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The goal of this work is to present a new theoretical and simultaneously practical approach to the conception of improper integrals of type II and, consequently, to improper integrals of type I. Seven distinct definitions of improper integrals of type II are presented, each illustrated by suitable examples. Among them, particularly important are those defined on nets composed of sets symmetric with respect to points belonging to a given finite family of points. Connections between improper integrals of type II defined here are described in two main theorems. It should be emphasized that all definitions of improper integrals presented here are special cases of a very general definition of improper integrals on non-compact sets, which is presented at the end of this work. This definition is inspired by a concept introduced by Maurin. [ABSTRACT FROM AUTHOR]

The uncountability of $\mathbb {R}$ is one of its most basic properties, known far outside of mathematics. Cantor's 1874 proof of the uncountability of $\mathbb {R}$ even appears in the very first paper on set theory, i.e., a historical milestone. In this paper, we study the uncountability of ${\mathbb R}$ in Kohlenbach's higher - order Reverse Mathematics (RM for short), in the guise of the following principle: $$\begin{align*}\mathit{for \ a \ countable \ set } \ A\subset \mathbb{R}, \mathit{\ there \ exists } \ y\in \mathbb{R}\setminus A. \end{align*}$$ An important conceptual observation is that the usual definition of countable set—based on injections or bijections to ${\mathbb N}$ —does not seem suitable for the RM-study of mainstream mathematics; we also propose a suitable (equivalent over strong systems) alternative definition of countable set, namely union over ${\mathbb N}$ of finite sets ; the latter is known from the literature and closer to how countable sets occur 'in the wild'. We identify a considerable number of theorems that are equivalent to the centred theorem based on our alternative definition. Perhaps surprisingly, our equivalent theorems involve most basic properties of the Riemann integral, regulated or bounded variation functions, Blumberg's theorem, and Volterra's early work circa 1881. Our equivalences are also robust , promoting the uncountability of ${\mathbb R}$ to the status of 'big' system in RM. [ABSTRACT FROM AUTHOR]

First, we prove a new alternative fixed point theorem in generalized gauge (or generalized uniformizable) spaces. This is a generalization of a famous result of Diaz-Margoli. Next, using this theorem, we show the stability of a delay differential equation where the unknown mapping takes its values in a locally convex space (in particular into a Banach space). Examples are given to support our results. We also point out some gaps in the literature and fix them. [ABSTRACT FROM AUTHOR]

Abstract This study establishes Newton-type inequalities for third differentiable and s-convex functions that use the Riemann integral. New Newton-type inequalities are also introduced using a summation parameter p ≥ 1 $p\geq 1$ for various convexity cases.

Elliptic polylogarithms can be defined as iterated integrals on a genus-one Riemann surface of a set of integration kernels whose generating series was already considered by Kronecker in the 19th century. In this article, we employ the Schottky parametrization of a Riemann surface to construct higher-genus analogues of Kronecker's generating series, which we refer to as Schottky–Kronecker forms. Our explicit construction generalizes ideas from Bernard's higher-genus construction of the Knizhnik-Zamolodchikov connection. Integration kernels generated from the Schottky–Kronecker forms are defined as Poincaré series. Under technical assumptions, related to the convergence of these Poincaré series on the underlying Riemann surface, we argue that these integration kernels coincide with a set of differentials defined by Enriquez, whose iterated integrals constitute higher-genus analogues of polylogarithms. Enriquez' original definition is not well-suited for numerical evaluation of higher-genus polylogarithms. In contrast, the Poincaré series defining our integration kernels can be evaluated numerically for real hyperelliptic curves, for which the above-mentioned convergence assumptions can be verified. We numerically evaluate several examples of genus-two polylogarithms, thereby paving the way for numerical evaluation of hyperelliptic analogues of polylogarithms. [ABSTRACT FROM AUTHOR]

This study establishes Newton-type inequalities for third differentiable and s-convex functions that use the Riemann integral. New Newton-type inequalities are also introduced using a summation parameter p ≥ 1 for various convexity cases. [ABSTRACT FROM AUTHOR]

