theory.

Improved dynamic sliding mode control for plate hoisting of cable crane under wind load

In order to quickly eliminate the swing angle of plate hoisting under wind load, an improved dynamic sliding mode control method is proposed in this paper. Firstly, the plate hoisting model under wind load is simplified into a double pendulum system, and the dynamic model of a double pendulum cable crane with wind disturbance is established. To enhance the coupling relationship between trolley displacement and double swing angle, a new coupling state vector for the double pendulum system is defined. On this basis, a dynamic sliding surface is constructed and then an integral function of the system’s discontinuous term is established, essentially ensuring the continuity of the sliding surface and reducing the system’s chattering characteristics. Finally, the stability of the system is rigorously proved by Lyapunov theorem and Barbalat lemma. The simulation and experimental results show that the controller proposed in this paper has a good effect on the anti-swing problem of plate hoisting under wind load.

Adaptive neural network control of non‐affine multi‐agent systems with actuator fault and input saturation

AbstractAn adaptive neural network fault‐tolerant control method is proposed for the non‐affine multi‐agent systems with actuator failure and input saturation. Through the transformation of the original multi‐agent systems, an equivalent model is proposed. Considering that the signal input of multi‐agent systems is limited, this article will solve the input saturation problem. But in a real system, the actuator would probably fail. Therefore, a fault tolerant control algorithm is proposed to solve the fault problem in the systems. In this article, an adaptive distributed controller is designed to eliminate the uncertainty of the systems. At the same time, the integral barrier Lyapunov function is selected to verify the stability. Finally, a simulation example is given to demonstrate the effectiveness of the proposed method.

The strong fuzzy variational Henstock multiple integral on n-dimensional fuzzy number space

In this paper, we define strongly fuzzy variational Henstock multiple integral and fuzzy Henstock integral by support functions on n-dimensional fuzzy number space. And we prove these two types of integrals are equivalent. After that, we propose the support-based derivative for strongly fuzzy variational Henstock integral functions and obtain the necessary and sufficient conditions for this differential. By the properties of ACGδ⁎⁎, we use the concepts of inner small variation and singular point covering to derive some characterization theorems of the primitives for strongly fuzzy variational Henstock integral. Finally, the Dominated Convergence Theorem for SFVH integral is obtained in the sense of UACGδ⁎⁎.

Discrete complementary exponential and sine integral functions

Abstract Discrete analogues of the sine integral and complementary exponential integral functions are investigated. Hypergeometric representation, power series, and Laplace transforms are derived for each. The difficulties in extending these definitions to other common trigonometric integral functions are discussed.

Convolution kernel determination problem in the third order Moore–Gibson–Thompson equation

This article is concerned with the study of the inverse problem of determining the difference kernel in a Volterra type integral term function in the third-order Moore–Gibson–Thompson (MGT) equation. First, the initial-boundary value problem is reduced to an equivalent problem. Using the Fourier spectral method, the equivalent problem is reduced to a system of integral equations. The existence and uniqueness of the solution to the integral equations are proved. The obtained solution to the integral equations of Volterra-type is also the unique solution to the equivalent problem. Based on the equivalence of the problems, the theorem of the existence and uniqueness of the classical solutions of the original inverse problem is proved.

DIAGNOSTIC STUDY: TRANSFORMING RESEARCH GOVERNANCE AT UNIVERSITAS JAMBI – A COMPARATIVE ANALYSIS WITH LPPM OF SEVERAL INDONESIAN UNIVERSITIES

The Tridharma of Higher Education, encompassing Education, Research, and Community Service, serves as the cornerstone of higher education's tripartite mission. These integral functions are interdependent and mutually reinforcing, positioning universities as pivotal institutions for fulfilling societal needs in terms of quality human resources, the generation of new knowledge, and providing solutions to a myriad of societal challenges. This study aims to engage in a comparative analysis with other higher education institutions involved in Research and Community Service (LPPM) - specifically USK, UB, UNPAD, and UNHAS - to enhance the performance and governance of UNJA's LPPM. The methodology employed in this research is an interview survey. The findings indicate that USK excels in supporting researchers and offering substantial incentives. UNPAD is noted for its robust foundational data, which forms a critical basis for decision-making. Additionally, UNPAD ensures efficient facilitation of research activities and publication. The standout feature of UB is its user-friendly system, greatly simplifying processes for stakeholders. Finally, UNHAS is commended for its clear direction in research and service execution, alongside well-defined annual targets. This comparative study is instrumental in identifying best practices and potential areas for improvement in UNJA's LPPM, thereby contributing to the broader goal of enhancing the effectiveness of higher education institutions in fulfilling their Tri Dharma. Keywords: Diagnostics, Comparison, LPPM Governance

