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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]

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]

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]

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]

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]

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]

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.

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]

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]

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]

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]

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 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]

The paper considers a game theory approach to calculating the centrality value of the vertices in a directed graph, based on the number of vertex occurrences in fixed length paths. It is proposed to define vertex centrality as a solution of a cooperative game, where the characteristic function is given as the number of simple paths of fixed length in subgraphs corresponding to coalitions. The concept of integral centrality is introduced as the value of a definite integral of the payoff function. It is shown that this centrality measure satisfies the Boldi–Vigna axioms. [ABSTRACT FROM AUTHOR]

Recently Masjed-Jamei and Koepf obtained generalizations of various classical summation theorems for the 2F1, 3F2, 4F3, 5F4 and 6F5 generalized hypergeometric series. We evaluate a new class of integrals involving generalized hypergeometric function by employing the results given by Masjed-Jamei and Koepf and MacRobert integral, and we give several special cases. [ABSTRACT FROM AUTHOR]

The article investigates a class of polynomials known as convolved Pell polynomials. This class generalizes the standard class of Pell polynomials. New formulas related to convolved Pell polynomials are established. These formulas may be useful in different applications, in particular in numerical analysis. New expressions are derived for the high-order derivatives of these polynomials, both in terms of their original polynomials and in terms of various well-known polynomials. As special cases, connection formulas linking the convolved Pell polynomials with some other polynomials can be deduced. The new moments formula of the convolved Pell polynomials that involves a terminating hypergeometric function of the unit argument is given. Then, some reduced specific moment formulas are deduced based on the reduction formulas of some hypergeometric functions. Some applications, including new specific definite and weighted definite integrals, are deduced based on some of the developed formulas. Finally, a matrix approach for this kind of polynomial is presented. [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]

Background and Objectives: The issue of a missing variable precludes the external validation of many prognostic models. For example, the Liverpool score predicts the survival of patients undergoing surgical therapy for colorectal liver metastases, but it includes the neutrophil–lymphocyte ratio, which cannot be measured retrospectively. Materials and Methods: We aimed to find the most appropriate replacement for the neutrophil–lymphocyte ratio. Survival analysis was performed on data representing 632 liver resections for colorectal liver metastases from 2000 to 2020. Variables associated with the Liverpool score, C-reactive protein, albumins, and fibrinogen were ranked. The rankings were performed in four ways: The first two were based on the Kaplan-Meier method (log-rank statistics and the definite integral I S between two survival curves). The next method of ranking was based on univariate and multivariate Cox regression analyses. Results: The ranks were as follows: the radicality of liver resection (rank 1), lymph node infiltration of primary colorectal cancer (rank 2), elevated C-reactive protein (rank 3), the American Society of Anesthesiologists Classification grade (rank 4), the right-sidedness of primary colorectal cancer (rank 5), the multiplicity of colorectal liver metastases (rank 6), the size of colorectal liver metastases (rank 7), albumins (rank 8), and fibrinogen (rank 9). Conclusions: The ranking methodologies resulted in almost the same ranking order of the variables. Elevated C-reactive protein was ranked highly and can be considered a relevant replacement for the neutrophil–lymphocyte ratio in the Liverpool score. These methods are suitable for ranking variables in similar models for medical research. [ABSTRACT FROM AUTHOR]

The method of brackets is a symbolic approach to the computation of integrals over Rn{{\mathbb{R}}}^{n} based on a deep result by Ramanujan. Its usefulness to obtain new and difficult integrals has been demonstrated many times in the last few years. This article shows that this method is consistent with most classical rules for the computation of integrals, such as the fundamental theorem of calculus, the Laplace transform, the reduction formula for the integration of functions with spherical symmetry, the Cauchy-Schlömilch transformation, and explicit evaluations for multivariate integrals of product of Bessel functions as obtained by Exton and Srivastava. This work is part of a program dedicated to the derivation of solid theoretical grounds for the use of this attractive integration method.

