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ABSTRACT In this paper, we present a computational method for solving the second‐order impulsive differential equations with loadings subject to integral boundary conditions based on the Dzhumabaev parametrization method. The idea of this method involves introducing additional parameters, reducing the original problem to solving a system of linear algebraic equations. The system's coefficients and right‐hand side are determined by solving Cauchy problems for ODEs and by calculating definite integrals. Four examples are provided to show the effectiveness and feasibility of the main results. [ABSTRACT FROM AUTHOR]

This work delves deeply into convolved Fibonacci polynomials (CFPs) that are considered generalizations of the standard Fibonacci polynomials. We present new formulas for these polynomials. An expression for the repeated integrals of the CFPs in terms of their original polynomials is given. A new approach is followed to obtain the higher-order derivatives of these polynomials from the repeated integrals formula. The inversion and moment formulas for these polynomials, which we find, are the keys to developing further formulas for these polynomials. The derivatives of the moments of the CFPs in terms of their original polynomials and different symmetric and non-symmetric polynomials are also derived. New product formulas of these polynomials with some polynomials, including the linearization formulas of these polynomials, are also deduced. Some closed forms for definite and weighted definite integrals involving the CFPs are found as consequences of some of the introduced formulas. [ABSTRACT FROM AUTHOR]

We study the fracture of nanobeams containing central cracks using nonlocal stress gradient integral model (NSGIM) with discontinuous and symmetrical conditions. Compared with the traditional NSGIM, the present model includes some definite integral terms in the constitutive boundary constraints and the newly introduced constitutive continuity conditions. Subsequently, the Mode I and Mode II fracture problems are studied under given uniform loading (or concentrated force) conditions in opposite and same directions. In the numerical simulations, the effects of nonlocal and gradient length scale parameters on normalized energy release rates and works by external loads are investigated in detail. [ABSTRACT FROM AUTHOR]

ABSTRACT As in the polynomial case, non‐polynomial divided differences can be viewed as a discrete analog of derivatives. This link between non‐polynomial divided differences and derivatives is defined by a generalization of the derivative operator. In this study, we obtain a generalization of Taylor series using the link between non‐polynomial divided differences and derivatives, and state generalized Taylor theorem. With the definition of a definite integral, the relation between the non‐polynomial divided difference and non‐polynomial B‐spline functions is given in terms of integration. Also, we derive a general form of the Peano kernel theorem based on a generalized Taylor expansion with the integral remainder. As in the polynomial case, it is shown that the non‐polynomial B‐splines are in fact the Peano kernels of non‐polynomial divided differences.MSC2020 Classification: 65D05, 65D07 [ABSTRACT FROM AUTHOR]

Definition: This entry first provides an overview of the historical, cultural and epistemological background that is key for Hamilton's positions on mechanics. We consider the investigations on geometrical optics in the 17th and 18th centuries, Euler's and Lagrange's foundations of variational calculus in the 18th century to find extrema of physical quantities expressed as infinite sums of infinitesimals (today, we would say 'definite integrals'), and Lagrange's introduction of a revolutionary analytical mechanics, all of which are all fertile grounds for Hamilton's steps—first, in what we could call analytical optics, then in an advanced form of analytical mechanics. Having provided such an overview, we run through some of Hamilton's original papers to highlight how he posed his principle(s) in the wake of his forerunners and how his principles are linked with the search for a unitary view of physics. [ABSTRACT FROM AUTHOR]

Bearing in mind an increasing popularity of the fractional calculus the main aim of this paper is to derive several new representation formulae for the cumulative distribution function (cdf) of the McKay I ν Bessel distribution including the Grünwald-Letnikov fractional derivative; also, two connection formulae between cdf of the McKay I ν random variable and the so–called Neumann series of modified Bessel functions of the first kind are established, providing, consequently, a new integral representation for such cdf in terms of a definite integral. Another fashion expression for the given cdf is derived in terms of the Grünwald-Letnikov fractional derivative of the widely applicable Marcum Q–function, which represents a certain simplification of the already existing relationship between McKay I ν random variable and a Marcum Q–functions. The exposition ends with some open questions, drawing the interested reader's attention, among others, to the summation of some Neumann series. [ABSTRACT FROM AUTHOR]

