First Derivative Research Articles

Theory on New Fractional Operators Using Normalization and Probability Tools

We show how a rescaling of fractional operators with bounded kernels may help circumvent their documented deficiencies, for example, the inconsistency at zero or the lack of inverse integral operator. On the other hand, we build a novel class of linear operators with memory effects to extend the L-fractional and the ordinary derivatives, using probability tools. A Mittag–Leffler-type function is introduced to solve linear problems, and nonlinear equations are addressed with power series, illustrating the methods for the SIR epidemic model. The inverse operator is constructed, and a fundamental theorem of calculus and an existence-and-uniqueness result for differintegral equations are proven. A conjecture on deconvolution is raised, which would permit completing the proposed theory.

Field theory for superconducting branes and generalized particle-vortex duality

Abstract We propose a field theory of closed p-brane Cp interacting with a (p + 1)-form gauge field Ap+1. This is a generalization of the Ginzburg-Landau theory (Abelian-Higgs model) for superconducting particles to higher-dimensional superconducting branes. A higher-form gauge invariant action is constructed by utilizing the Area derivative, which is a higher-dimensional generalization of the ordinary derivative. We find that the fundamental phenomena of superconductivity, such as the Meisser effect, topological defects, topological order, are naturally extended in the brane-field theory. We explicitly construct a topologically non-trivial static configuration that is characterized by the first homotopy group. Then, we calculate the low-energy effective theory in the presence of the topological defect and find that it is described by a BF-type topological field theory coupled with the world-volume of the topological defect. We also discuss a potential duality between the superconducting brane-field model and a brane-field model with a global U(1) higher-form symmetry as a generalization of the Particle-Vortex duality.

Piecewise second kind Chebyshev functions for a class of piecewise fractional nonlinear reaction–diffusion equations with variable coefficients

In this paper, a new type of piecewise fractional derivative (PFD) is introduced. The ordinary and distributed-order fractional derivatives in the Caputo sense are used to define this type of PFD. A new version of nonlinear reaction–diffusion equations with variable coefficients is defined using this type of PFD. The orthonormal piecewise second kind Chebyshev functions (CFs), as a new family of basic functions, are generated. An explicit formula is extracted for PFD of these piecewise functions. A hybrid method based on the orthonormal piecewise second kind CFs and orthonormal second kind Chebyshev polynomials is proposed to solve the aforementioned problem. The established approach transforms solving the expressed problem into solving an algebraic system of equations. To illustrate the accuracy of the developed method, some numerical examples are considered.

Concept image framework for derivatives from contour graphs

Physics experts and students commonly use a variety of representations when working with partial derivatives, including symbols, graphs, and words. One especially powerful representation is the contour graph. In open-ended problem-solving interviews with nine upper-division physics students, we asked students to determine derivatives from contour graphs with electrostatic and thermodynamic contexts. Students undertook three overarching actions: (i) orienting to the given graph, (ii) finding one or more derivatives, and (iii) reflecting on the task. In general, students tended to have more difficulty with the thermodynamic task than the electrostatic one. Students made use of a variety of representational features throughout the interviews, namely by annotating the provided contour graphs in ways that highlighted what is held constant. We use these results to generalize the existing concept image framework for ordinary derivatives to create a framework useful for understanding student ideas about multivariable derivatives. In particular, we add a new layer to the framework—the layer—that calls attention to how students do or do not attend to which variables are changing (or remaining constant) when finding a derivative. We view the change layer as a powerful central idea for both further research and the development of instructional tasks for physics students. Published by the American Physical Society 2024

New generalized Jacobi–Galerkin operational matrices of derivatives: an algorithm for solving the time-fractional coupled KdV equations

The present paper investigates a new method for computationally solving the time-fractional coupled Korteweg–de Vries equations (TFCKdVEs) with initial boundary conditions (IBCs). The method utilizes a set of generalized shifted Jacobi polynomials (GSJPs) that adhere to the specified initial and boundary conditions (IBCs). Our approach involves constructing operational matrices (OMs) for both ordinary derivatives (ODs) and fractional derivatives (FDs) of the GSJPs we employ. We subsequently employ the collocation spectral method using these OMs. This method successfully converts the TFCKdVEs into a set of algebraic equations, greatly simplifying the task. In order to assess the efficiency and precision of the proposed numerical technique, we utilized it to solve two distinct numerical instances.

