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Whole-class mathematical discussion in a problem-solving activity is recognized as a powerful pedagogical activity but also a challenge for teachers who must consider several difficulties that learners might face, particularly in terms of an overload of Working Memory and Executive Functions. This study investigates how the use of a digital platform (Padlet) can support participatory and inclusive mathematical classroom discussion. We proposed a teaching experiment based on graphical tasks anticipating integral calculus to grade 13 students, and we examined how the use of the digital platform plays a role in the construction and interpretation of new mathematical objects emerging from the activity. The use of Instrumental Genesis and Double Instrumental Genesis frameworks allowed us to make the affordances of the tool emerge. As a result, we got evidence of how mathematical discussion may develop as a network of interactions, feedback, and connection of input and discuss examples of how active participation and inclusion are enhanced by the tool affordances. Indeed, the digital platform allowed easy interaction, with many ways to represent and express the ongoing evolution of personal and shared meanings and the possibility to manage the time of the activity. This fostered students' participation and students which did not participate in previous discussions were actively engaged in it. [ABSTRACT FROM AUTHOR]

The paper is devoted to the line integral topic belonging to the section of vector calculus in Mathematical Analysis applied to the undergraduate Mechanical Engineering program. An efficient way of teaching line integrals is proposed and developed based on the elementary work/force principles. By this way, the mathematical concept of the line integral is supposed to be learned in harmony with the elementary mechanics to appreciate its diversity and to set up the right idea about the scientific area covered by Mechanical Engineering and related academic and technical fields. In terms of practical training, the research also offers two new techniques of analytical calculus for line integrals containing singularities and provides a new coherent engineering approach to deal with vector fields as integrands in line integrals. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to define Katugampola fractional integrals in multiplicative calculus. A novel method for generalizing the multiplicative fractional integrals is the Katugampola fractional integrals in multiplicative calculus. The multiplicative Hadamard fractional integrals are also novel findings of this research and may be derived from the special situations of Katugampola fractional integrals. These integrals generalize to multiplicative Riemann–Liouville fractional integrals and multiplicative Hadamard fractional integrals. Moreover, we use the Katugampola fractional integrals to prove certain new Hermite–Hadamard and trapezoidal-type inequalities for multiplicative convex functions. Additionally, it is demonstrated that several of the previously established inequalities are generalized from the newly derived inequalities. Finally, we give some computational analysis of the inequalities proved in this paper. [ABSTRACT FROM AUTHOR]

Most introductory physics courses derive the energy densities of the static electric and magnetic field for the simple cases of parallel plate capacitors and infinitely long solenoids. Authors then inform readers, usually without proof, that the energy density equations derived for the simple cases are actually the correct equations for all capacitors and inductors, regardless of their shape. The proof of the general case is typically omitted because it involves the differential calculus of vector fields. In view of this, we provide a derivation for the energy density only based on integral calculus for capacitors and inductors of any kind. The derivation, albeit seemingly complicated at first, is conceptually simple enough for introductory physics courses and does not require any knowledge of the differential calculus of vector fields. Editor's Note: The authors present a clever derivation of the static electric and magnetic field energy densities for capacitors and inductors of any shape. The derivations generalize the simple and often discussed cases of infinite parallel plate capacitors and infinitely long solenoids while requiring only a conceptual understanding of integral vector calculus. The discussion is suitable for introductory physics courses and will be of interest for teachers seeking new ways to introduce the challenging concept of field energy density. [ABSTRACT FROM AUTHOR]

The problem of equivalence between differential and integral problems is absolutely crucial when applying solution methods based on operators and their properties in function spaces. In this paper, we complement the solution of this important problem by considering the case of general derivatives and integrals of fractional order for vector functions for weak topology. Even if a Caputo differential fractional order problem has a right-hand side that is weakly continuous, the equivalence between the differential and integral forms may be affected. In this paper, we present a complete solution to this problem using fractional order Pettis integrals and suitably defined pseudo-derivatives, taking care to construct appropriate Hölder-type spaces on which the operators under study are mutually inverse. In this paper, we prove, in a number of cases, the equivalence of differential and integral problems in Hölder spaces and, by means of appropriate counter-examples, investigate cases where this property of the problems is absent. [ABSTRACT FROM AUTHOR]