In 2012, Chu investigated the generalization of classical Watson–Whipple–Dixon summation theorems in the form of analytical formulas. By employing four generalized Watson summation formulas, the objective of this paper is to evaluate a new class of several Eulerian-type integrals (single and double) and Laplace-type integrals involving a hypergeometric function. Several interesting special cases are also given. Symmetry arises spontaneously in the hypergeometric function. [ABSTRACT FROM AUTHOR]

Milne's formula is a mathematical expression used to approximate the value of a definite integral. This formula is especially useful for problems encountered in physics, engineering, and various other scientific disciplines. We establish an equality for conformable fractional integrals. With the help of this equality, we obtain error bounds for one of the open Newton–Cotes formulas, namely, Milne's formula for the case of differentiable convex functions within the framework of fractional and classical calculus. Furthermore, we provide our results by using special cases of the obtained theorems. [ABSTRACT FROM AUTHOR]

We present a size-dependent formulation for the longitudinal vibration study of cracked thick rods based on both the strain and stress-driven local/nonlocal mixture theories of elasticity with discontinuity. Due to the presence of the crack, the rod is divided into two segments connected by a linear spring, and compatibility conditions are given to describe the geometric discontinuity caused by the crack. The equations of motion of the discrete rods are formulated based on Rayleigh rod theory, and the two classes of local–nonlocal constitutive equations are integrated into an equivalent differential form, equipped with a set of constitutive boundary conditions at two ends of the whole structure and a set of constitutive continuity conditions at the junction of the sub-structures. The differential quadrature method (GDQM), together with the interpolation quadrature formula, is introduced to solve all the equations of motion of the sub-rods, the above constraint condition and the definite integrals occurring therein, simultaneously, through which we extract the dimensionless frequencies of the cracked rods with different boundary edges. After conducting comparison studies with the existing literature, numerical studies reveal that the present local–nonlocal model with discontinuity can effectively address the softening (or hardening) phenomenon as the structure's size reduces. Moreover, the influence of crack location, crack severity, inertia of lateral motions and nonlocal parameters on the rods' vibration frequencies is examined in detail. [ABSTRACT FROM AUTHOR]

In this article, the weighted three-point integral quadrature formula is estimated by the new bounds using the new method of calculating estimates for quadrature rules applying the weighted Hermite-Hadamard inequality for higher-order convex functions and weighted version of the integral identity expressed by w-harmonic sequences of functions. The importance of those results lies in providing new estimates of the definite integral values by using weighted three-point formula for numerical integration. The obtained results are employed in establishing new estimates for the Legendre-Gauss three-point quadrature formula with the use of specific form of the weight function w. [ABSTRACT FROM AUTHOR]

The first goal of this paper is to prove that if is a finite noncyclic simple group and is a prime divisor of ; then , where is the size of a Sylow p-subgroup of . The invariant is a definite integral of the Sylow polynomial of a finite group . Our second goal is to show that is the only finite simple group with . [ABSTRACT FROM AUTHOR]

For a positive integer m , a (positive definite integral) quadratic form is called primitively m -universal if it primitively represents all quadratic forms of rank m. It was proved in [9] that there are exactly 107 equivalence classes of primitively 1-universal quaternary quadratic forms. In this article, we prove that the minimal rank of primitively 2-universal quadratic forms is six, and there are exactly 201 equivalence classes of primitively 2-universal senary quadratic forms. [ABSTRACT FROM AUTHOR]

Definite integrals containing gamma functions and other special functions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we express a generalization of the Ramanujan integral I ⁢ (α) with the analytical solutions, using the Laplace transform technique and some algebraic relation or the Pochhammer symbol. Moreover, we evaluate some consequences of a generalized definite integral ϕ * ⁢ (υ , β , a) . The well-known special cases appeared, whose solutions are possible by Cauchy's residue theorem, and many known applications of the integral I ⁢ (a , β , υ) are discussed by the Leibniz rule for differentiation under the sign of integration. Further, one closed-form evaluation of the infinite series of the F 0 1 ⁢ (⋅) function is deduced. In addition, we establish some integral expressions in terms of the Euler numbers, which are not available in the tables of the book of Gradshteyn and Ryzhik. [ABSTRACT FROM AUTHOR]