Kinetic models are relevant to describe heterogeneous kinetic processes; a number of kinetic models and their mathematical expressions have been reported in the literature, many of these based on idealistic conditions in terms of geometrical constrain and driving forces. Alternatively, the semi-empirical Sestak–Berggren (SB) conversion function, which was proposed as a general equation, encompasses a large variety of equations corresponding to different kinetic models. Despite the fact that the SB equation does not provide any physical meaning, it is extremely useful for kinetic analysis as it offers a good fit to experimental data even when they do not follow the ideal conditions assumed for the conventional kinetic models. One limitation of the SB kinetic model is the fact that its conversion function cannot be analytically integrated to provide an exact solution; thus, it cannot be directly applied in kinetic integral methods. The objective of this study aims to propose some solutions for some specific cases, while the mathematical limits for the values of the kinetic exponents m, n, p of the SB model and their validity are also explored. Further ideas for improving the SB equation or finding an alternative for a superior conversion function were explored in this work. [ABSTRACT FROM AUTHOR]

This work was developed to present constructive steps of multiple integrals using the open-source software Geogebra. The main focus was directed towards creating three-dimensional graphs of integrals through Riemann sums in two variables. Some practical examples are developed to demonstrate the reliability of the presented results, which are compared using traditional algebraic methods and computed in Geogebra. With this, we aim to highlight the potential of Geogebra in teaching integral calculus and make the graphical visualization process less exhaustive. [ABSTRACT FROM AUTHOR]

Differential and Integral Calculus (CDI) is an essential subject for teaching Mathematics and other sciences. Despite this importance, we observed student failure and high rates of failure or dropout, which justifies the relevance of considering epistemological aspects that make it possible to understand different phenomena in the teaching of this discipline. In this sense, we propose a study of epistemological elements of density and mass knowledge, through multivariational integrals, since integral is essential knowledge for the area of exact sciences. The objective of this study is to investigate the development and implementation of an intervention proposal, centered on study and research activities, that offers CDI students opportunities to explore the concept of the integral of one and more variables. To this end, we carried out an investigation through the creation and implementation of an intervention based on work with task-solving episodes, in order to analyze generalization movement(s) that students performed to define a multivariational defined integral from defined integrals of a variable. The results showed that the students mobilized a set of knowledge of multiple integrals, from the context of calculating mass in one, two and three dimensions. Expansive generalization was used to expand procedural issues in the calculation of an integral, while reconstructive generalization was used to understand structural aspects of the Riemann integral of more than one variable. [ABSTRACT FROM AUTHOR]

We show two elementary versions of the Change of Variable Theorem (for the unidimensional Riemann integral) that are not a particular case of the influential general version due to Hyman Kestelman. Examples are given. [ABSTRACT FROM AUTHOR]

Fractional calculus and convex inequalities combine to form a comprehensive mathematical framework for understanding and analyzing a variety of problems. This note develops Hermite-Hadamard, Fej'er, and Pachpatte type integral inequalities within pseudo left-right order relations by applying fractional operators with non-singular kernels. Recently, results have been developed using classical Riemann integral operators in addition to various other partial order relations that have some defects that we explained in literature in order to demonstrate the unique characteristics of pseudo order relations. To verify the developed results, we constructed several interesting examples and provided a number of remarks that demonstrate that this type of fractional operator generalizes several previously published results when different things are set up. This work can lead to improvements in mathematical theory, computational methods, and applications across a wide range of disciplines. [ABSTRACT FROM AUTHOR]

In the current research, some midpoint-type inequalities are generalized for co-ordinated convex functions with the help of generalized conformable fractional integrals. Moreover, some findings of this paper include results based on Riemann–Liouville fractional integrals and Riemann integrals. [ABSTRACT FROM AUTHOR]

This article presents a proof of the bounded convergence theorem for Riemann integrals. An effort has been made to keep the exposition concise and self-contained. [ABSTRACT FROM AUTHOR]