The training of mathematics teachers in universities in Colombia has as a transversal axis the resolution of problems based on their social and cognitive mission pillars. In this sense, this study relates to the re-signification and construction of the concept of definite integral (DI) (integral calculus) through socio-epistemological studies, action researches, and the typology of didactic situations. The results are obtained through content analysis, didactic sequences (GeoGebra), and a discussion group. The above allows us to conclude that the validation of meanings, historical contexts, and associated social practices leads to the construction of the concept of DI as a model of mathematical analysis. This structuring of knowledge from its epistemological framework enables the exploration of mathematical objects from the basic notions that emerge in the history of humanity and didactic processes that reconstruct the evolution of the concept in society. [ABSTRACT FROM AUTHOR]

We propose a simple geometric optics analog of a gravitational lens with a refractive index equal to one at large distances and scaling like n (r) 2 = 1 + C 2 / r 2 , where C is a constant. We obtain the equation for ray trajectories from Fermat's principle of least time and the Euler equation. Our model yields a very simple ray trajectory equation. The optical rays bending, reflecting, and looping around the lens are all described by a single trigonometric function in polar coordinates. Optical rays experiencing fatal attraction are described by a hyperbolic function. We use our model to illustrate the formation of Einstein rings and multiple images. Editor's Note: This article describes a simple theoretical model for gravitational lensing. The authors analyze a graded index of refraction that reproduces the behavior for light passing near the event horizon of a black hole. The mathematical simplicity of the model permits exploration of the effects of gravitational lensing—including bending, reflection, and the formation of Einstein rings—using only integral calculus and Fermat's principle. The authors illustrate many interesting lensing phenomena with 2D and 3D graphics. The model described in this paper could be introduced as a "theoretical toy model" to complement classroom demonstrations of gravitational lensing such as a "logarithmic lens" or the stem of a wine glass, making gravitational lensing and its use in modern astrophysics accessible to introductory physics students. [ABSTRACT FROM AUTHOR]

Interval-valued maps adjust integral inequalities using different types of ordering relations, including inclusion and center-radius, both of which behave differently. Our purpose was to develop various novel bounds and refinements for weighted Hermite-Hadamard inequalities as well as their product form by employing new types of fractional integral operators under a cr-order relation. Mostly authors have used inclusion order to adjust inequalities in interval maps, but they have some flaws, specifically they lack the property of comparability between intervals. However, we show that under cr-order, it satisfies all relational properties of intervals, including reflexivity, antisymmetry, transitivity, and comparability and preserves integrals as well. Furthermore, we provide numerous interesting remarks, corollaries, and examples in order to demonstrate the accuracy of our findings. [ABSTRACT FROM AUTHOR]

In this paper, we study the singularities of Feynman integrals by compactifying the integration domain as well as the ambient space of these integrals, by embedding them in higher-dimensional space. In this compactified space, the singularities occur due to the meeting of compactified propagators at non-general position. The present analysis, which had been previously used only for the singularities of second type, is used to study other kinds of singularities viz threshold, pseudo-threshold and anomalous threshold singularities. We study various one-loop and two-loop examples and obtain their singularities. We also present observations based on results obtained, that allow us to determine whether the singularities lie on the physical sheet or not for some simple cases. Thus, this work at the frontier of our knowledge of Feynman integral calculus sheds insight into the analytic structure. [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]

This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wiener functional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index H < 1 / 2 , without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method. [ABSTRACT FROM AUTHOR]

The correct derivation of integral inequalities on fuzzy-number-valued mappings depends on applying fractional calculus to fuzzy number analysis. The purpose of this article is to introduce a new class of convex mappings and generalize various previously published results on the fuzzy number and interval-valued mappings via fuzzy-order relations using fuzzy coordinated ỽ-convexity mappings so that the new version of the well-known Hermite–Hadamard (H-H) inequality can be presented in various variants via the fractional integral operators (Riemann–Liouville). Some new product forms of these inequalities for coordinated ỽ-convex fuzzy-number-valued mappings (coordinated ỽ-convex FNVMs) are also discussed. Additionally, we provide several fascinating non-trivial examples and exceptional cases to show that these results are accurate. [ABSTRACT FROM AUTHOR]