As a tool to deal with the phenomenon in layered or porous media, many theoretical problems need to be improved urgently. In this paper, the definitions of two kinds of fractal derivatives based on scale transformation are developed from the perspective of mathematics, and the four operation rules, chain rules, Rolle theorem, mean value theorem, Taylor formula, variable upper limit integral derivation rule, definite integral basic principle, partial integral formula and double partial integral formula of the fractal version are given. Better still, we derive the fractal Euler-Lagrangian equation by using the variational method, and demonstrate the steps and effectiveness of deriving the fractal nonlinear wave equation based on the fractal Euler-Lagrangian equation through the example of the functionally graded beam. [ABSTRACT FROM AUTHOR]

From the algebraic solution of xm - x + t = 0 for m = 2, 3, 4 and the corresponding solution in terms of hypergeometric functions, we obtain a set of reduction formulas for hypergeometric functions. By differentiation and integration of these results, and applying other known reduction formulas of hypergeometric functions, we derive new reduction formulas of special functions as well as the calculation of some definite integrals in terms of elementary functions. [ABSTRACT FROM AUTHOR]

In this manuscript, the electromagnetic (EM) radiation force (RF) exerted on a perfect electromagnetic conductor (PEMC) sphere by a vortex electromagnetic (VEM) wave with spiral phase distribution had been investigated. The analytical formulation of EM fields is being done in the framework of Mie theory, while the field expressions are being modeled considering the features of VEM wave for PEMC sphere. Initially, the incident field coefficients are evaluated using definite integrals. The scattering coefficients are then determined by imposing boundary conditions at the surface of PEMC sphere i.e., at r = a, leading to a linear system of equations computed via solving matrix. So, a lengthy calculation yields undetermined scattered field coefficients relative to incident field coefficients. The cross-section (Q) factors i.e., scattering (Qsca) and extinction (Qext) have been computed. The influence of sphere size parameter (p) and beam waist radius (wo) versus scattering angle (a) for the RF along with Qsca has been numerically analyzed. As no loss of energy occurs inside PEMC sphere so absorption ((Qabs) = 0), by virtue of the energy conservation principle then, Qext = Qsca. Under specific condition, we implemented present results on Qsca towards light without orbital angular momentum (OAM) i.e., (10) and plane wave for the PEMC sphere. The research work has potential applications = towards particle manipulation, optical technology, and optical tweezers. [ABSTRACT FROM AUTHOR]

In this article, the error analysis for designed test and definite integral in employing uniform experimental design is analogically developed on basis of midpoint rule in rectangle method for assessing definite integral through conducting discretized sampling approximately. It concluded that the discrete sampling-point by means of good lattice point in the evaluations of definite integral and maximum value of a function is promised with higher accuracy, and the predicted entire error of this uniform sampling point method for above problems decreases with the number of sampling points significantly. [ABSTRACT FROM AUTHOR]

Simpson's rule is a numerical method used for approximating the definite integral of a function. In this paper, by utilizing mappings whose second derivatives are bounded, we acquire the upper and lower bounds for the Simpson-type inequalities by means of Riemann–Liouville fractional integral operators. We also study special cases of our main results. Furthermore, we give some examples with graphs to illustrate the main results. This study on fractional Simpson's inequalities is the first paper in the literature as a method. [ABSTRACT FROM AUTHOR]

We calculate some infinite sums containing the digamma function in closed form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as parameter differentiation formulas for the beta incomplete function, reduction formulas of 3 F 2 hypergeometric functions, or a definite integral which does not seem to be tabulated in the most common literature. As an application of certain sums involving the digamma function, we calculated some reduction formulas for the parameter differentiation of the Mittag–Leffler function and the Wright function. [ABSTRACT FROM AUTHOR]

This work proposes a new numerical approach for dealing with fractional stochastic differential equations. In particular, a novel three-point fractional formula for approximating the Riemann–Liouville integrator is established, and then it is applied to generate approximate solutions for fractional stochastic differential equations. Such a formula is derived with the use of the generalized Taylor theorem coupled with a recent definition of the definite fractional integral. Our approach is compared with the approximate solution generated by the Euler–Maruyama method and the exact solution for the purpose of verifying our findings. [ABSTRACT FROM AUTHOR]