Fixed-Time Stabilization of a Class of Stochastic Nonlinear Systems

This paper investigates an improved fixed-time stability theory together with a state feedback controller for a class of nonlinear stochastic systems. First, a delicate transformation is performed, and next, a Gamma function is utilized to directly derive the value of the integral function, which ultimately yields a fixed-time stabilization theorem with a higher precision upper bound for the settling time. Unlike the existing estimation process of amplifying twice, we only performed one amplification, which weakens the effect of amplification. Then, a state feedback controller is constructed for stochastic systems by the method of adding a power integrator. Utilizing the proposed stochastic fixed-time stability theory, simulations show that the intended controller ensures that the trivial solution of the suggested system is fixed-time stable in probability. The results of the simulation demonstrate that the suggested control scheme is meaningful.

The nonleptonic decays of b-flavored mesons to S-wave charmonium and charm meson states

The detection of radially excited charmonium and charm meson states, and measurement of heavy meson decays, particularly Bc+→J/ψDs+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c^+\\rightarrow J/\\psi D_s^+$$\\end{document} and Bc+→J/ψDs∗+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c^+\\rightarrow J/\\psi D_s^{*+}$$\\end{document}, by the LHCb and ATLAS Collaborations, have aroused a lot of theoretical interest in the nonleptonic decays of b-flavored mesons. In this paper, we study the exclusive two-body nonleptonic B¯0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\bar{B}}^0$$\\end{document}, Bs0¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{B_s^0}$$\\end{document}, B-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^-$$\\end{document} and Bc-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c^-$$\\end{document}-meson decays to two vector meson (V1(nS)V2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_1(nS)V_2$$\\end{document}) states. Assuming the factorization hypothesis, we calculate the weak-decay form factors from the overlapping integrals of meson wave functions, in the framework of the relativistic independent quark (RIQ) model. We find a few dominant decay modes: B-→D∗0ρ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^-\\rightarrow D^{*0}\\rho ^-$$\\end{document}, B0¯→D∗+ρ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{B^0}\\rightarrow D^{*+}\\rho ^-$$\\end{document}, Bs0¯→Ds∗+ρ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{B_s^0}\\rightarrow D_s^{*+}\\rho ^-$$\\end{document}, B-→J/ψK∗-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^-\\rightarrow J/\\psi K^{*-}$$\\end{document} and Bc-→J/ψDs∗-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c^-\\rightarrow J/\\psi D_s^{*-}$$\\end{document} with predicted branching fractions of 1.52, 1.40, 1.16, 0.46 and 0.27 (in %\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\%$$\\end{document}), which are experimentally accessible. The predicted branching fractions for corresponding decay modes to excited (2S) states, obtained in O(10-3-10-4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathscr {O}} }(10^{-3}-10^{-4})$$\\end{document} lie within the detection accuracy of the current experiments at LHCb and Tevatron. Our predicted CP-odd fractions (R⊥\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_\\perp $$\\end{document}) for color-favored Bc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_c$$\\end{document}-decays to two charmful states (D∗D(s)∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^*D^*_{(s)}$$\\end{document}) are found significant as compared to similar decay modes of other b-flavored mesons.

Integral barrier Lyapunov functions based sliding mode observer‐controller for nonlinear systems with performance constraints

AbstractThis article presents an observer‐controller structure for nonlinear systems with external disturbances, uncertainties and performance constraints. Firstly, considering the unmeasurable state issue, a sliding mode observer is designed to estimate the unknown state. Then an integral barrier Lyapunov function (iBLF) is established to ensure that the observer estimation errors are held within the performance constraints. With the objective of utilizing iBLF, an equivalent output injection technique is adopted. Combining with backstepping technique, an observer‐controller scheme is given. The controller can hold all signals in the closed‐loop system bounded with the help of iBLF, and the tracking error will converge to a bounded tight set without violating the corresponding constraints. The effectiveness of the proposed strategy is clarified via a vessel heading control example.

A New Integral Function Algorithm for Global Optimization and Its Application to the Data Clustering Problem

The filled function method is an approach to finding global minimum points of multidimensional unconstrained global optimization problems. The conventional parametric filled functions have computational weaknesses when they are employed in some benchmark optimization functions. This paper proposes a new integral function algorithm based on the auxiliary function approach. The proposed method can successfully be used to find the global minimum point of a function of several variables. Some testing global optimization problems have been used to show the ability of this recommended method. The integral function algorithm is then implemented to solve the center-based data clustering problem. The results show that the proposed algorithm can solve the problem successfully.