This study aims to investigate the properties of fractional calculus theory (FCT) in the complex domain. We focus on the relationship between the theories of special functions (SFT) and FCT, which have seen recent advancements and have led to various successful applications in fields such as engineering, mathematics, physics, biology, and other allied disciplines. Our main contribution is the development of a special function, specifically the confluent hypergeometric function (CHF) on the complex domain. By deriving various implementations of fractional order derivatives and integral operators using this function, we present a new class of special functions combining certain cases of Mittag-Leffler and confluent hypergeometric functions. Moreover, a new numerical technique for solving linear and nonlinear multi-order fractional differential equations has been developed using the proposed class of functions and the point collocation method. Graphical results are shown to demonstrate the efficacy of this proposed technique and its applicability. [ABSTRACT FROM AUTHOR]

In this paper we study the everywhere Hölder continuity of the minima of the following class of vectorial integral funcionals: with some general conditions on the density . We make the following assumptions about the function . Let be a bounded open subset of , with , and let be a Carathéodory function, where and with . We make the following growth conditions on : there exists a constant such that for a.e. , for every and every with , and with , for a.e. , , and . Assuming that the previous growth hypothesis holds, we prove the following regularity result. If is a local minimizer of the previous functional, then for every , with . The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude Hölder continuity. [ABSTRACT FROM AUTHOR]

This study identifies the reasons for geodynamics variability of the coastal system within two cliff-shore sections of the southern Baltic Sea (SBS). The comparative analysis included distinct moraines and their foregrounds near the open sea (S1) and within the Gulf of Gdańsk (S2). Short-term trends indicate a direct link between landslide occurrence and increased cliff retreat. Long-term (total) values were obtained by developing the 4F MODEL for large-scale applications, based on the analysis of remote sensing and hydroacoustic data (to determine the extent of shore platforms), the modelling of higher-order polynomial functions describing their extent, followed by the integral calculus of the indicated functions within the open-source Desmos environment. The retreat dynamics for individual landslides (S1) was an order of magnitude higher (m/yr) than the average for the whole cliff section (0.17 ± 0.008 m/yr), which correlates well with medium- and long-term development tendencies and recession dynamics, revealed by the numerical modelling method, since approximately 8 ka b2k, years before 2000 CE (at S1 = 0.17 ± 0.020 m/yr, at S2 = 0.11 ± 0.005 m/yr). While the approach described in this paper can reveal, project, and simulate the dynamics of past and future trends within other cliffed coasts shaped in tideless conditions, it also proves stable moraine erosional responses to sea-level rise since the Mid-Holocene. [ABSTRACT FROM AUTHOR]

Nonlinear Fractional Volterra integral equations (FVIEs) of the first kind present challenges due to their intricate nature, combining fractional calculus and integral equations. In this research paper, we introduce a novel method for solving such equations using Leibniz integral rules. Our study focuses on a thorough analysis and application of the proposed algorithm to solve fractional Volterra integral equations. By using Leibniz integral rules, we offer a fresh perspective on handling these equations, shedding light on their fundamental properties and behaviours. As a result of this study, we anticipate contributing distinctively to the broader development of analytical tools and techniques. By bridging the gap between fractional calculus and integral equations, our approach not only offers a valuable computational methodology but also paves the way for new insights into the application domains in which such equations arise. [ABSTRACT FROM AUTHOR]

In this work, we study the existence of solutions of nonlinear fractional coupled system of $\varphi $ -Hilfer type in the frame of Banach spaces. We improve a property of a measure of noncompactness in a suitably selected Banach space. Darbo's fixed point theorem is applied to obtain a new existence result. Finally, the validity of our result is illustrated through an example. [ABSTRACT FROM AUTHOR]

We give explicit presentations of the integral equivariant cohomology of the affine Grassmannians and flag varieties in type A, arising from their natural embeddings in the corresponding infinite (Sato) Grassmannian and flag variety. These presentations are compared with results obtained by Lam and Shimozono, for rational equivariant cohomology of the affine Grassmannian, and by Larson, for the integral cohomology of the moduli stack of vector bundles on. [ABSTRACT FROM AUTHOR]

We discuss the class of functions, which are well approximated on compacta by the geometric mean of the eigenvalues of a unital (completely) positive map into a matrix algebra or more generally a type $II_1$ factor, using the notion of a Fuglede–Kadison determinant. In two variables, the two classes are the same, but in three or more noncommuting variables, there are generally functions arising from type $II_1$ von Neumann algebras, due to the recently established failure of the Connes embedding conjecture. The question of whether or not approximability holds for scalar inputs is shown to be equivalent to a restricted form of the Connes embedding conjecture, the so-called shuffle-word-embedding conjecture. [ABSTRACT FROM AUTHOR]