Localization by Autonomous Underwater Vehicle (AUV) is one of the main methods for locating target nodes in underwater acoustic sensor networks. Aiming at the problems of low localization coverage area and large localization error when AUV locates target nodes, a Double AUVs Cooperative Localization Based on Relative Heading Angle Optimization (DA-RHAO) algorithm is proposed in this paper. Firstly, Riemann integral is used to derive the relationship between localization coverage area and the relative heading angle during AUV movement. Secondly, the 3D target water is divided into multiple depth layers, and the relative heading angle between AUVs is selected according to the regional node density in each depth layer, which effectively enlarges localization coverage area of AUVs. Then, an AUV cooperative localization model is established to solve the coordinates of target nodes, which improves the localization efficiency and accuracy of the algorithm. Finally, the simulation results show that compared with the single AUV localization algorithm and AUV cooperative localization algorithm, the localization accuracy of DA-RHAO algorithm is improved by 26.89% and 12.56%, respectively. [ABSTRACT FROM AUTHOR]

The notion of statistical convergence has fascinated many researchers due mainly to the fact that it is more general than the well-established hypothesis of ordinary (classical) convergence. This work aims to investigate and present (presumably new) the statistical versions of deferred weighted Riemann integrability and deferred weighted Riemann summability for sequences of fuzzy functions. We first interrelate these two lovely theoretical notions by establishing an inclusion theorem. We then state and prove two fuzzy Korovkin-type theorems based on our proposed helpful and potential notions. We also demonstrate that our results are the nontrivial extensions of several known fuzzy Korovkin-type approximation theorems given in earlier works. Moreover, we estimate the statistically deferred weighted Riemann summability rate supported by another promising new result. Finally, we consider several interesting exceptional cases and illustrative examples supporting our definitions and the results presented in this paper.

In this article we present a method of integration inspired by an example from Proofs Without Words II by Nelsen. Riemann rectangles under a curve are transformed into triangles. The limit of the collection of triangles gives us a new shape. The area under the original curve and the area of the new shape are the same. In this way, interesting connections between areas under different curves are established and integration formulae are obtained. Only elementary concepts like similarity and Riemann rectangles are used. A variety of examples is worked out to show the strength of the method. The method may be used for first year calculus students to deepen their understanding of the Riemann integral. [ABSTRACT FROM AUTHOR]

In this work we start developing a Riemann-type integration theory on spaces which are equipped with a fractal structure. These topological structures have a recursive nature, which allows us to guarantee a good approximation to the true value of a certain integral with respect to some measure defined on the Borel σ -algebra of the space. We give the notion of Darboux sums and lower and upper Riemann integrals of a bounded function when given a measure and a fractal structure. Furthermore, we give the notion of a Riemann-integrable function in this context and prove that each μ -measurable function is Riemann-integrable with respect to μ. Moreover, if μ is the Lebesgue measure, then the Lebesgue integral on a bounded set of R n meets the Riemann integral with respect to the Lebesgue measure in the context of measures and fractal structures. Finally, we give some examples showing that we can calculate improper integrals and integrals on fractal sets. [ABSTRACT FROM AUTHOR]

We investigate and study on mathematical structures involving mathematical models and others associated with seismic waves in an earthquake. Our first aim is to give some novel formulas and certain finite sums including the Bernoulli numbers and the Hermite polynomials with the aid of generating functions, the Riemann integral, and the Volkenborn integral. The second aim is to examine the seismic wave propagation in different geological units with the help of special polynomials containing the Hermite polynomials and their graph fitting of functions. To evaluate the shape of the seismic waves propagating within the ground (rock and/or soil), we use comparing method with the graph of the Hermite polynomials and functions and the polynomial Rocking Bearings. Furthermore, we also define generating function for the polynomial type Rocking Bearings. We give open problems on this generating function and earthquake facts. By applying partial derivative operator to the generating function for the m-parametric Hermite type polynomials, we give a novel recurrence relation and derivative formulas for these polynomials. We also give a new general formula for monomials in terms of these polynomials. Moreover, for the purpose of visualizing curve fitting approach to the seismic waves, we draw many plots of the Hermite functions with Mathematica (Version 12.0.0) with their codes. Finally, with the aid of these graphs, we give useful evaluation on the shapes of the seismic waves propagated in the ground (rock and/or soil). [ABSTRACT FROM AUTHOR]