Copper overload induces apoptosis and impaired proliferation of T cell in zebrafish

Copper is an essential biometal for cell development and function, however, unbalanced copper homeostasis in T cell development and the underlying mechanisms are largely unexplored. Here, we use a zebrafish model to investigate the effect of copper overload in T cell development. We show that copper stressed zebrafish larvae exhibit a significant reduction in T cells with increased cell apoptosis and impaired cell proliferation. T cell progenitors, hematopoietic stem and progenitor cells, also exhibit increased cell apoptosis. Copper overload induces production of ROS and the down-regulations of its resistance genes foxos, and ectopic expression of foxo3a, ROS scavenger GSH, could both effectively rescue the reduction of T cells in copper overload larvae. Moreover, foxm1-cytoskeleton axis, parallel to ROS-foxo axis, also mediates the copper overload induced T cell developmental defects. Meanwhile, ROS destroys expression of cytoskeleton rather than of foxm1 in the cells to induce cell apoptosis and the impaired proliferation. The functional integrity of copper transporters cox17 and atp7b are required for copper stress in inducing T cell apoptosis and proliferation impairment. Our findings demonstrate that the down-stream ROS-foxo/cytoskeleton and foxm1-cytoskeleton signaling pathways contribute jointly to copper overload induced T cell apoptosis and proliferation defects, which are depend on the integral function of Cox17 and Atp7b, and provide new insight into the copper homeostasis in T lymphocyte development.

Characterization of the Mean First-Passage Time Function Subject to Advection in Annular-like Domains

Cell migration in a biological medium towards a blood vessel is modeled, as a random process, sucessively inside an annulus (two-dimensional domain) and an annular cylinder (three-dimensional domain). The conditional probability function u for the cell moving inside such domains (tissue) fulfills by assumption a diffusion–advection equation that is subject to a Dirichlet boundary condition on the outer boundary and a Robin boundary condition on the inner boundary. The mean first-passage time (MFPT) function determined by u estimates the average time for the travelling cell to reach various interesting targets. The MFPT function fulfills a Poisson equation inside a domain with suitable boundary conditions, which give rise to various mathematical problems. The main novelty of this study is the characterization of such an MFPT function inside an annulus and an annular cylinder, which is subject to a Robin boundary condition on the inner boundary and a Dirichlet boundary condition on the outer one, and these are integral functions whose densities are the solution of an inhomogeneous system of linear integral equations.

An Improved Fractional Moment Maximum Entropy Method With Polynomial Fitting

Abstract The moment method is commonly used in reliability analysis, in which the maximum entropy method (MEM) and polynomial fitting (PF) have been widely used due to their advantages in accuracy and efficiency, respectively. In this paper, we propose a novel reliability analysis method by combining MEM and PF. The probability density function is preliminarily estimated using the fractional moment maximum entropy method (FM-MEM), based on which PF is then used to further improve the accuracy. The proposed method can avoid the phenomenon of the negative probability density and function oscillations in PF effectively. Moreover, the order of the exponential polynomial in the FM-MEM is adaptively selected in the preliminary solution calculation process. An iterative process for the number of exponential polynomial terms is also proposed, using the integral of the moment error function and the integrals of the local and global negative probability density as the convergence criteria. Four numerical examples and one engineering example are tested, and the results are compared with those of the Monte Carlo simulation and the classical FM-MEM results, respectively, demonstrating the good performance of the proposed method.

On some sums involving the integral part function

Denote by [Formula: see text], [Formula: see text] and [Formula: see text] the number of representations of [Formula: see text] as a product of [Formula: see text] natural numbers, the number of distinct prime factors of [Formula: see text] and the characteristic function of the square-free integers, respectively. Let [Formula: see text] be the integral part of real number [Formula: see text]. For [Formula: see text], we prove that [Formula: see text] for [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is an arbitrarily small positive number. These improve the corresponding results of Bordellès.

Monotonicity and Convexity Properties and Some Inequalities Involving $E_{n,p}(x)$

In this paper, we established some monotonicity and convexity properties of the p-analogue of the exponential integral function. The increasing and decreasing, positive and negative, and convexity and concavity properties of the function were established and proved. Complete monotonicity of the function was also considered.