In this article, we construct some examples of noncommutative projective Calabi–Yau schemes by using noncommutative Segre products and quantum weighted hypersurfaces. We also compare our constructions with commutative Calabi–Yau varieties and examples constructed in Kanazawa (2015, Journal of Pure and Applied Algebra 219, 2771–2780). In particular, we show that some of our constructions are essentially new examples of noncommutative projective Calabi–Yau schemes. [ABSTRACT FROM AUTHOR]

If X is a topological space and Y is any set, then we call a family $\mathcal {F}$ of maps from X to Y nowhere constant if for every non-empty open set U in X there is $f \in \mathcal {F}$ with $|f[U]|> 1$ , i.e., f is not constant on U. We prove the following result that improves several earlier results in the literature. If X is a topological space for which $C(X)$ , the family of all continuous maps of X to $\mathbb {R}$ , is nowhere constant and X has a $\pi $ -base consisting of connected sets then X is $\mathfrak {c}$ -resolvable. [ABSTRACT FROM AUTHOR]

The structure of Borel reducibility for equivalence relations on $\omega _1$ is determined. [ABSTRACT FROM AUTHOR]

Following Faber–Pandharipande, we use the virtual localization formula for the moduli space of stable maps to $\mathbb {P}^{1}$ to compute relations between Hodge integrals. We prove that certain generating series of these integrals are polynomials. [ABSTRACT FROM AUTHOR]

The application of fractional calculus to obtain dynamic models for supercapacitors represents an alternative approach to obtaining simpler and more accurate models. This paper presents a model for the supercapacitor in the time domain, based on the use of the fractional or non-integer order integral. This fractional model is compared with the conventional simple model, which is typically used in industrial applications. This fractional integral-based model provides satisfactory fits in relation to the number of parameters used in the model. Furthermore, an interpretation of the effect of the application of fractional integration is presented for constant current charging and discharging processes at constant current, using the Riemann–Liouville definition for the non-integer order integral. Supercapacitors are devices that exhibit non-linear behavior, with a distinct charging and discharging operation. There are several methods of dynamic analysis for the characterization of supercapacitors. The information extracted from these methods is essential for understanding the behavior of supercapacitors and, thus, ensuring that processes involving supercapacitors are as efficient as possible. This paper presents a dynamic analysis based on charge and discharge operations with constant currents. The conclusion is that the fractional model provides fairly accurate fits. [ABSTRACT FROM AUTHOR]

Local minimizers of integral functionals of the calculus of variations are analyzed under growth conditions dictated by different lower and upper bounds for the integrand. Growths of non-necessarily power type are allowed. The local boundedness of the relevant minimizers is established under a suitable balance between the lower and the upper bounds. Classical minimizers, as well as quasi-minimizers are included in our discussion. Functionals subject to so-called p , q -growth conditions are embraced as special cases and the corresponding sharp results available in the literature are recovered. [ABSTRACT FROM AUTHOR]

We consider the anisotropic integral functional of the calculus of variations ... where ci≥0 and 2 ≤pi≤pi+1 for every i = 1 ,...n-1. We exhibit a minimizer functional, for an opportune choice of the exponents pi, which turns out to be bounded everywhere and Lipschitz continuous (or even of class C¹) in an opportune subset of Ω. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study to fractional calculus for class of Ψ function. The present study is designed to study generalized fractional derivatives and find their generalized transforms called Ψ-Laplace transform and Ψ-Sumudu transform. Moreover, find the analytical solutions of some applications in physics the form of generalized fractional derivatives by transform technique. [ABSTRACT FROM AUTHOR]

This paper deals with the quasi-instantaneous determination (in a single-step response time) of apparent, active, and reactive (i.e., blind and distortion) power mean values including the total power factor, total harmonic distortion, and phase shift of fundamentals of a power electronic and electrical system (PEES) using the ip-iq method, which is the main contribution of the paper. The power components' mean values are investigated during the transient and steady states. The power components' mean values can be determined directly from phase current and voltage quantities, using an integral calculus over one period within the next calculation step and using moving average and moving rms techniques (or digital filtering). Consequently, the power factor can be evaluated with known values of a phase shift of fundamentals (using a Fourier analysis). The results of this study show how a distortion power component during transients is generated even under a harmonic supply and linear resistive–inductive load. The paper contains a theoretical base, modeling, and simulation for the three and single phases of the transients in power electronic systems. The worked-out results can be used to determine and size any PES. The presented approach brings a detailed time waveform verified by simulations in Matlab/Simulink 2022a and the Real-time HW Simulator Plecs RT Box 1. [ABSTRACT FROM AUTHOR]