Novel method in mice to evaluate left atrial (LA) systolic contribution to left ventricular volume (LV) using pressure-volume (PV) loops

Abstract Background LA myopathy is a critically important non-LV contributor to the progression of HFpEF (heart failure with preserved ejection fraction) in patients and is associated with a severe phenotype, leading to impaired cardiac output reserve and reduced exercise capacity. In normal humans LA systole contributes ~20% of total LV filling, but with HFpEF, this contribution decreases to <16% with a greater reliance on this contribution to LV filling. Purpose We sought to investigate LA function as a critical component of cardiac function analysis by utilizing the LV pressure signal to identify and quantify the contribution of volume during LA systole to the total LV volume in mice with HFpEF. Methods Hypertension (HTN)-associated HFpEF C57BL/6J mice were subjected to chronic aldosterone infusion and 1% NaCl for 4 weeks (SAUNA model). Obesity after a high fat diet (HFD: 60% fat diet) for 16 weeks followed by the SAUNA protocol, induced HTN and obesity associated HFpEF. Blood pressure was monitored weekly. Mice were anesthetized with isoflurane, intubated, chest opened to expose the heart and allow placement of a PV catheter (Transonic) which was introduced into the LV via apical stab with a 27G needle. Calibration of the PV catheter was performed according to the manufacturer’s instructions. Atrial systole was identified using 2nd order derivative of the LV pressure signal. Contribution of LA systole was calculated using volume at the beginning of LA systole identified by the 2nd order derivative of the LV pressure signal and subtracting from the LV end diastolic volume. The % of LA systole contribution was calculated as a % of the total LV stroke volume. Data was analyzed with LabChart pro software (AD Instruments). Results SAUNA mice had normal LVEF, moderate HTN and lung congestion. PV relationships demonstrated Tau was impaired in HFpEF vs. control mice (14±1 vs. 10±0.5 ms) with a shift of the PV loop to the left (Fig. A.) In control mice, the atrial systole contributed 39± 4% to LV volume and was impaired in HFpEF mice 20± 3% a 1.95-fold reduction (P<0.01). Importantly atrial systole contribution was increased in obese mice 53±5 %. However, the combination of obesity and HTN, both common comorbidities in HFpEF, resulted in a reduced LA systole contribution by 1.43-fold to 37± 9%, (P<0.005; Fig B). Conclusion A novel method of % atrial systole from LV PV loop using 2nd order derivative to quantify the volume contribution of atrial systole to total LV stroke volume was identified in mice. LA dysfunction was present in murine HTN-associated HFpEF with/without obesity.

(In)stability of de Sitter quasinormal mode spectra

We consider how the quasinormal spectrum for the conformal wave operator on the static patch of de Sitter changes in response to the addition of a small potential. Since the quasinormal modes and co-modes are explicitly known, we are able to give explicit formulae for the instantaneous rate of change of each frequency in terms of the perturbing potential. We verify these exact computations numerically using a novel technique extending the spectral hyperboloidal approach of Jaramillo et al. (2021). We propose a definition for a family of pseudospectra that we show capture the instability properties of the quasinormal frequencies.