The question posed in the title is positively answered for any continuous, real function defined on any crowded subset of real numbers. [ABSTRACT FROM AUTHOR]

Using the example of scalar and vector Wiener–Hopf equations, we consider two methods for combining the options for the Riemann integral and Lebesgue functional spaces in problems of studying and solving integral convolution equations. The method of nonlinear factorization equations and the kernel averaging method are used. A generalization of the direct Riemann integrability is introduced and applied. [ABSTRACT FROM AUTHOR]

We consider a boundary value problem for a system of integro-differential equations arising when modeling the influence of an electric field on a plasma layer. The paper presents an analytical solution of this problem, which is constructed with the use of theory of the Fourier transform of generalized functions and the Gakhov–Muskhelishvili method for solving singular integral equations and the Riemann boundary value problem. [ABSTRACT FROM AUTHOR]

The paper is devoted to the approximate calculation of Riemann definite integrals, singular and hypersingular integrals over closed and open non-rectifiable curves and fractals. The conditions of existence for the Riemann definite integrals over non-rectifiable curves and fractals are provided. We give a definition of a singular integral over non-rectifiable curves and fractals which generalizes the known one. We define hypersingular integrals over non-rectifiable curves and fractals. We construct quadratures for the calculation of Riemann definite integrals, singular and hypersingular integrals over non-rectifiable curves and fractals and the corresponding error estimates for various classes of functions. Singular and hypersingular integrals are defined up to an additive constant (or a combination of constants) that are subject to a convention that depends on the actual problem being solved. We illustrate our theoretical results with numerical examples for Riemann definite integrals, singular integrals and hypersingular integrals over fractals. [ABSTRACT FROM AUTHOR]

This paper aims to introduce the notions of deferred Nörlund statistical Riemann integrability and statistical deferred Nörlund Riemann summability for sequence of real-valued functions and to apply them in Korovkin-type new approximations. First, we present an inclusion theorem to understand the connection between these new notions. Then, based on these potential notions we establish new versions of Korovkin-type theorems with three algebraic test functions. Finally, we compute an example, under the consideration of a positive linear operator in association with the Bernstein polynomials to exhibit the effectiveness of our findings. [ABSTRACT FROM AUTHOR]

We determine the first curvature correction for the string amplitude of two supergravity states and two Kaluza-Klein modes on AdS5×S5, which is dual to the correlator O 2 O 2 O p O p of half-BPS operators in N = 4 SYM theory. The result has the form of an integral over the Riemann sphere as for the usual Virasoro-Shapiro amplitude, with the insertion of single-valued multiple polylogarithms of weight three. The result fixes OPE data of single-trace operators in N = 4 SYM theory at strong coupling, including operators with non-zero R-charge and odd spin. We successfully check our results by comparing to data available from integrability, localisation and consistency with a 10d effective action. [ABSTRACT FROM AUTHOR]

We present a streamlined, slightly modified version, in the two-variable situation, of a beautiful, but not so well known, theory by Bögel, already from the 1930s, on an alternative higher dimensional calculus of real functions, a double calculus, which includes many two-variable extensions of classical results from single variable calculus, such as Rolle's theorem, Lagrange's mean value theorem, Cauchy's mean value theorem, Fermat's extremum theorem, the first derivative test, and the first and second fundamental theorems of calculus.

