Definite Integrals Research Articles

Evaluation of the Second Virial Coefficient for the Mie Potential Using the Method of Brackets

The second virial coefficient for the Mie potential is evaluated using the method of brackets. This method converts a definite integral into a series in the parameters of the problem, in this case this is the temperature T. The results obtained here are consistent with some known special cases, such as the Lenard-Jones potential. The asymptotic properties of the second virial coefficient in molecular thermodynamic systems and complex fluid modeling are described in the limiting cases of T → 0 and T → ∞ .

Performance Analysis and Parallel Scalability of Numerical Methods for Fractional-in-Space Diffusion Problems with Adaptive Time Stepping

We study numerical methods and algorithms for time-dependent fractional-in-space diffusion problems. The considered anomalous diffusion is modelled by the fractional Laplacian (−Δ)α, 0<α<1, following the integral definition. Fractional diffusion is non-local, and the finite element method (FEM) discretization in space leads to a dense stiffness matrix. It is well known that numerically solving such non-local boundary value problems is expensive. Difficulties increase significantly when the problem is time-dependent. The aim of the article is to develop computationally efficient methods and algorithms. There are two main features of our approach. Hierarchical semi-separable (HSS) compression is applied for an approximate solution of the arising linear systems. For time discretization, we use an adaptive forward–backward Euler scheme. The properties of the composite algorithm thus obtained are investigated. In particular, the block representation of HSS compression allowed us to upgrade the HSS solver to efficiently handle varying diagonally perturbed transition matrices corresponding to changing time steps. The contribution of the paper is threefold. The methods are completely constructive, which allows for a clearly structured description of the algorithms. A theoretical estimate of the computational complexity is presented. It shows the advantages of the adaptive time stepping in combination with the HSS solver. Theoretical results are supported by representative numerical experiments. Both sequential and parallel scalability and efficiency are analyzed. The presented results provide convincing proof of the concept of the proposed methods and algorithms.

Z22-graded supersymmetry via superfield on minimal Z22-superspace

Abstract A superfield formalism for the minimal Z2 2-graded version of supersymmetry is developed. This is done by using the recently introduced definition of integration on the minimal Z2 2-superspace. It is shown that one may construct Z2 2-supersymmetric action by the procedure similar to the standard supersymmetry. 
However, the Lagrangian obtained has very general interaction terms, which give rise to a Z2 2-graded extension of many known theories defined in two-dimensional spacetime. 
As an illustration, we will give a Z2 2-extension of the sine-Gordon model different from the one already discussed in the literature.

2022 개정 교육과정에서 넓이로서의 정적분 정의에 따른 교수학적 난점 및 대안 탐색

This study aims to analyze the expected didactic difficulties and prepare complementary measures for the concept of definite integral, defined as 'the area between the curve and the -axis' in the 2022 Curriculum, following the 2015 Curriculum. To this end, the development of the area concept was examined by comparing the 2022 and 2015 Curricula and analysing textbooks. The analysis revealed difficulties in understanding definite integral as the area due to the limited development of the area concept, such as only justifying the method for calculating the area of a rectangle when the sides are natural numbers. Additionally, it was found that the situation material for questioning that had already been developed in the 2015 Curriculum, introducing the 'relationship between definite integral and sum of series', were ineffective in the 2022 Curriculum. The material were revised and supplemented for use in the 2022 <CalculusII>. Based on this discussion, improvement measures for the area concept development in the 2022 Curriculum were proposed.

SCATTERING OF THE E-POLARIZED ELECTROMAGNETIC WAVE BY SOLITUDE CONDUCTIVE STRIP WITH IMPEDANCE

The key problem of scattering of a plane E - polarized electromagnetic wave by a solitude electrically conductive impedance strip is studied in detail. A well-posed full-wave mathematical model of scattering in the form of the first order singular integral equation was obtained by applying a modifica- tion of a classical method if integral equations. To examine this singular integral equation two basic methods were used. The first one is remarkable Lord Rayleigh’s asymptotic method, which delivers an explicit analytical solution in a narrow frequency range. It deals with the long-wave case of the plane E - polarized electromagnetic wave scattering problem. The second method is a direct numerical method, dealing with approximate computa- tion of definite integrals, which gives an approximate solution as well, but in the full-frequency range. A series of numerical experiments on scattering a E-polarized electromagnetic wave by impedance strip were carried out and are presented in this paper. Comparing of computation results shows good agreement in long-wave case of the plane E - polarized electromagnetic wave scattering problem, which indicates the correctness of numeric compu- tation in full frequency range as well.