In this study, a new generalized fractal–fractional (FF) derivative is proposed. By applying this definition to some elementary functions, we show its compatibility with the results of the FF derivative in the Caputo sense with the power law. The main elements of classical differential calculus are introduced in terms of this new derivative. Thus, we establish and demonstrate the basic operations with derivatives, chain rule, mean value theorems with their immediate applications and inverse function's derivative. We complete the theory of generalized FF calculus by proposing a notion of integration and presenting two important results of integral calculus: the fundamental theorem and Barrow's rule. Finally, we analytically solve interesting FF ordinary differential equations by applying our proposed definition. [ABSTRACT FROM AUTHOR]

It is the purpose of the present paper to obtain certain fixed point outcomes in the sense of C*-algebra valued metric spaces. Here, we present the definitions of the gauge function, the Bianchini-Grandolfi gauge function, a-admissibility, and (α, β)-admissible Geraghty contractive mapping in the sense of C*-algebra. Using these definitions, we define (α, β)-Bianchini-Grandolfi gauge contraction of type I and type II. Next, we prove our primary results that the function satisfying our contraction condition has to have a unique fixed point. We also explain our results using examples. Additionally, we discuss some consequent results that can be easily obtained from our primary outcomes. Finally, there is a useful application to integral calculus. [ABSTRACT FROM AUTHOR]

In this paper, we first construct an integral identity associated with tempered fractional operators. By using this identity, we have found the error bounds for Simpson's second formula, namely Newton–Cotes quadrature formula for differentiable convex functions in the framework of tempered fractional integrals and classical calculus. Furthermore, it is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature. [ABSTRACT FROM AUTHOR]

We study Γ-convergence of nonconvex integrals of the calculus of variations in strongly connected sets when the integrands do not have polynomial growth and can take infinite values. Applications to homogenization of unbounded integrals in strongly perforated sets are also developed. [ABSTRACT FROM AUTHOR]

It has been shown that the findings of d -metric spaces may be deduced from S -metric spaces by considering d ϖ , ϰ = Λ ϖ , ϖ , ϰ . In this study, no such concepts that translate to the outcomes of metric spaces are considered. We establish standard fixed point theorems for integral type contractions involving rational terms in the context of complete S -metric spaces and discuss their implications. We also provide examples to illustrate the work. This paper's findings generalize and expand a number of previously published conclusions. In addition, the abstract conclusions are supported by an application of the Riemann-Liouville calculus to a fractional integral problem and a supportive numerical example. [ABSTRACT FROM AUTHOR]

In this paper, we use differential forms to prove a number of theorems of integral vector calculus that are rarely found in textbooks. Two of them, as far as the author knows, have not been published before. Some possible applications to problems in physics are shared including a general approach for computing net forces and torques on current-carrying loops that yields insights that are not evident from the standard approach. Readers who are familiar with differential forms will enjoy seeing how they can be employed to prove several new vector calculus identities. But even readers who do not follow those derivations will benefit from seeing how these identities can be employed to find forces and torques on current-carrying loops. [ABSTRACT FROM AUTHOR]

In undergraduate calculus of several variables, double and triple integrals are usually defined as limits of certain Riemann sums. The existence of the integral, as well as the integration formulas, are stated without proof since they require more advanced mathematics. In this article, an alternative and straightforward way of defining multiple integrals is proposed where the usual integration formulas follow as a direct application of the definition. The underlying idea is to map the arbitrary region of integration to an n-dimensional open interval, and integration over the latter is defined via the usual iterated integral. Moreover, the substitution formula is taken as a definition. Numerous illustrative examples are provided. [ABSTRACT FROM AUTHOR]

For learning mathematics, mathematical skills are needed, some of which are mathematical reasoning skills, mathematical critical thinking skill, and mathematical literacy skills. This research aims to obtain information regarding the impact of mathematical reasoning and critical thinking skills on mathematical literacy skills. This research used a quantitative approach using an associative method with a correlational technique. The sample of this research was comprised 51 students who took integral calculus course in the Department of Mathematics and Mathematics Education, Faculty of Mathematics and Science in Universitas Negeri Jakarta, which were collected randomly using simple random sampling. The statistical analysis used in this research was multiple regression analysis. The results of this research showed that: 1) There was a positive impact of mathematical reasoning on mathematical literacy. 2) There was a positive impact of mathematical critical thinking skill on mathematical literacy. 3) There was an impact of both mathematical reasoning skill and mathematical critical thinking skill together on mathematical literacy. Further research is needed related to the impact of mathematical reasoning and critical thinking skills on mathematical literacy skills reviewed from the student's initial mathematical skill. [ABSTRACT FROM AUTHOR]