The Hadamard condition on a Cauchy surface and the renormalized stress-energy tensor

Given a Cauchy surface in a curved spacetime and a suitably defined quantum state on the CCR algebra of the Klein-Gordon quantum field on that surface, we show, by expanding the squared spacetime geodesic distance and the 'U' and 'V' Hadamard coefficients (and suitable derivatives thereof) in sufficiently accurate covariant Taylor expansions on the surface that the renormalized expectation value of the quantum stress-energy tensor on the surface is determined by the geometry of the surface and the first 4 time derivatives of the metric off the surface, in addition to the Cauchy data for the field's two-point function. This result has been anticipated in and is motivated by a previous investigation by the authors on the initial value problem in semiclassical gravity, for which the geometric initial data corresponds, a priori, to the spatial metric on the surface and up to 3 time derivatives off the surface, but where it was argued that the fourth derivative can be obtained with aid of the field equations on the initial surface.

Investigation of new analysis methods for simultaneous and rapid identification of five different microplastics using ATR-FTIR spectroscopy and chemometrics

Microplastic (MP) pollution in water has become one of the most important global problems of our time. The development of appropriate and rapid analysis techniques is of great importance at the beginning of the studies aimed at solving this problem. In the presented study, in order to perform the qualitative and quantitative analysis of MP forms of polyamide (PA), polyethylene (PE), polypropylene (PP), polystyrene (PS), and polyethylene terephthalate (PET), which are known to be most abundant in water, in a fast and easy way, new Attenuated Total Reflectance-Fourier Transform Infrared (ATR-FTIR) spectroscopy methods were tried to be developed by utilizing chemometric methods. While principal component analysis (PCA) was applied for qualitative analyses, partial least squares (PLS) models were created for quantitative analyses. Raw, 1st, and 2nd order derivatives of all spectra and their spectra with different levels of smoothing points were taken and 24 different chemometric models were created for each MP. In interpreting the statistical performances of the developed PCA and PLS models, different parameters were used. According to the obtained results, the qualitative discrimination of all polymer types was successfully achieved. It was determined that the PLS models developed for the quantitative determination of mixtures consisting of different concentrations of MP types could not be at the desired level. However, it was determined that the PLS models developed for PA, PE, PP, and PET, where the normal spectrum was used, could give quantitatively accurate results, albeit partially.

Updating Calculus Teaching with AI: A Classroom Experience

In the context of mathematics education, the integration of artificial intelligence (AI) in teaching calculus is revolutionizing instructional methodologies and enhancing learning experiences both inside and outside the classroom. This study explores the use of specific AI tools, including ChatGPT, MathGPT, Gemini, and Wolfram Alpha, to deepen students’ understanding of key mathematical concepts such as derivatives and rates of change through continuous interaction with a virtual tutor. By employing well-designed prompts, these tools facilitated problem-solving exercises that were verified and refined by AI, fostering both precision in calculations and conceptual clarity. Observations from the classroom implementation reveal that students not only improved their accuracy in performing derivative calculations but also developed a clear understanding of the distinctions between average and instantaneous rates of change. The AI tools created a dynamic, adaptive learning environment, providing immediate feedback and simulations that significantly boosted student engagement and motivation. These findings underscore the potential of AI to transform mathematics education by making learning more personalized and accessible, ultimately enhancing educational outcomes and preparing students for future academic and professional challenges. Furthermore, this study introduces an innovative approach to refining AI prompts and interactions, highlighting the importance of iterative improvement to enhance the quality of AI feedback. This approach is crucial for developing better problem-solving skills and ensuring a comprehensive understanding of mathematical concepts.

Estimating Cadmium Concentration in Agricultural Soils with ZY1-02D Hyperspectral Data: A Comparative Analysis of Spectral Transformations and Machine Learning Models