This article is devoted to establishing new formulas concerning generalized Hermite polynomials (GHPs) that generalize the classical Hermite polynomials. Derivative expressions of these polynomials that involve one parameter are found in terms of other parameter polynomials. Some other important formulas, such as the linearization and connection formulas between these polynomials and some other polynomials, are also given. Most of the coefficients are represented in terms of hypergeometric functions that can be reduced in some specific cases using some standard formulas. Two applications of the developed formulas in this paper are given. The first application is concerned with introducing some weighted definite integrals involving the GHPs. In contrast, the second is concerned with establishing the operational matrix of the integer derivatives of the GHPs. [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]

Owing to the remarkable success of the hypergeometric functions of one variable, the authors present a study of some families of hypergeometric functions of two or more variables. These functions include (for example) the Kampé de Fériet-type hypergeometric functions in two variables and Srivastava's general hypergeometric function in three variables. The main aim of this paper is to provide several (presumably new) transformation and summation formulas for appropriately specified members of each of these families of hypergeometric functions in two and three variables. The methodology and techniques, which are used in this paper, are based upon the evaluation of some definite integrals involving logarithmic functions in terms of Riemann's zeta function, Catalan's constant, polylogarithm functions, and so on. [ABSTRACT FROM AUTHOR]

Let Q (x) be a positive definite integral quadratic form with the determinant D being squarefree, and r (n , Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function d (n) and Fourier coefficients r (n , Q). With more efforts, our method should have a number of applications for other multiplicative functions. [ABSTRACT FROM AUTHOR]

The signature of a rectifiable path is a tensor series in the tensor algebra whose coefficients are definite iterated integrals of the path. The signature characterizes the path up to a generalized form of reparameterization. It is a classical result of Chen that the log‐signature (the logarithm of the signature) is a Lie series. A Lie series is polynomial if it has finite degree. We show that the log‐signature is polynomial if and only if the path is a straight line up to reparameterization. Consequently, the log‐signature of a rectifiable path either has degree one or infinite support. Though our result pertains to rectifiable paths, the proof uses rough path theory, in particular that the signature characterizes a rough path up to reparameterization. [ABSTRACT FROM AUTHOR]

In the Cartesian two-dimensional space, this note applies a mixed quadrature rule in conjunction with an adaptive scheme across the circular surface. The two mathematical processes that result in a regular square area from the circular surface. The mixed quadrature rule was assessed in an adaptive scheme using five numerical tests. It was discovered to be more effective than Boole's rule. [ABSTRACT FROM AUTHOR]

Hundreds of special functions have been employed in applied mathematics and computing sciences for many centuries due to their outstanding features and wide range of applications. When considering the relevance of these consequences in the evaluation of generalized integrals, applied physics, and many engineering areas, the illustration of image formulas involving one or more variable special functions is significant under various definite integrals. In this paper, it is devoted to study the various integral identities involving incomplete Fox-Wright functions and Srivastava’s polynomials. It is shown that the integrals of the Fox-Wright functions are also the Fox-Wright functions but of greater order. Due to the fact that our results are unified, a substantial number of new results can be constructed as special instances from our leading results. The results obtained in this work are general in nature and very useful in science, engineering and finance. [ABSTRACT FROM AUTHOR]

Due to the great success of hypergeometric functions, we provide the analytical solutions of certain definite logarithmic integrals and Nielsen-type integrals in terms of multi-variable Kampé de Fériet functions with suitable convergence conditions and higher harmonic sums by using series rearrangement technique and incomplete Gamma function. Further we also obtain the solution of other related logarithmic integrals in terms of generalized hypergeometric functions and Kummer’s confluent hypergeometric functions by using series rearrangement technique. The results presented in the paper and comparable outcomes are hoped to be supplied by the use of computer-aid programs, for example, Mathematica. [ABSTRACT FROM AUTHOR]

Motivated essentially by the fact that several interesting families of generalized elliptic-type integrals and many definite integrals of such families with respect to their modulus (or complementary modulus) are known to arise naturally in various physical contexts, we present a systematic account of the theory of these families of elliptic-type integrals in a unified and generalized manner. We also point out relevant connections of the general theory, which is discussed in this paper, with the results in numerous earlier publications on the theory and applications of the aforementioned families of elliptictype integrals. Several classes of hypergeometric functions of one, two and more variables are observed to emerge in our unified presentation of these families of elliptic-type integrals. [ABSTRACT FROM AUTHOR]