Comprehensive analysis of the performance of a microfluidic photoelectrochemical solar cell with heat exchanger

A dye-sensitized solar cell (DSSC) is a type of photoelectrochemical cell that converts sunlight directly into electricity using a photoanode with a dye-sensitized, nonporous semiconductor layer. This study investigates the impact of integrating microfluidic capabilities into DSSCs, resulting in a flow-cell configuration that introduces new electrical characteristics without compromising its performance. The primary goal of the work was to define and determine the values of heat, mass, and charge transport resistances, and to explore their relationships with the cell’s current–voltage characteristics, temperature, light intensity, and spectra. We employed both experimental methods and computational fluid dynamics using the Lattice-Boltzmann solver to analyze the cell’s dynamics. A novel multilayered microfluidic DSSC (µDSSC) was designed, featuring an innovative “gapless” configuration with a photoactive surface area exceeding 7 cm2, integrated directly with a microfluidic heat exchanger. This configuration allowed the µDSSC to achieve a power conversion efficiency (PCE) of at least 6 % under standard test conditions, with a fill factor reaching up to 68 %. By gradually covering the cell surface, the impact of series resistance (Rs) on PCE was minimized, enabling a maximum PCE of 19–22 %. A new integral definition of the fill factor (IFF) was proposed, which accurately describes the performance of cells constrained by ion mass transport resistance. For the gapless µDSSC, the IFF ranged from 58 % to 88 %. Additionally, we introduced a maximum cell efficiency (MCE) parameter to represent the cell’s efficiency at the theoretical limit of Rs = 0, determined to be 20.7 % for the gapless µDSSC. This MCE value is specific to the cell and is independent of the illuminated surface area and incident light intensity. The thermal efficiency of µDSSC cells is generally limited by heat conduction through the multilayer structure, but at practical cell surface temperatures of 50–80 °C, this limitation is 3 to 6 times less significant than the limitation imposed by the maximum solar energy flux reaching the cell surface. The integration of microchannels into the cell led to a 58 % reduction in hydraulic resistance along the main flow direction. While electrolyte flow typically reduces efficiency in wide-gap DSSCs, the gapless configuration minimizes mass transport resistance, ensuring that fluid flow does not adversely affect the performance of gapless µDSSCs.

Multidimensional signal detection modeling reveals Gestalt-like perceptual integration of face emotion and identity.

Numerous studies have tested the hypothesis that facial identity and emotional expression are independently processed, but a solid conclusion has been difficult to reach, with the literature showing contradictory results. We argue that this is partly due to different researchers using different definitions of perceptual integration and independence, usually vague and/or simply operational, and also due to lack of proper stimulus control. Here, we performed a study using three-dimensional realistic computer-generated faces for which the discriminability of identities and expressions, the intensity of the expressions, and low-level features of the faces were controlled. A large number of participants, distributed across twelve experimental groups, performed identification tasks for the six basic emotional expressions and the neutral expression (between 2018 and 2019). A multidimensional signal detection model was utilized to analyze the data, which allowed us to dissociate between multiple formally defined notions of independence and holism. Results showed strong and robust violations of perceptual independence that were consistent across all experiments and suggest Gestalt-like perceptual integration of face identity and expression. To date, our results provide the strongest evidence for holistic/Gestalt processing found among face perception studies that have used formal definitions of independence and holism. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

A family of quadrature formulas with their error bounds for convex functions and applications

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.

Fractional calculus for distributions

AbstractFractional derivatives and integrals for measures and distributions are reviewed. The focus is on domains and co-domains for translation invariant fractional operators. Fractional derivatives and integrals interpreted as "Equation missing"-convolution operators with power law kernels are found to have the largest domains of definition. As a result, extending domains from functions to distributions via convolution operators contributes to far reaching unifications of many previously existing definitions of fractional integrals and derivatives. Weyl fractional operators are thereby extended to distributions using the method of adjoints. In addition, discretized fractional calculus and fractional calculus of periodic distributions can both be formulated and understood in terms of "Equation missing"-convolution.