In this paper, we aim to present a novel n-point composite fractional formula for approximating a Riemann–Liouville fractional integral operator. With the use of the definite fractional integral's definition coupled with the generalized Taylor's formula, a novel three-point central fractional formula is established for approximating a Riemann–Liouville fractional integrator. Such a new formula, which emerges clearly from the symmetrical aspects of the proposed numerical approach, is then further extended to formulate an n-point composite fractional formula for approximating the same operator. Several numerical examples are introduced to validate our findings. [ABSTRACT FROM AUTHOR]

Mathematics has much negative stigma in society caused by the delivery of less interactive learning materials that makes many people not interested in studying mathematics. Educational games can be an option to create interactive learning media. This study aimed to develop a mobile visual novel game for mathematics subjects, especially definite integral, to increase interest in learning mathematics. The visual novel emphasises interactivity by delivering a story inside a game. This research investigates the level of immersion and behavioural intention to use using the Hedonic-Motivation System Adoption Model (HMSAM). An evaluation was conducted using questionnaires with 46 students as respondents. The results showed that the game increased students' interest in learning mathematics. The students feel immersed when playing this game and desire to play again. This is indicated by the behavioural intention to use level of 84.28%, which states that students strongly agree that the game can increase the application's reusability. In addition, the game obtained 82.09% for immersion, showing that students strongly agree that the game can make the players get carried away. [ABSTRACT FROM AUTHOR]

This paper shows how to apply Leibniz's integral rule to calculate the action variables for the Kepler problem. This method offers an attractive alternative to the usual technique of complex contour integration. The method presented here to calculate definite integrals has a broad scope and is especially suitable for undergraduates who are unfamiliar with complex analysis. [ABSTRACT FROM AUTHOR]

The standard method to evaluate many definite integrals that are encountered in physics is contour integration. Here, we show how these can be evaluated by other means, enlarging the toolbox available to students and enabling the discussion of physical problems where these integrals arise before contour integration is introduced. [ABSTRACT FROM AUTHOR]

In recent years, numerous scholars have investigated the relationship between symmetry and generalized convexity. Due to this close relationship, generalized convexity and symmetry have become new areas of study in the field of inequalities. With the help of fuzzy up and down relation, the class of up and down λ-convex fuzzy-number valued mappings is introduced in this study; and weighted Hermite-Hadamard type fuzzy inclusions are demonstrated for these functions. The product of two up and down λ-convex fuzzy-number valued mappings also has Hermite-Hadamard type fuzzy inclusions, which is another development. Additionally, by imposing some mild restrictions on up and down λ-convex (λ-concave) fuzzy number valued mappings, we have introduced two new significant classes of fuzzy number valued up and down λ-convexity (λ-concavity), referred to as lower up and down λ-convex (lower up and down λ-concave) and upper up and down λ-convex (λ-concave) fuzzy number valued mappings. Using these definitions, we have amassed many classical and novel exceptional cases that implement the key findings. Our proven results expand and generalize several previous findings in the literature body. Additionally, we offer appropriate examples to corroborate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this article, the main objective is construction of fractional Newton–Simpson-type inequalities with the concept of majorization. We established a new identity on estimates of definite integrals utilizing majorization and this identity will lead us to develop new generalized forms of prior estimates. Some basic inequalities such as Hölder's, power-mean, and Young's along with the Niezgoda–Jensen–Mercer inequality have been used to obtain new bounds and it has been determined that the main findings are generalizations of many results that exist in the literature. Applications to the quadrature rule are given as well. We make links between our findings and a number of well-known discoveries in the literature. [ABSTRACT FROM AUTHOR]

The present study is characterized as qualitative and has as background a teaching approach based on a learning environment of Model Analysis and Digital Technologies for the teaching of Differential and Integral Calculus. Its objective is to analyse potentialities and limitations of the mentioned environment to the development of advanced mathematical thinking processes, based on an assignment that aimed to guide the students in analysing a model for the uniformly varied rectilinear motion, in order to approach the concept of definite integral using the softwares Modellus and GeoGebra. Therefore, we analysed the transcriptions of the discussions made by one of the pairs of students participating in an extension course offered in a federal university in 2019. Such analysis identified potentialities of the proposed environment related to the processes of moving between mental representations, translating and generalizing, in which the technologies used played a fundamental role, as well as the design of the tasks. [ABSTRACT FROM AUTHOR]