The accumulation of cadmium (Cd) in agricultural soils presents a significant threat to crop safety, emphasizing the critical necessity for effective monitoring and management of soil Cd levels. Despite technological advancements, accurately monitoring soil Cd concentrations using satellite hyperspectral technology remains challenging, particularly in efficiently extracting spectral information. In this study, a total of 304 soil samples were collected from agricultural soils surrounding a tungsten mine located in the Xiancha River basin, Jiangxi Province, Southern China. Leveraging hyperspectral data from the ZY1-02D satellite, this research developed a comprehensive framework that evaluates the predictive accuracy of nine spectral transformations across four modeling approaches to estimate soil Cd concentrations. The spectral transformation methods included four logarithmic and reciprocal transformations, two derivative transformations, and three baseline correction and normalization transformations. The four models utilized for predicting soil Cd were partial least squares regression (PLSR), support vector machine (SVM), bidirectional recurrent neural networks (BRNN), and random forest (RF). The results indicated that these spectral transformations markedly enhanced the absorption and reflection features of the spectral curves, accentuating key peaks and troughs. Compared to the original spectral curves, the correlation analysis between the transformed spectra and soil Cd content showed a notable improvement, particularly with derivative transformations. The combination of the first derivative (FD) transformation with the RF model yielded the highest accuracy (R2 = 0.61, RMSE = 0.37 mg/kg, MAE = 0.21 mg/kg). Furthermore, the RF model in multiple spectral transformations exhibited higher suitability for modeling soil Cd content compared to other models. Overall, this research highlights the substantial applicative potential of the ZY1-02D satellite hyperspectral data for detecting soil heavy metals and provides a framework that integrates optimal spectral transformations and modeling techniques to estimate soil Cd contents.

Fractional Derivative of Hyperbolic Function

Fractional derivative is a generalization of ordinary derivative with non-integer or fractional order. This research presented fractional derivative of hyperbolic function (hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cotangent, hyperbolic secant, and hyperbolic cosecant) with order constraint . The hyperbolic function is presented in Maclaurin series form. Then, the fractional derivative can be determined by using definition of Riemann-Liouville fractional derivative. The result is simulated by using Matlab software

Clot waveform analysis of prolonged activated partial thromboplastin time in various disorders.

The prolongation in APTT reflects various disorders affecting the blood coagulation system. Automated coagulation analyzers such as ACL TOP, help in evaluating the clotting cascade by displaying clot reaction curves or waveforms. The fibrin formation curve forms S-shaped curve showing the changes in the absorbance on X-axis and time in seconds on Y-axis. The clotting data is processed to obtain derivatives. This observational study analyzed 80 samples including 20 samples from normal healthy individuals as control. The parameters studied were time (secs), height (mAbs), and width (secs) of first derivative (FD), fibrin clot time at ½ height, height (mAbs), and width (secs) of both first and second peaks of second derivative (SD). Hemophilia (n = 13) that consisted of factor VIII deficiency (n = 11) and factor IX deficiency (n = 2) showed biphasic patterns in both first DC and second DC. Cases with thrombosis (n = 8) showed marked elevation in first DC. Various other disorders (n = 39) that consisted of mainly hepatic dysfunction (n = 24) showed prolongation in fibrin formation, first DC, and second DC peak time. The atypical derivative curve analysis helps in determining the interferences in clotting cascade. It gives significant information and provides direction for further testing.

A novel image denoising technique with Caputo type space–time fractional operators

A novel image denoising model, namely Full Fractional Total Variation (TVFF), based on the Rudin-Osher-Fatemi (ROF) and the fractional total variation models is presented. The leading advantage of TVFF model is that it uses fractional derivatives with length scale parameters instead of ordinary derivatives with respect to both time and spatial variables in the diffusion equation. The Riesz–Caputo fractional derivative operator is used to disperse nonlocal influence throughout all directions, whereas the Caputo fractional derivative concept is employed for time fractional derivatives. Therefore, the influence of neighboring pixels is given greater weight compared to those situated farther away and this reflects the consideration behind denoising process better. Moreover, the numerical approach is constructed, and its stability and convergence properties are thoroughly examined. To show the superiority of our model, the denoised images are subjected to visual and numerical comparisons using metrics such as the Signal-to-Noise Ratio (SNR), the Structural Similarity Index Measure (SSIM) and the Edge-Retention Ratio (ERR). The performance of the TVFF method is evaluated under various types of noise, including Poisson, Speckle, and Salt & Pepper, and the results are compared with those obtained using Gauss and Median Filters. Furthermore, the proposed method is applied to both blind and synthetic images, thereby showcasing its versatility and applicability across diverse datasets. The outcomes showcase the substantial potential of our enhanced model as a versatile and efficient tool for image denoising.