The present work investigates Bloch equations arising in magnetic resonance with Caputo and Caputo Fabrizio derivatives. Banach's fixed point approach is used to construct the existence theory for the model's solution. Also, the stability of the solution is established by Ulam–Hyers conditions. A novel 3-step iterative method is used for the considered model's numerical simulation with Caputo and Caputo Fabrizio derivative. This iterative method is formulated by combining Lagrange's interpolation with the fundamental theorem of integral calculus. The proposed method's error estimate is provided. The simulation results are displayed in tabulated and graphical form for distinct values of fractional order. The results demonstrate how the proposed method is accurate and appropriate for analysing fractional Bloch model. Further, this technique can also approximate the solution of other equations arising in engineering physics and quantum fields. [ABSTRACT FROM AUTHOR]

The fractional Lindley distribution is used to model the distribution of perturbations in count data regressions, which allow for dealing with widely dispersed data. It is obtained from the non-fractional Lindley distribution by replacing the support $\mathbb{T} = {\mathbb{R}^ + }$ T = R + by ${\mathbb{T}} = {\mathbb{N}}\backslash \{ 0\} $ T = N ∖ { 0 } and applying time scale theory, whose ambition is to unify the theories of difference equations and differential equations, integral and differential calculus, and the calculus of finite differences. It thus provides a framework for the study of dynamical systems in discrete-continuous time. Delta moments are discrete-time Laplace transforms of the frequency function of the fractional Lindley distribution. The parameter of the fractional Lindley distribution is estimated by least squares, weighted least squares, maximum likelihood, moments, and proportions. The moment estimator always exists, so that delta moments result from the nabla Laplace transform of the frequency function of the fractional Lindley distribution. The maximum likelihood estimates have the least mean-square errors. The proportion method works satisfactorily only when the mode of the distribution is null and the proportion of zeros is high. A simulation allows for quantifying the mean-square errors associated with the estimators. A count regression based on the fractional Lindley distribution with data on the total number of stays after hospital admission among U.S. residents aged 65 and over shows that the Akaike information criteria is significantly lower than with the uniform Poisson and Poisson regressions. [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]

In this study, the analysis focuses on a transient thermoelastic problem in an isotropic homogeneous elastic plate that is exposed to heat loading within the framework of the fractional-order theory. The sectional heat supply is applied on both the front edge and the farthest edge of the rectangular plate. The integral transformation was considered as a means to solve the main governing equations. The Mittag-Leffler function is utilized to express the analytical solution for temperature change, displacement, and stress response. The investigation also encompasses the study of thermoelastic behaviors in a plate featuring a central crack. The stress intensity factors at the fracture tip are determined numerically using the weight function method in this proposed solution. The findings are depicted using numerical computations, considering the material as a media, and visually represented in graphical form. [ABSTRACT FROM AUTHOR]

In this paper, we first establish a version of the Feynman-Kac formula for the tempered fractional general diffusion equation$ \begin{align} \partial^{\beta, \eta}_{t} u(t,x) = \mathfrak{L}u(t,x) +b(t)u(t,x),\; \; x\in\mathcal{X},\; t\geq0, \end{align} $with initial value $ f $ belonging to a Banach space $ (\mathbb{B}, \|\cdot\|) $, where $ \partial^{\beta, \eta}_{t} $ denotes the Caputo tempered fractional derivative with order $ \beta\in(0,1) $ and tempered parameter $ \eta>0 $, $ b(t) $ is a bounded and continuous external potential on $ [0, \infty) $, $ \mathfrak{L} $ is the infinitesimal generator of a general time-homogeneous strong Markov process $ \{X_{t}\}_{t\geq0} $, and $ \mathcal{X} $ denotes a Lusin space that is a topological space being homeomorphic to a Borel subset of a compact metric space. By using the properties of the tempered $ \beta $-stable subordinator $ S_{\beta,\eta}(t) $ and the inverse tempered $ \beta $-stable subordinator $ D_{\beta,\eta}(t) $, and the stochastic calculus for the stochastic integral driven by $ D_{\beta,\eta}(t) $, we show that the Feynman-Kac representation $ u(t,x) $ defined by$ \begin{align} u(t,x) = {\mathbb{E}}^{x}\bigg[f(X_{D_{\beta,\eta}(t)}) e^{\int_{0}^{t}b(r)dD_{\beta,\eta}(r)}\bigg] \end{align} $is the unique mild and weak solutions to the tempered fractional general diffusion equation. From the Feynman-Kac formula, we further show the continuity of the solution with respect to time based on the integral properties of the Mittag-Leffler function and differential formula of covariance for $ D_{\beta,\eta}(t) $. By exploring the scaling property of $ D_{\beta,\eta}(t) $, the explicit order is also presented for the continuity of the solution with respect to tempered parameter $ \eta $. [ABSTRACT FROM AUTHOR]