Proof and Application of Cauchy’s Residue Theorem

Complex analysis is a major subfield of mathematics that is concerned with investigating complex functions and their behaviors. The Cauchy’s residue theorem plays an important role in complex analysis. It is also the main focus for this paper. The residue theorem connects complex integrals of functions with their residues at singular points. It is widely used to find integrals of complex functions that may be insolvable otherwise. This paper first looks at the definition of Cauchy’s Residue Theorem. Next, it provides proof for the theorem from Laurent theorem and Cauchy-Goursat theorem. Finally, it looks at some applications of Cauchy’s Residue theorem, including the theorem’s use in evaluating integrals, in the proof of related theorems important to the field of complex analysis, and in its use in the formation of Fourier and Laplace transforms. Through these examples, the efficacy of the theorem, and the use of it in many areas, can be seen clearly.

The Relationship between the Box Dimension of Continuous Functions and Their (k,s)-Riemann–Liouville Fractional Integral

This article is a study on the (k,s)-Riemann–Liouville fractional integral, a generalization of the Riemann–Liouville fractional integral. Firstly, we introduce several properties of the extended integral of continuous functions. Furthermore, we make the estimation of the Box dimension of the graph of continuous functions after the extended integral. It is shown that the upper Box dimension of the (k,s)-Riemann–Liouville fractional integral for any continuous functions is no more than the upper Box dimension of the functions on the unit interval I=[0,1], which indicates that the upper Box dimension of the integrand f(x) will not be increased by the σ-order (k,s)-Riemann–Liouville fractional integral ksD−σf(x) where σ>0 on I. Additionally, we prove that the fractal dimension of ksD−σf(x) of one-dimensional continuous functions f(x) is still one.

Thermodynamics of antimony—selenium alloys formation and evaporation

The thermodynamic functions of alloy formation and evaporation were considered for two particular systems — Sb – Sb2Se3 and Sb2Se3 – Se in connection with the presence of congruently melting compound Sb2Se3 in the antimony—selenium system. The calculations are based on the partial vapor pressure values of the components forming the particular systems. The thermodynamic activity of antimony selenide and selenium as the most volatile components in the systems was calculated based on the saturated vapor pressure values of antimony selenide over the Sb – Sb2Se3 and selenium melts over Sb2Se3 – Se liquid alloys determined by the boiling point method (isothermal variant). Similar functions of the low volatile components in the above systems: Sb in the first system and Sb2Se3 in the latter one was calculated by numerical integration of the Gibbs—Duhem equation using the substitution proposed by Darken. The partial pressures of antimony selenide and antimony over Sb – Sb2Se3 and Sb2Se3 – Se melts were approximated by temperature—concentration relationships. The system is distinguished with a positive deviation from ideality due to the presence of a delamination region in the first system. The partial and integral entropies and enthalpies of the formation of liquid alloys were calculated based on the values of component activities found as the ratio of the partial vapor pressure of an element or compound above the solution to the saturated vapor pressure of a pure element or compound. The partial and integral functions of alloy formation are presented in the form of graphical dependences on the selenium amount in the melt. The obtained thermodynamic constants will replenish the physical and chemical data base and will be used to calculate the boundaries of the vapor— liquid equilibrium fields on the diagram of state, allowing to determine the possibility and completeness of distillation separation of molten systems.

INFORMATION CAPABILITIES OF THE TRANSITION FUNCTION OF THE HYDROGEN STORAGE AND SUPPLY SYSTEM GAS GENERATOR TO ASSESS ITS FIRE HAZARD LEVEL

To estimate the level of fire hazard of hydrogen storage and supply systems, it is advisable to use reliability indicators, in particular, of their core element, which is a gas generator. Such indicators include indicators of amplitude and phase reliability of the gas generator, which are determined through the Laplace function. The arguments of this function are the relative errors of the parameters of the gas generator, which are determined by its transition function. Three options for determining the parameters of the gas generator have been developed, the feature of which is obtaining information about its parameters at two a priori given moments. The magnitudes of these moments do not exceed the time of the transient process. In the a priori specified moments, the values of the transient fraction are measured (option 1), and the values of its derivative (option 2) or the value of the integral of the transient function (option 3) are additionally measured. Algorithms for processing the received information have been developed for all options, the implementation of which ensures the determination of gas generator parameters. In terms of its structure, the second option for determining gas generator parameters is the simplest, which has 1.5 times fewer arithmetic operations compared to the first option. In addition, the second option for determining the parameters of the gas generator is invariant to the value of the time parameters that determine the measurement moments. The results of determining the parameters of the gas generator and their nominal values are used to determine the arguments of the Laplace functions. We have specified that when determining the parameters of the gas generator, the complexing method can be used, which involves the implementation of all three information processing algorithms. As an example of the second option of determining gas generator parameters, the structural diagram of the algorithm is presented. We have emphasised that the implementation of the developed algorithms for determining the parameters of the gas generator has no subjective factor associated with the use of expert judgments. Keywords: gas generator, fire hazard level, transition function.