Integrating Hyperspectral Reflectance and Physiological Parameters to Detect Urban Tree Stress: A Study of Drought and Simulated Acid Rain

With urbanization and climate change worsening, urban trees are constantly exposed to environmental stress. To enhance the functionality and health of trees, it is crucial to rapidly and non-destructively detect and respond to tree stress. Research utilizing hyperspectral characteristics for detecting various stresses has recently been actively pursued. This study conducted comparative analysis using various leaf physiological parameters (chlorophyll content, chlorophyll fluorescence, leaf water, and gas exchange status) and hyperspectral data (VIS: visible ray; SWIR: short-wave infrared) to diagnose stress in Prunus yedoensis, commonly grown urban trees, by subjecting them simultaneously to different stresses (drought and simulated acid rain). The findings suggest that hyperspectral reflectance proved more responsive in identifying stress compared to the physiological parameters. Initially, VIS was more effective in detecting two stress responses than SWIR through a classification model (PLS-DA: partial least squares-discriminant analysis). Although SWIR initially faced challenges in simulated acid rain stress detection, spectral preprocessing (SNV: standard normal variate, + S.G 2nd: Savitzky–Golay 2nd derivative) enhanced its stress classification accuracy. Over time, the SWIR bands (1437 nm, 1667 nm, and 1949 nm) exhibited characteristics (such as moisture detection) more closely aligned with stress responses compared to VIS, as determined through PCA (principal component analysis). Hyperspectral reflectance also revealed the potential to measure chlorophyll fluorescence (Fo: minimum fluorescence). Building upon the foundational data of this study, the future potential of diagnosing urban tree stress using portable spectrometers is strong.

Analytical approach to structural chemistry origins of mechanical, acoustical and thermal properties.

Crystalline matters with periodically arranged atoms found wide applications in modern science and technology. To facilitate the design of new materials and the advancement of existing ones, accurate and efficient models without relying too much on known inputs for predicting the functionalities are essential. Here, we propose an analytical approach for such a purpose, with only the knowledge of the structural chemistry of crystals. Based on the electrostatic interaction between periodically arranged atoms, the 1st, 2nd and 3rd derivatives of interatomic potential, respectively, enable a prediction of ten kinds in total of mechanical, acoustical and thermal properties. Over a thousand measurements are collected from ∼500 literatures, this results in the symmetric mean percentage error (SMPE) within ±25% and the symmetric mean absolute percentage error (SMAPE) ranging from 22%∼74% across all properties predicted, which further enables a revelation of bond characteristics as the most important but implicit origin for functionalities.

Exploration of melting heat transfer and entropy generation in a magnetized hybrid nanoliquid over an extending sheet of varying thickness

The analysis of melting heat transfer over an expansive sheet of variable thickness is of the utmost importance in various industrial and engineering sectors, such as injection molding of polymers and composites, cooling of nuclear reactors, hot rolling, etc. Thus, the current work examines the flow dynamics, melting heat transport, and entropy generation of a magneto-hybrid nanofluid over a stretching device with variable thickness in porous media. A water-based hybridized nanofluid is formed using copper (Cu) and aluminium oxide (Al2O3) nanoparticles in the presence of thermal radiation, viscous dissipation, and Joulean heating. The transport partial derivatives were transmuted to ordinary derivatives using appropriate similarity quantities. These equations were numerically solved through shooting techniques combined with the Runge–Kutta–Fehlberg (RKF) method. Diverse figures and tables are sketched to illustrate the outcomes of the various parameters involved in the study. The investigation reveals an enlargement in the heat-bounding surface with an escalation in the magnitude of volume fraction, melting heat transfer, and magnetic field terms. In contrast, the momentum-bounding surface depletes with these parameters. Furthermore, as thermal radiation and Eckert numbers rise, the thermal gradient increases. The entropy generation increases with higher Brikman number and porosity term, but the melting heat parameter causes a decline in the entropy profiles.