The numerical integration of definite integrals is essential in fundamental and applied sciences. The accuracy of approximate integral calculations is contingent upon the initial data and specific requirements, leading to the imposition of diverse conditions on the resultant computations. Classical methods for the numerical analysis of definite integrals are known, such as the quadrature formulas of Gregory, Newton-Cotes, Euler, Gauss, Markov, etc. Since the middle of the last century, the theory of constructing optimal formulas for numerical integration based on variational methods began to develop. It should be noted that there are optimal quadrature formulas in the sense of Nikolsky and Sard. In this paper, we study the problem of constructing an optimal quadrature formula in the sense of Sard. When constructing a quadrature formula, the method of φ-functions is used. The error of the formula is estimated from above using the integral of the square of the function from a specific Hilbert space. Next, such a φ function is selected, and the integral of the square in this interval takes the smallest value. The coefficients of the optimal quadrature formula are calculated using the resulting φ function. The optimal quadrature formula in this work is exact on the functions eσx and e-σx, where σ is a nonzero real parameter. [ABSTRACT FROM AUTHOR]

This work aims to provide a new Galerkin algorithm for solving the fractional RayleighStokes equation (FRSE). We select the basis functions for the Galerkin technique to be appropriate orthogonal combinations of the second kind of Chebyshev polynomials (CPs). By implementing the Galerkin approach, the FRSE, with its governing conditions, is converted into a matrix system whose entries can be obtained explicitly. This system can be obtained by expressing the derivatives of the basis functions in terms of the second-kind CPs and after computing some definite integrals based on some properties of CPs of the second kind. A thorough investigation is carried out for the convergence analysis. We demonstrate that the approach is applicable and accurate by providing some numerical examples. [ABSTRACT FROM AUTHOR]

In numerical analysis, the quadrature formulas serve as a pivotal tool for approximating definite integrals. In this paper, we introduce a family of quadrature formulas and analyze their associated error bounds for convex functions. The main advantage of these new error bounds is that from these error bounds, we can find the error bounds of different quadrature formulas. This work extends the traditional quadrature formulas such as the midpoint formula, trapezoidal formula, Simpson's formula, and Boole's formula. We also use the power mean and Hölder's integral inequalities to find more general and sharp bounds. Furthermore, we give numerical example and applications to quadrature formulas of the newly established inequalities. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for the unified Chebyshev polynomials (UCPs). It is first necessary to introduce the three basic formulas of these polynomials, namely analytic form, moments, and inversion formulas, which will later be utilized to derive further formulas of the UCPs. We will prove the basic formula that shows that these polynomials can be expressed as a combination of three consecutive terms of Chebyshev polynomials (CPs) of the second kind. New derivatives and connection formulas between two different classes of the UCPs are established. Some other expressions of the derivatives of UCPs are given in terms of other orthogonal and non-orthogonal polynomials. The UCPs are also the basis for additional derivative expressions of well-known polynomials. A new linearization formula (LF) of the UCPs that generalizes some well-known formulas is given in a simplified form where no hypergeometric forms are present. Other product formulas of the UCPs with various polynomials are also given. As an application to some of the derived formulas, some definite and weighted definite integrals are computed in closed forms. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for the unified Chebyshev polynomials (UCPs). It is first necessary to introduce the three basic formulas of these polynomials, namely analytic form, moments, and inversion formulas, which will later be utilized to derive further formulas of the UCPs. We will prove the basic formula that shows that these polynomials can be expressed as a combination of three consecutive terms of Chebyshev polynomials (CPs) of the second kind. New derivatives and connection formulas between two different classes of the UCPs are established. Some other expressions of the derivatives of UCPs are given in terms of other orthogonal and non-orthogonal polynomials. The UCPs are also the basis for additional derivative expressions of well-known polynomials. A new linearization formula (LF) of the UCPs that generalizes some well-known formulas is given in a simplified form where no hypergeometric forms are present. Other product formulas of the UCPs with various polynomials are also given. As an application to some of the derived formulas, some definite and weighted definite integrals are computed in closed forms.