In this university study, sections of an integral calculus course were randomly assigned to either a control or treatment group for a lesson on volumes of revolution. Both groups were given similar collaborative tasks, but only the students in the treatment group were given access to a set of 3D manipulatives. Pre- and post-assessments were administered to measure student achievement and confidence on the tasks comparing results for the control and treatment groups as well as for the male and female students. No statistically significant differences in student achievement were detected for the control or treatment group or by reported gender on either an immediate posttest or a delayed posttest. There was statistical significance in confidence after engaging in the 3D manipulative tasks favoring the treatment group. However, further inspection by gender revealed that while males in the treatment group were more likely to report higher confidence ratings than males in the control group, the reverse was true for females. [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]

We investigated the effects of a major revision of the differential and integral calculus curriculum, the primary goal of which was to improve STEM retention. The revamped curriculum has greater emphasis on the power of computing to help visualize patterns and gain insights to better prepare students for STEM majors, and less emphasis on traditional and theoretical calculus content, such as the limit definition of the derivative. We investigated a comparison between 338 students who had taken the traditional sequence and 328 who had taken the revamped sequence during the Fall 2020 and Spring 2021 semesters. STEM retention, i.e., following through with the initial intent to major in a STEM field, was enhanced by 7% in the revamped group. STEM majors in the traditional group were found to over-perform in their other STEM classes by a 0.18 GPA margin, while non-STEM majors in the revamped group over-performed in their other STEM classes by a 0.17 GPA margin. Both differences are statistically significant. Focus groups were also conducted to gather and synthesize student perceptions. Results may encourage similar innovations to core math curricula in other universities to foster enhanced STEM retention. [ABSTRACT FROM AUTHOR]

Henri Bergson's philosophy is centered on forming a concept of lived time or durée, which he saw as a process of continuous variation and flux. He believed that the study of time should be the foundation of philosophy. By studying time, we find an integration of concrete, infinite, qualitative multiplicity within consciousness that we should use to understand the essence of reality. I show that his insights into the reality of duration come directly from a metaphysical or phenomenological interpretation of integral and differential calculus. Drawing on the recently published lectures from 1902–1905, I show how An Introduction to Metaphysics schematizes the methodology of Matter and Memory and Time and Free Will by means of this mathematical analogy. To Bergson, the concepts of calculus are not mere metaphors; they unleash certain metaphysical implications that problematize the mind's relation to movement, life, consciousness, time, and consequently, reality itself. [ABSTRACT FROM AUTHOR]

This paper revealed new fractional calculus applications of special functions in the geometric function theory. The aim of the study presented here was to introduce and begin the investigations on a new fractional calculus integral operator defined as the fractional integral of order λ for the Bessel function of the first kind. The focus of this research was on obtaining certain geometric properties that give necessary and sufficient univalence conditions for the new fractional calculus operator using the methods associated to differential subordination theory, also referred to as admissible functions theory, developed by Sanford S. Miller and Petru T. Mocanu. The paper discussed, in the proved theorems and corollaries, conditions that the fractional integral of the Bessel function of the first kind must comply in order to be a part of the sets of starlike functions, positive and negative order starlike functions, convex functions, positive and negative order convex functions, and close-to-convex functions, respectively. The geometric properties proved for the fractional integral of the Bessel function of the first kind recommend this function as a useful tool for future developments, both in geometric function theory in general, as well as in differential subordination and superordination theories in particular. [ABSTRACT FROM AUTHOR]