A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite patch of the wedge-shaped, which meets the interface at a right angle and is loaded with tangential and normal forces is considered. Using methods of the theory of analytic functions, the problem is reduced to the system of singular integro-differential equations (SIDE) with fixed singularity. Under tension-compression of patch using an integral transformation a Riemann problem is obtained, the solution of which is presented in explicit form. The tangential contact stresses along the contact line are determined and their asymptotic behavior in the neighborhood of singular points is established. [ABSTRACT FROM AUTHOR]

We propose the concept of Riemann diamond-alpha integrals for time scales interval-valued functions. We first give the definition and some properties of the interval Riemann diamond-alpha integral that are naturally investigated as an extension of interval Riemann nabla and delta integrals. With the help of the interval Riemann diamond-alpha integral, we present interval variants of Jensen inequalities for convex and concave interval-valued functions on an arbitrary time scale. Moreover, diamond-alpha Hölder's and Minkowski's interval inequalities are proved. Also, several numerical examples are provided in order to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, firstly, we study the continuity of Cauchy-type integral operator C Γ infra associated with inframonogenic functions and give the Plemelj formula. Secondly, we prove the properties of the Teodorescu operator related to the inframonogenic functions, including its boundness, continuity and differentiability. Finally, we give the related integral representation of Riemann boundary value problems for inframonogenic functions and generalized inframonogenic functions. [ABSTRACT FROM AUTHOR]

In this paper, using hypergeometric functions, we provide sharp estimates of the remainder of the alternating p-series, ∑n≥1/(−1)n−1 np, where p ≥ 2 is an integer. We show that the largest ρ and the largest σ such that the inequalities 1/2(n+1)p−ρ≤| ∑k=n+1∞ (−1)k−1/kp|≤ 1/2np +σ, hold for any integer n ≥ 1 are ρ (p) = 2p+1− 1/1−(1−2¹−p)ζ (p) and σ (p) = 1/1−(1−21−p)ζ (p) −2, where ζ (p) = ∑k=1∞ 1/kp, the Riemann zeta function. [ABSTRACT FROM AUTHOR]

In this paper, we prove new unitarily invariant norm inequalities for positive semidefinite matrices. Some of these inequalities represents a generalization of earlier work due to Kittaneh who refines an inequality due to Davidson and Power which is useful in best approximation of C* -algebras. [ABSTRACT FROM AUTHOR]

In this study, we apply fractional calculus on certain convex functions and derive many novel mean-type inequalities by employing fractional calculus and convexity theory. In order to investigate fractional mean inequalities, we first build an identity in this study. Then, with its help, we derive many mean-type inequalities and estimate the error of HH inequality using a generalized version of RL-fractional integrals and certain classes of convex functions. The results obtained are validated by taking specific functions. Many mean-type inequalities that exist in the literature are generalized by the main results of this study. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new Bullen-type inequalities for differentiable convex functions using the generalized fractional integrals. The main advantage of the inequalities and operators used to obtain them is that these inequalities can be turned into some existing inequalities for Riemann integrals and new inequalities for Riemann–Liouville fractional integral inequalities and k -fractional integrals. Finally, we add some applications of special means of real numbers using the newly established inequalities to make these results more interesting. [ABSTRACT FROM AUTHOR]

We provide two novel ways to compute the surface gravity (κ ) and the Hawking temperature (T H) of a stationary black hole: in the first method T H is given as the three-volume integral of the Gauss–Bonnet invariant (or the Kretschmann scalar for Ricci-flat metrics) in the total region outside the event horizon; in the second method it is given as the surface integral of the Riemann tensor contracted with the covariant derivative of a Killing vector on the event horizon. To arrive at these new formulas for the black hole temperature (and the related surface gravity), we first construct a new differential geometric identity using the Bianchi identity and an antisymmetric rank-2 tensor, valid for spacetimes with at least one Killing vector field. The Gauss–Bonnet tensor and the Gauss–Bonnet scalar play a particular role in this geometric identity. We calculate the surface gravity and the Hawking temperature of the Kerr and the extremal Reissner–Nordström holes as examples. [ABSTRACT FROM AUTHOR]

Using generating functions with their functional equations method, a great number of novel combinatorial sums, formulas, and recurrence relation including Fubini type polynomials and numbers, Stirling type numbers, and Apostol type polynomials are given. Applying Riemann integral to this generating function with their functional equations, some identities involving Cauchy and Stirling numbers are obtained. Moreover, some interpretations about the results are given. [ABSTRACT FROM AUTHOR]