Modeling the Dynamics of Supercapacitors by Means of Riemann–Liouville Integral Definition

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.

The identity conflict process: Appraisal theory as an integrative framework for understanding identity conflict at work.

Identity conflict-the experience of perceiving incompatibilities between aspects of one's identity content that call into question the individual's ability to meet the identity standard of at least one of these identities-can significantly impact individuals' work experiences. As individuals navigate experiences of identity conflict at work, managers and organizations also grapple with how to support employees' multiple identities while mitigating the primarily negative outcomes of identity conflict. However, the scholarship on work-relevant identity conflict faces several challenges, including disciplinary fragmentation, conceptual imprecision, and diverse but deficient theoretical perspectives, which together have limited our ability to accumulate knowledge about this experience and to develop useful management tools. To overcome these, we conducted a thorough review of the cross-disciplinary literature, allowing us to offer a refined integrative definition of identity conflict and a reconceptualization of identity conflict as the result of an appraisal process. As we delineate what we know about the appraisal process of identity conflict, we provide a detailed theoretical explanation of its antecedents, outcomes, and responses and shed light on the mechanisms that drive the process. This approach not only enhances theoretical depth and guides new research directions but also equips managers to address and reduce identity conflict experienced by their employees. This research contributes to the literature by offering clarity and coherence to the identity conflict domain, providing theoretical and practical guidance, and outlining promising directions for future inquiry. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

FATA: An efficient optimization method based on geophysics

An efficient swarm intelligence algorithm is proposed to solve continuous multi-type optimization problems, named the fata morgana algorithm (FATA). By mimicking the process of mirage formation, FATA designs the mirage light filtering principle (MLF) and the light propagation strategy (LPS), respectively. The MLF strategy, combined with the definite integration principle, drives the algorithmic population to enhance FATA’s exploration capability. The LPS strategy, combined with the trigonometric principle, drives the algorithmic individual to improve the algorithm's convergence speed and exploitation capability. These two search strategies can better use FATA’s population and individual search capabilities. The FATA is compared with a broad array of competitive optimizers on 23 benchmark functions and IEEE CEC 2014 to verify the optimization capability. This work is designed separately for qualitative analysis, exploration and exploitation competence analysis, the analysis of avoiding locally optimal solutions, and comprehensive comparison experiments. The experimental results demonstrate the comprehensiveness and competitiveness of FATA for solving multi-type functions. Meanwhile, FATA is applied to three practical engineering optimization problems to evaluate its performance. Then, the algorithm obtains better results than its counterparts in engineering problems. According to the results, FATA has excellent potential to be used as an efficient computer-aided tool for dealing with practical optimization tasks. Source codes and related files are available at https://aliasgharheidari.com/FATA.html and other websites.

Sustaining quantitatively-grounded meanings for definite integrals in high school calculus

ABSTRACT Although the importance of developing students’ quantitatively-grounded meanings for integrals is widely acknowledged, there is limited research into how this can be accomplished in classroom settings. To address this limitation, I conducted a classroom design study in which I analyzed the collective development of classroom mathematical practices for establishing quantitatively-grounded meanings for integrals by a Year 12 calculus class. The findings suggest that the class established ways of approximating the value of definite integrals and integrals with a variable upper bound by progressing through the six layers of the Adding Up Pieces framework in ways that were largely similar to previous findings from studies conducted in clinical settings. The findings from this study also extend previous research by documenting the reasoning that students used to partition the domain and calculate target quantities using the trapezoidal rule, as well as the difficulty they encountered in developing the accumulation function.

The Assessment of Sperm DNA Integrity: Implications for Assisted Reproductive Technology Fertility Outcomes across Livestock Species.

Sperm DNA integrity is increasingly considered a useful measure of semen quality in mammalian reproduction. However, the definition of DNA integrity, the ideal means by which it should be measured, and its predictive value for fertility remain a topic of much discussion. With an emphasis on livestock species, this review discusses the assays that have been developed to measure DNA integrity as well as their correlation with in vitro and in vivo fertility.