Building on the successes of local kernel methods for approximating the solutions to partial differential equations (PDEs) and the evaluation of definite integrals (quadrature/cubature), a local estimate of the error in such approximations is developed. This estimate is useful for determining locations in the solution domain where increased node density (equivalently, reduction in the spacing between nodes) can decrease the error in the solution. An adaptive procedure for adding nodes to the domain for both the approximation of derivatives and the approximate evaluation of definite integrals is described. This method efficiently computes the error estimate at a set of prescribed points and adds new nodes for approximation where the error is too large. Computational experiments demonstrate close agreement between the error estimate and actual absolute error in the approximation. Such methods are necessary or desirable when approximating solutions to PDEs (or in the case of quadrature/cubature), where the initial data and subsequent solution (or integrand) exhibit localized features that require significant refinement to resolve and where uniform increases in the density of nodes across the entire computational domain is not possible or too burdensome. [ABSTRACT FROM AUTHOR]

Even with the powerful statistical parameters derived from the Extreme Gradient Boost (XGB) algorithm, it would be advantageous to define the predicted accuracy to the level of a specific case, particularly when the model output is used to guide clinical decision-making. The probability density function (PDF) of the derived intracranial pressure predictions enables the computation of a definite integral around a point estimate, representing the event's probability within a range of values. Seven hold-out test cases used for the external validation of an XGB model underwent retinal vascular pulse and intracranial pressure measurement using modified photoplethysmography and lumbar puncture, respectively. The definite integral ±1 cm water from the median (DIICP) demonstrated a negative and highly significant correlation (-0.5213±0.17, p< 0.004) with the absolute difference between the measured and predicted median intracranial pressure (DiffICPmd). The concordance between the arterial and venous probability density functions was estimated using the two-sample Kolmogorov-Smirnov statistic, extending the distribution agreement across all data points. This parameter showed a statistically significant and positive correlation (0.4942±0.18, p< 0.001) with DiffICPmd. Two cautionary subset cases (Case 8 and Case 9), where disagreement was observed between measured and predicted intracranial pressure, were compared to the seven hold-out test cases. Arterial predictions from both cautionary subset cases converged on a uniform distribution in contrast to all other cases where distributions converged on either log-normal or closely related skewed distributions (gamma, logistic, beta). The mean±standard error of the arterial DIICP from cases 8 and 9 (3.83±0.56%) was lower compared to that of the hold-out test cases (14.14±1.07%) the between group difference was statistically significant (p<0.03). Although the sample size in this analysis was limited, these results support a dual and complementary analysis approach from independently derived retinal arterial and venous non-invasive intracranial pressure predictions. Results suggest that plotting the PDF and calculating the lower order moments, arterial DIICP, and the two sample Kolmogorov-Smirnov statistic may provide individualized predictive accuracy parameters. [ABSTRACT FROM AUTHOR]

Telescopers for a function are linear differential (resp. difference) operators annihilating the definite integral (resp. definite sum) of this function. They play a key role in Wilf–Zeilberger theory and algorithms for computing them have been extensively studied in the past 30 years. In this paper, we introduce the notion of telescopers for differential forms with D-finite function coefficients. These telescopers appear in several areas of mathematics, for instance parametrized differential Galois theory and mirror symmetry. We give a sufficient and necessary condition for the existence of telescopers for a differential form and describe a method to compute them if they exist. Algorithms for verifying this condition are also given. [ABSTRACT FROM AUTHOR]

New expanding spherical cavity model (ECM) for conical indentation is proposed. For polymeric materials description, the model incorporates isotropic non-monotonic strain hardening. For capturing the indentation size effect (ISE), the model incorporates the strain gradient dependence in yield strength based on lower-order strain gradient plasticity assumptions. Specifically, the forward gradient of the equivalent (accumulated) plastic strain is utilized as a non-local part of the yield strength. To predict the indentation depth-dependent hardness based on the proposed model, it is sufficient to numerically integrate one nonlinear ODE of the first order, and then calculate the definite integral. For the local perfect plasticity model, the hardness is obtained as an analytical expression that differs from known ECMs. The hardness estimate obtained numerically using the proposed model is compared with the experimental ISE data for polycarbonate (PC) and polymethyl methacrylate (PMMA). For the local perfect plasticity model, the formula obtained in the study is compared with the experimental data on the hardness of preliminary work-hardened materials. In both cases, the model shows good agreement with the experimental data. Fitting the experimental data on ISE, we found that intrinsic length scale of PMMA should be near 3 microns and near 9 microns for PC. [ABSTRACT FROM AUTHOR]

In this work, we investigate a class of generalized affine fractal interpolation functions (FIF) with variable parameters, where ordinate scaling is substituted by a real-valued control function. Let S be an iterated function system (IFS) with the attractor G∆, where ∆ is a given data set. We consider an affine transformation ω(∆) of ∆, and we define the IFS Ŝ with the attractor Gω(∆). We give a sufficient condition so that Gω(∆) = ω(G∆). In addition, we compare the definite integrals of the corresponding FIF and study the additivity property. Some examples will be given, highlighting the effectiveness of our results. [ABSTRACT FROM AUTHOR]

