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A new algorithm that describes the faradaic current for elementary redox reactions in the cyclic voltammetric responses of persistently adsorbed species on metal electrodes at any scan rate is presented. This work does not assume electrochemical reversibility and instead demonstrates a set of equations that encapsulate how the forward and back charge-transfer rate constants influence the data as a function of the experimental time scale. The method presented here is compared against other approaches that rely on either finite-difference calculations or that require numerical approximation of improper integrals (i.e., ±infinity as a bound). The method here demonstrates that the current-potential data can be described by incomplete gamma functions, whose two arguments capture the relevant kinetic variables. Following the notation for the Butler-Volmer model of charge transfer, exact solutions are presented for the cases of the charge-transfer coefficient, α, equal to 1 or 0. A related algorithm based on these results affords calculation of current-potential data for 0 < α < 1, allowing comprehensive analysis (i.e., point by point) of voltammetric data throughout the reversible, quasi-reversible, and irreversible regimes. Accordingly, this work represents an alternative to the method of Laviron, i.e., analyzing just the peak splitting values, for experimentalists to understand and interpret their voltammetric data in totality.

This manuscript brings the derivation of influence functions for a three-dimensional full-space under bilinearly-distributed time-harmonic loads. The differential equations describing the medium are decomposed in terms of uncoupled vector fields. A double Fourier transform allows the system of equations to be solved algebraically in the transformed space, where the bilinear-loading boundary conditions are imposed. The resulting displacement and stress solutions are presented in terms of double improper integrals to be evaluated numerically. The manuscript brings selected results from the evaluation of these solutions. These influence functions can be used in boundary element models of elastodynamic problems to yield computationally-efficient solutions and improved representation of sharply-varying contact traction fields.

Joint models allow survival outcomes of a patient to be dynamically predictable based on intermediate events observed after treatment. The existing dynamic prediction methods need some numerical integration over frailty or random-effects distributions since the integral is implicit and improper. For a joint frailty-copula model, the Clayton copula and the gamma frailty model have been used to derive a dynamic prediction formula based on meta-analytic data. However, the prediction formulas under the Clayton copula involve an improper integral and need to be approximated numerically, as in all the joint models. In this paper, we consider the Gumbel copula and the FGM copula to obtain the exact (true) prediction formula without any numerical integration. The proposed formula also provides a tool for assessing approximation errors occurring to the approximation by numerical integration. Our numerical assessments show some approximation errors of the true prediction formula especially when the frailty distribution is heavy-tailed. A real data example illustrates how the proposed formulas can be used for assessing the approximation error. Our study may suggest some assessments for approximation errors in other types of joint models using more complex random-effects and frailty distributions.

This article is primarily concerned with finite-time convergence (FTC) and predefined-time convergence (PTC) design for a class of general zeroing neural network (ZNN) by constructing different activation functions (AFs). Based on the limit comparison test for improper integrals, some useful theoretical criteria are proposed to determine whether a nonlinear-activated ZNN model has FTC, PTC, or not. This novel method can avoid the valuation loss of the zoom method and the unsolvable barrier of the direct integration method that are widely used in the previous ZNN design. According to these convergence criteria, some instructive corollaries are derived to design valuable AFs to make ZNN models with FTC or PTC more easily. By taking a matrix-inversion ZNN model, some commonly used AFs are used to verify the usability of the criteria. In addition, some new AFs are constructed to further design some better ZNN models with superior FTC or PTC. Finally, convergence types of the ZNN model based on different AFs are visualized in numerical experiments.

This paper presents a method that finds the expected values of self- and mutual impedances of overhead lines with the uncertainties such as the ground resistivity, conductor characteristics, and transmission line structures. For this purpose, this study designs a stochastic random sampling method that removes the uncertainties. A finite element analysis method that includes a single logarithmic closed-form solution to Carson’s improper integral is adopted. The impact of an increase in sags and an imbalance in phase conductors on self- and mutual impedances is examined. As a result, the expected values of the self- and mutual impedances of an overhead line are evaluated. It is also shown that as the sag ratio increases, the self- and mutual impedances decrease.

Distributed energy resources with unbalanced phases increase the power imbalance in a grid, and compensating for the imbalance requires accurate knowledge of the impedance of the transmission and distribution lines. To achieve such compensation, many studies have evaluated the series impedance of the lines. The previous studies can be classified into three categories: (a) studies that solved Carson’s original equations (COEs), (b) those that approximated these equations by ignoring high-order terms, and (c) those that provided a closed-form solution for the equations. Solving the COEs requires the expansion of the improper integrals and infinite series. Therefore, the last approach is preferable for easier calculations and a small error. Thus, the objective of this study is to present a more accurate and robust closed-form solution. Toward this end, this study improves the single-logarithmic-approximation method by adding a fourth compensation term. That is, this study proposes the correction term $(2x/(x + \sqrt {1+x^{2}}) \approx 1- e^{-2x} - (x^{3}e^{-2x})/3 + (x^{5}e^{-5x})/5)$ . This study substitutes the correction term to the original derivatives and finds their correct integrations to improve a single-logarithmic-approximation solution. The proposed solution was verified through case studies, and it showed fewer errors than the previous solutions. Additionally, the proposed solution can be also used to estimate the expected value of the self- and mutual impedance of overhead lines via stochastic simulations (e.g., Monte Carlo simulations), which will be presented in the second paper of this study.