Integrasi Nilai-Nilai Islam dalam Pembelajaran Matematika di Madrasah Ibtidaiyyah

The purpose of this research is to find out the integration of Islamic values in mathematics learning at Madrasah Ibtidaiyyah. This study uses a library research method or library research, literature study is a research method that utilizes library sources as a source in research activities. The data collection technique used in this literature research is to look for data on matters or variables in the form of notes, books, papers, or articles and journals. After the data is collected, content analysis is carried out to get the right and accurate conclusion formulation. Data analysis consists of data reduction, data presentation, and conclusion drawing. The results of this study found several discussions regarding the Definition of Integration of Islamic Values, Mathematics Learning, Principles of Knowledge Integration, Strategies for the Integration of Islamic Values in Mathematics Learning, and Integration of Islamic Values in Mathematics Learning. The conclusion of this study is that the integration of Islamic values in the mathematics learning process at Madrasah Ibtidaiyah initially felt difficult to implement. Especially for mathematics subjects that have a lot to do with numbers, formulas, and geometric construction, it will feel easy to apply if we as educators always try to find gaps to instill Islamic religious values in classroom learning

Probability density and information entropy of machine learning derived intracranial pressure predictions.

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.

Effects of surface tension on the collapse time of an empty bubble

The collapse of an empty spherical bubble in an ideal liquid, in the absence of viscosity and surface tension, was studied by Lord Rayleigh. Using energy conservation, he derived an exact expression for the total collapse time as a function of the initial radius of the bubble, the density of the liquid, and the far-field pressure. In the present work, we extend Rayleigh's expression to include surface tension effects. Results are found to depend on a dimensionless parameter ϵ that measures the ratio between the work done by surface tension and that done by pressure during the collapse. This parameter is small for large bubbles but can be of order unity or larger for bubbles of small radius and, eventually, small pressure. We show that the ratio between the collapse time in the presence of surface tension and Rayleigh's collapse time is proportional to a definite integral that is a smooth, monotonically decreasing function of ϵ. This function can be easily bounded analytically for any value of ϵ, yielding a simple and accurate approximation for the collapse time that, for all practical purposes, provides a complete analytical solution to the problem at hand. We finally extend results to the case of a hyperspherical collapsing empty bubble.

Investigating the Structural Composition of Contemporary Sustainable Organic Agriculture Supply Chains: A Case Study from Thailand

The global demand for organic produce has surged, with notable growth among Thai consumers. However, within Thailand's organic agricultural sectors, supply chain obscurity has impeded the understanding and resolution of emerging challenges associated with meeting this rising demand. Public interest has traditionally centered on mainstream economic sectors, resulting in inadequate attention to the sustainable organic farming supply chains. This study aims to explore the complexities of current sustainable organic supply chain configurations in Thailand, addressing gaps that lead to suboptimal planning and non-sustainable practices. Utilizing the Integration Definition for Function Modeling (IDEF0) framework, this research collected data from key stakeholders through in-depth interviews. Participants included representatives from each stage of the supply chain, providing comprehensive insights into the operational dynamics. The analysis identified several critical challenges within the supply chain, including insufficient collaborative planning, reduced productivity, inaccurate demand forecasting, ineffective budgetary planning, and inadequate cold chain management. These challenges highlight the fragmented nature of the current supply chain and the need for strategic improvements. To address these shortcomings, the study suggests integrating technological advancements in demand planning, operations, and budgeting, fostering collaboration among supply chain members, and forming strategic partnerships with cold chain management service providers. These recommendations aim to enhance the efficiency and sustainability of Thailand's organic produce supply chain, ensuring it can meet increasing global and domestic demands effectively.

A probabilistic proof of some integral formulas involving incomplete gamma functions

The theory of normal variance mixture distributions is used to provide elementary derivations of closed-form expressions for the definite integrals ∫ 0 ∞ x − 2 ν cos ⁡ ( b x ) γ ( ν , α x 2 ) d x (for ν > 1 / 2 , b > 0, α > 0) and ∫ 0 ∞ x 2 ν − 1 cos ⁡ ( b x ) Γ ( − ν , α x 2 ) d x (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.

Primitively 2-universal senary integral quadratic forms

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.