AbstractThe theory of normal variance mixture distributions is used to provide elementary derivations of closed-form expressions for the definite integrals ∫0∞x−2νcos⁡(bx)γ(ν,αx2)dx (for ν>1/2, b > 0, α > 0) and ∫0∞x2ν−1cos⁡(bx)Γ(−ν,αx2)dx (for ν > 0, b > 0, α > 0), where γ(a, x) and Γ(a, x) are the lower and upper incomplete gamma functions, respectively. The method of proof is of independent interest and could be used to derive further new definite integral formulas. [ABSTRACT FROM AUTHOR]

Cauchy's residue theorem gives a relatively general form for a complex integral along a simple closed contour. With the help of Cauchy's residue theorem, an appropriate closed contour can be chosen to calculate some abnormal definite integrals that might be very complicated and difficult to solve by conventional methods. This study focuses on four distinct types of definite integrals: integrals involving sine and cosine functions, polynomial functions, exponential functions, and logarithmic functions. The contours chosen are a sector of a circle that involves one or several isolated singularities of the function. The residue at the isolated singularities of the function is then calculated. The value of the residues is substituted in the formula deducted from Cauchy's residue theorem. The integral along the simple closed contour can be expressed in two parts, one along the real axis and the other along the circle. This study demonstrates that Cauchy's Residue Theorem is superior to conventional real analysis methods for evaluating the integrals of different types of complex functions. [ABSTRACT FROM AUTHOR]

Diameter at breast height (DBH) is a crucial parameter for forest inventory. However, accurately estimating DBH remains challenging due to the noisy and incomplete cross-sectional points. To address this, this paper proposed a reliable DBH estimation method using terrestrial LiDAR points through polar coordinate transformation and progressive outlier removal. In this paper, the initial center was initially detected by rasterizing the convex hull, and then the Cartesian coordinates were transformed into polar coordinates. In the polar coordinate system, the outliers were classified as low and high outliers according to the distribution of polar radius difference. Both types of outliers were then removed using adaptive thresholds and the moving least squares algorithm. Finally, DBH was estimated by calculating the definite integral of arc length in the polar coordinate system. Twenty publicly available individual trees were adopted for the test. Experimental results indicated that the proposed method performs better than the other four classical DBH estimation methods. Furthermore, several extreme cases scanned using terrestrial LiDAR in practice, such as cross-sectional points with lots of outliers or larger data gaps, were also tested. Experimental results demonstrate that the proposed method accurately calculates DBH even in these challenging cases. [ABSTRACT FROM AUTHOR]

In this write-up, we focus on pseudo-Hilfer-type fractional order delayed differential equations with bounded definite integral initial conditions on the time interval [0, T]. We begin by establishing relevant lemmas. Then, we derive the solution to the homogeneous Hilfer-type pseudo-fractional order retarded differential equation that satisfies the appropriate initial condition using classical methods. Next, we obtain explicit formulas for solutions to linear inhomogeneous Hilfer-type pseudo-fractional time retarded differential equations with constant coefficients, employing classical ideas. Furthermore, we investigate the existence and uniqueness of the solution of the Hilfer-type pseudo-fractional order delayed differential equation and demonstrate the stability of the given differential equation in the Ulam-Hyers sense on the time interval [0, T]. [ABSTRACT FROM AUTHOR]

This paper draws attention to an underappreciated method for evaluating certain types of definite integrals. The method relies on a substitution in the Eulerian integral for the Legendre gamma function, and has become known in some quarters as a Schwinger parametrization. We present some examples to illustrate the utility of this technique in the hope that by doing so we may convince the reader that it makes a valuable addition to one's integration toolkit. [ABSTRACT FROM AUTHOR]