The article considers multidimensional improper integrals of functions that are the product of generalized polynomials in some degrees. Such integrals are found in many branches of mathematics and theoretical physics. In particular, they include Feynman integrals arising in the study of various objects of quantum field theory. The exact calculation of these integrals is a difficult and not always possible task; therefore, determining the conditions for their convergence and obtaining their asymptotic expansion in one of the parameters is of considerable practical interest. The convergence conditions for the integrals considered in the article can still be used, for example, in the study of multiple series representing the sum of the values of a rational function at the nodes of an integer lattice. The article considers the problem when the integration area is R+^n, and the generalized polynomials included in the integrand are either positive everywhere except zero or have positive coefficients. The convergence set of these integrals is described and the equivalence of the convergence condition to the condition on the Newton polytopes of polynomials in integrands is proved. The convergence criterion proved in the paper coincides in formulation with the corresponding result of the work of A. K. Tsikh and T. O. Ermolaeva, but it was obtained by other methods and for a slightly wider set of integrands. The proofs of the statements in the paper are based on the simplest properties of convex polytopes and basic facts from the theory of improper multiple integrals.

Relevance. Different models of contact between bodies are used in determining the stressed and deformed state in the strip lying on the base. It is necessary to evaluate the qualitative and quantitative nature of the change in stress in the strip depending on the coupling of the strip and base. The aim of the work - to analyze the effect of the coefficient of friction on the value of stresses in an anisotropic band when interacting with a rigid base. Methods. The solution is based on the equations of the plane problem of the theory of elasticity of an anisotropic body under the conditions that the band is closely adjacent to the base and the tangent force on the contact plane is proportional to the normal pressure. Displacements and stresses at any point of the strip are written in the form of the method of initial functions through the functions of displacements and forces on the lower plane, which depend on the nature of the load applied on the upper plane and the conditions of contact between the strip and the base. After the transformations, the calculation formulas for displacements and stresses are expressed using the Fourier integral transform through the normal surface load in the form of improper integrals. Results. Formulas for determining displacements and stresses are obtained for the variant of loading a strip with a concentrated force. These formulas are used to construct influence functions for the problem of equilibrium of an anisotropic strip lying on a rigid base, taking into account friction. Graphs of the effect of the coefficient of friction and the direction of the anisotropy axes of the material on the stress state of the strip are presented. The results of stress calculation are compared using anisotropic and isotropic models.

This study discusses the use of numerical integration in evaluating the improper integrals appearing as inverse integral transforms of non-analytic functions. These transforms appear while studying the response of various sources in an elastic medium through integral transform method. In these studies, the inverse Fourier transforms are solved numerically without bothering about the singularities and branch points in the corresponding integrands. References on numerical integration cited in relevant papers do not support such an evaluation but suggest contrary. Approximation of inverse Laplace transform integral into a series is used without following the essential restrictions and assumptions. Volume of the published papers using these dubious procedures has reached to an alarming level. The discussion presented aims to draw the attention of researchers as well as journals so as to stop this menace at the earliest possible.Keywords: Inverse Fourier transforms, inverse Laplace transforms, Romberg integration, improper integral, elastic waves

In this paper we prove converse theorems to obtain usual convergence of improper integrals from Cesàro summability.

In this paper Green?s function for the semi-infinite strip (which is the two-dimensional Green?s function for a groove of infinite depth and length) is determined in the form of an improper integral, as opposed to the standard summation form. The integral itself, although rather complex, is found in a closed form. By using the derived Green?s function simple formulas are obtained for a single and two-wire line configurations inside the groove.

Calculation of hadronization, decay or scattering processes at non-zero temperatures and densities within the Nambu-Jona-Lasinio-like models requires some techniques for computation of Feynmann diagrams. Decomposition of Feynman diagrams at the one loop level leads to the appearance of elementary integrals with one, two, three, and four fermion lines. For example, evaluation of the $\pi\pi$ scattering amplitude requires calculating a box diagram with four fermion lines. In this work, the real and imaginary parts of the box integral at the one loop level are provided in the form suitable for numerical evaluation. The obtained expressions are applicable to any value of temperature, particle mass, and chemical potential. We pay special attention to the conditions for the existence of the appearing improper integrals and correct the results \cite{Klevansky} for the three fermion lines. As a result, we have obtained constraints on possible values of particle momenta. Among the expressions for the box integral, the general formulas for the integral with an arbitrary number of lines are derived for the case with zero or collinear fermion momenta., Comment: 22 pages, no figures. The second version of the paper is essentialy distinct from the first one. The puzzle with different values of the three line integral for different coordinate systems is solved

Background: There is little clarity in the application of content related to improper integrals in university students, due to the absence of meaning, which prevents them from making a connection with everyday problem situations. Methods: we designed a mathematical modelling proposal where a specific situation involving the instrumentation, use and application of this type of integrals is experimented and solved with a population of engineering students, who learn to use them. Results: The importance of using mathematical modelling as a didactic-dynamic resource is highlighted because it helps students to reach an understanding of real situations involving improper integrals in different contexts. Conclusions: Despite the numerous errors detected in the students, this strategy made it possible to demonstrate the development of advanced mathematical thinking skills in young people.