Copyright of Mendive - Revista de Educacion is the property of Universidad de Pinar del Rio and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Gaceta de la Real Sociedad Matematica Espanola is the property of Real Sociedad Matematica Espanola and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

A positive definite ternary integral quadratic form is called regular if it represents all the positive integers that are represented by its genus, and spinor-regular if it satisfies the weaker condition of representing all the positive integers represented by its spinor genus. There are known to be exactly 29 equivalence classes of primitive ternary forms that are spinor-regular, but not regular. In this paper, an analysis of the positive integers that are represented everywhere locally, but not globally, by forms in each of these 29 equivalence classes is obtained via the general theory of spinor exceptions for genera of ternary quadratic forms. [ABSTRACT FROM AUTHOR]

The development of information measures associated with interval-valued intuitionistic fuzzy values (IVIFVs) has been an important research area over the past few decades. In the literature, the existing decision -making method using IVIFVs has some drawbacks, and the identification degree and information utilization suffer from a gap in the evaluation of alternatives. Therefore, the need for a reliable, useful, and comprehensive decision method is obvious. To obtain more accurate and reliable evaluation results, multiattribute group decision-making (MAGDM) problems, where the same attribute weights given by different decision-makers are different, are studied in this paper. First, the novel operational laws of IVIFVs and a new interval-valued intuitionistic fuzzy weighted arithmetic aggregation operator are defined to overcome the drawbacks of the IIFWA aggregation operator and avoid losing or distorting the original decision information in the process of aggregation. Second, the mean and variance of the possibility degrees of IVIFVs are defined based on the concept of a definite integral. Third, a novel MAGDM method based on the new aggregation operator and the mean and variance of the possibility degrees of IVIFVs is proposed to improve the identification of the evaluation results and ensure the effectiveness of the ranking order. Finally, the effectiveness and practicability of the proposed method are verified by an air combat training accuracy assessment example. This example can be used to assist decision-makers in evaluating air combat training hits in a timely and efficient manner, providing an objective, scientific basis for the realization and application of air combat training hit assessment and a new method and idea for MAGDM problems in an interval-valued intuitionistic fuzzy environment. [ABSTRACT FROM AUTHOR]

The major goal of the current article is to create new formulas and connections between several well-known polynomials and the Euler polynomials. These formulas are developed using some of these polynomials' well-known fundamental characteristics as well as those of the Euler polynomials. In terms of the Euler polynomials, new formulas for the derivatives of various symmetric and non-symmetric polynomials, including the well-known classical orthogonal polynomials, are given. This leads to the deduction of several new connection formulas between various polynomials and the Euler polynomials. As an important application, new closed forms for the definite integrals for the product of various symmetric and non-symmetric polynomials with the Euler polynomials are established based on the newly derived connection formulas. [ABSTRACT FROM AUTHOR]

This paper deals with Lopatinskii type boundary value problem (bvp) for the (poly) harmonic differential operators. In the case of Robin bvp for the Laplace equation in the ball B 1 a Green function is constructed in the cases c > 0 , c ∉ − N , where c is the coefficient in front of u in the boundary condition ∂ u ∂ n + c u = f . To do this a definite integral must be computed. The latter is possible in quadratures (elementary functions) in several special cases. The simple proof of the construction of the Green function is based on some solutions of the radial vector field equation Λ u + c u = f . Elliptic boundary value problems for Δ m u = 0 in B 1 are considered and solved in Theorem 2. The paper is illustrated by many examples of bvp for Δ u = 0 , Δ 2 u = 0 and Δ 3 u = 0 in B 1 as well as some additional results from the theory of spherical functions are proposed. [ABSTRACT FROM AUTHOR]

This paper presents new results of Bernoulli polynomials. New derivative expressions of some celebrated orthogonal polynomials and other polynomials are given in terms of Bernoulli polynomials. Hence, some new connection formulas between these polynomials and Bernoulli polynomials are also deduced. The linking coefficients involve hypergeometric functions of different arguments that can be summed in some cases. Formulas that express some celebrated numbers in terms of Bernoulli numbers are displayed. Based on the new connection formulas between different polynomials and Bernoulli polynomials, along with some well-known integrals involving these polynomials, new closed forms for some definite integrals are given. [ABSTRACT FROM AUTHOR]