In the present paper, we focus on the reducibility of an almost-periodic linear Hamiltonian system... ...where J is an anti-symmetric symplectic matrix, A is a symmetric matrix, Q(t) is an analytic almostperiodic matrix with respect to t, and e is a parameter which is sufficiently small. Using some nonresonant and non-degeneracy conditions, rapidly convergent methods prove that, for most sufficiently small e, the Hamiltonian system is reducible to a constant coefficients Hamiltonian system through an almost-periodic symplectic transformation with similar frequencies as Q(t). At the end, an application to Schr?odinger equation is given. [ABSTRACT FROM AUTHOR]

To create various kinds of inequalities, the idea of convexity is essential. Convexity and integral inequality hence have a significant link. This study's goals are to introduce a new class of generalized convex fuzzy-interval-valued functions (convex FIVFs) which are known as (p, J)-convex FIVFs and to establish Jensen, Schur and Hermite-Hadamard type inequalities for (p, J)-convex FIVFs using fuzzy order relation. The Kulisch-Miranker order relation, which is based on interval space, is used to define this fuzzy order relation level-wise. Additionally, we have demonstrated that, as special examples, our conclusions encompass a sizable class of both new and well-known inequalities for (p, J)-convex FIVFs. We offer helpful examples that demonstrate the theory created in this study's application. These findings and various methods might point the way in new directions for modeling, interval-valued functions and fuzzy optimization issues. [ABSTRACT FROM AUTHOR]

In this paper, the Gaussian quadrature formula is used to approximately calculate two-dimensional hypersingular integrals of the form ∬ S f (x , y) r 3 d S on the circular domain. The two-dimensional hypersingular integrals are changed into iterated integrals by polar transformation. The inner layer of the integral has singularity and Gaussian quadrature formula is constructed for this part. The outer layer without singularity is calculated by Gauss–Legendre quadrature formula of Riemann integral. The corresponding theoretical analysis is given. Furthermore, this type of two-dimensional hypersingular integral is extended to the form ∬ S f (x , y) r p + 1 d S (p > 2) and an error estimate is also derived. Several numerical examples are presented to verify the effectiveness of the proposed algorithm, no matter the singularity is located at the center or near the boundary of the circular domain. [ABSTRACT FROM AUTHOR]

In this paper, several results and theorems about the high-order strongly generalized Hukuhara differentiability of function defined via the fuzzy Riemann improper integral (in the sense of Wu) have been established. Then, some properties dealing with the partial derivatives of fuzzy Laplace transform for a fuzzy function of two real variables have been proved. Afterwards, an algorithm of fuzzy Laplace transform for solving second-order fuzzy partial differential equations has been proposed. Finally, two numerical examples, including the heat equation under fuzzy initial conditions, have been studied to justify the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

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In this manuscript, we present the notion of improper Riemann integral and develop a methodology to solve nth-order fuzzy initial value problems (FIVPs) via fuzzy Laplace transform for linearly correlated processes. Our approach translates a fuzzy differential equation (FDE) problem from the fuzzy number space to a well-established Banach space. In this new approach, it is not necessary to consider several cases due to switch points, and we do not need to use the Stacking theorem because the obtained solution is always a fuzzy number. In order to illustrate our approach, we present several examples comparing the obtained solutions with those produced by other methods. [ABSTRACT FROM AUTHOR]

The role of integral inequalities can be seen in both applied and theoretical mathematics fields. According to the definition of convexity, it is possible to relate both concepts of convexity and integral inequality. Furthermore, convexity plays a key role in the topic of inclusions as a result of its definitional behavior. The importance and superior applications of convex functions are well known, particularly in the areas of integration, variational inequality, and optimization. In this paper, various types of inequalities are introduced using inclusion relations. The inclusion relation enables us firstly to derive some Hermite-Hadamard inequalities (H.H-inequalities) and then to present Jensen inequality for harmonical h-Godunova-Levin interval-valued functions (GL-IVFS) via Riemann integral operator. Moreover, the findings presented in this study have been verified with the use of useful examples that are not trivial. [ABSTRACT FROM AUTHOR]