Joint Probability Densities on Riemannian Manifolds are Symmetric Tensor Densities

This paper presents the tensor properties of joint probability densities on a Riemannian manifold. Initially, we develop a binary data matrix to record the values of a large number of particles confining in a closed system at a certain time in order to retrieve the joint probability densities of related variables. By introducing the particle-oriented coordinate and the generalized inner product as a multi-linear operation on the basis of this coordinate, we extract the set of joint probabilities and prove them to meet covariant tensor properties on a general Riemannian space of variables. Based on the Taylor expansion of scalar fields in Riemannian manifolds, it has been shown that the symmetrized iterative covariant derivatives of the cumulative probability function defined on Riemannian manifolds also give the set of related joint probability densities equivalent to the aforementioned multi-linear method. We show these covariant tensors reduce to classical ordinary partial derivatives in ordinary Euclidean space with Cartesian coordinates and give the formal definition of joint probabilities by partial derivatives of the cumulative distribution function. The equivalence between the symmetrized covariant derivative and the generalized inner product has been concluded. Some examples of well-known physical tensors clarify that many deterministic physical variables are presented as tensor densities with an interpretation similar to probability densities.

Exact exploration of time fractional-based magnetized flow of a generalized second grade fluid through an oscillating rectangular duct

To model several engineering and physical models, the approach of the fractional derivative is highly anticipated. As compared to the ordinary derivatives, the fractional derivatives with more flexibility can estimate the data due to the involvement of the fractional-order derivatives. Due to these advantages of the fractional approach, this study communicates with the determination of the fractional-based exact outcomes of an oscillatory rectangular duct problem of a generalized second-grade fluid. The approach of the fractional operator is involved in the relationship of the constitutive equations. For cosine oscillation of the rectangular duct, exact results of the magnetized unsteady flow problem are evaluated through the technique of Laplace transform with double finite Fourier sine transform. This study concludes that the velocity field exhibits escalating behavior relative to the improved fractional parameter. Moreover, the magnetic parameter with increasing values declines the flow field while the accelerating values of the fluid parameter enhance the velocity field.

Mixed convective flow analysis of a Maxwell fluid with double diffusion theory on a vertically exponentially stretching surface

It observers that an object is submerged in a liquid, more pressure is applied to its bottom surface than its top surface, as a result pressure rises within depth of fluids. A buoyancy force is generated due to the pressure difference. In the current investigation, the mathematical formulation of bio-convective stagnation point flow of a radiative Maxwell fluid with multiple slip effect on an exponentially porous stretching surface is analyzed thoroughly. The buoyancy assisting and opposing conditions with magnetic field are discussed in the current investigation. Moreover, the energy and concertation equations are formulated by the utilization of Cattaneo–Christov theory and non-uniform heat source/sink. The flow model is developed in the form of partial derivatives, then use the similarity variables to convert the physical system into nonlinear ordinary derivative. The nonlinear system is tackled numerically with the help of Bvp4c approach on the MATLAB. The graphical upshots for various emerging parameter are observed with two aspects: buoyancy assisting λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( {\\lambda > 0} \\right)$$\\end{document} and buoyancy opposing λ<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( {\\lambda < 0} \\right)$$\\end{document}. The observation shows that the greater values of mixed convection parameter improves the fluid velocity for assisting flow λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( {\\lambda > 0} \\right)$$\\end{document}, while declining trend is noticing for opposing flow situation λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( {\\lambda > 0} \\right)$$\\end{document}. Further, it is worth noticing that stronger inputs of thermal and concentration relaxation parameter yield lesser thermal and concentration diffusivity, which reduces the temperature and nanoparticles concentration.