The optimal tuning of high-order motion parameters has emerged as a promising strategy for actively controlling the kinematics/dynamics of high-speed cam mechanisms. However, accomplishing this task remains challenging with current low-order interpolation or tuning methods. This study proposes an integrated high-order interpolation and tuning methodology for the optimal construction of high-speed motion curves. Initially, an explicit C4-spline interpolant (C4SI) is developed. This interpolant utilizes four-order continuous (C4) splines to synthesize a high-fidelity motion curve that satisfies the predefined motion constraints up to the fourth order, including dimensionless displacement, velocity, acceleration, jerk, and quirk. Concerning the reduction of motion peaks, a unique C4SI-based global kinematics optimization strategy is designed, using the definite integral of the motion curve (free of the time variable) as the objective function. This facile time-free optimization strategy could yield a simultaneous reduction in multiple motion peaks (up to five), which is currently inaccessible with conventional motion tuning strategies. Concerning the improvement of dynamic characteristics, the C4SI-based time-free global dynamics optimization of variable motion parameters is further performed. The results indicate that the optimized fourth-order motion curve offers minimal high-speed transmission error and residual vibration over the whole rise-dwell-return-dwell cycle, which outperforms the standard motion curves and other low-order counterparts. [ABSTRACT FROM AUTHOR]

This study presents an analytical approximation of the definite Chapman integral, applicable to any zenith angle and finite integration limits. The author also presents the asymptotic expression for the definite Chapman integral, which enables an accurate and efficient implementation free of numerical overflows. The maximum relative error in our analytical solution is below 0.5%. [ABSTRACT FROM AUTHOR]

Making use of integration by parts and variable replacement methods, we derive some interesting explicit definite integral formulae involving trigonometric or hyperbolic functions, whose results are expressed in terms of Catalan's constant, Dirichlet's beta function, and Riemann's zeta function, as well as π in the denominator. [ABSTRACT FROM AUTHOR]

In this paper, new results on the α -fractal function with variable parameters are presented. The Weyl–Marchaud variable order fractional derivative of an α -fractal function with variable parameters is examined by imposing certain conditions on the scaling factors. Following the investigation of fractional derivative, the definite integral of the α -fractal function with variable parameters is evaluated for various intervals in the prescribed domain. Finally, an explicit structure for the α -fractal function is provided using the base q representation of numbers. [ABSTRACT FROM AUTHOR]

The given text discusses the use of Riemann approximating sums in calculus courses. It explains how Riemann products can be used to evaluate limits and emphasizes the importance of a specific condition for these products to approach 1 as N approaches infinity. The text also explores the convergence of infinite products and their relationship to infinite series. It provides a heuristic derivation of the limiting values of Riemann products and their connection to Riemann sums. The article concludes by discussing the validity of switching the order of sums and provides a reference for further exploration in real analysis. [Extracted from the article]

For land rent, it is characteristic that it arises as a consequence of capital investment in the purchase of land, which is not a production investment, because capital is not invested for the reason of organizing agricultural production, the main reason of investing capital is to acquire certain ownership of land areas. In this paper, we will present the possibility of solving the problem of rent calculations using the economic application of the definite integral. First, we will show if the integral calcusus is applied in the rent calculation and then in the domain of its calculation. [ABSTRACT FROM AUTHOR]

By computing definite integrals, we shall examine binomial series of convergence rate ± 1 / 2 and weighted by harmonic-like numbers. Several closed formulae in terms of the Riemann and Hurwitz zeta functions as well as logarithm and polylogarithm functions will be established, including a conjectured one made recently by Z.-W. Sun. [ABSTRACT FROM AUTHOR]

This study employs Richardson extrapolation on mixed quadrature rule which is imbraided by Lobatto-4-point rule (RLA(f)) with Gauss-Legendre-3-point rule (RGL3 (f)) to form the extrapolatory quadrature rule (RRLAG13 (f)) Of precision nine. The current rule is numerically verified with six test problems and the bound for the error is resolved with suitable examples. [ABSTRACT FROM AUTHOR]

This study explores calculus students' opportunities to learn the concepts of integral by examining one mathematician's videotaped lessons and the textbook. Results show that both lessons and the textbook introduce important cognitive resources briefly and focus on other units of knowledge. Implications to these results are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we present the analysis of the students' achievements in learning calculus in a dynamic software environment during the Covid19 crisis. Two groups of students, the experimental and the control one, were monitored. Blended learning was applied to the students in the experimental group, with the help of Microsoft Teams and dynamic software GeoGebra, in autumn 2020. All students in the control group learned in the classroom without using GeoGebra in 2019. The comparison between these two groups of first-year students, regarding their calculus test results, is described in this paper. It is interesting that the results of the experimental group were significantly better than the results of students in the control group, despite the fact that the students from the experimental group learned during the Covid19 crisis. [ABSTRACT FROM AUTHOR]