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While research on the opportunity to learn about mathematics concepts provided by textbooks at the secondary level is well documented, there is still a paucity of similar research at the undergraduate level. Contributing towards addressing this knowledge gap, the present study examined opportunities to engage in quantitative and covariational reasoning, in the context of the derivative concept, provided by two textbooks commonly used in the teaching of calculus in the United States-a regular calculus textbook and an applied calculus textbook. Analysis of expository sections, examples, and exercises related to the derivative provided in these textbooks revealed three main results. First, the applied calculus textbook provides plenty of opportunities to engage in quantitative reasoning. Similar opportunities are limited in the regular calculus textbook. Second, opportunities to engage in covariational reasoning are not only minimal in both textbooks, but also limited to low levels of covariational reasoning, namely coordination, direction, and quantitative coordination. Third, the applied calculus textbook consistently defines ordinary and partial derivatives as rates of change while the regular calculus textbook consistently defines these concepts as limits of a difference quotient. Findings of this research have implications for several stakeholders, including calculus textbook authors and calculus instructors. [ABSTRACT FROM AUTHOR]

Abstract College graduates’ job satisfaction is validated via their feedback, experiences and personal developments during their career progression. Validation is accomplished to ensure the students’ job satisfaction and retain them for a prolonged time. The traditional validation models have difficulties in analyzing the individual satisfaction level. The research issue is addressed by introducing the Quantitative Assessment Method (QAM) using Fuzzy Logic to validate the job satisfaction level of different newly placed students. The QAM approach assesses the student experience and personal development across various quarters. The fuzzy optimization uses two factors differentially using partial derivatives. The partial derivatives are extracted using the min–max functions of the fuzzification process such that the derivatives are halted after the maximum factors. The proposed method optimizes the validation using individual satisfaction levels and cumulative experience shared by the students. The available derivatives identify the best-afford job satisfaction level for different progression levels. This best-fit feature is handled using multiple min and max derivatives to extract optimal outputs. This proposed method is valid for improving satisfaction levels and experience analysis.

The purpose of the study is a theoretical justification (working hypothesis) of a possible approach to calculating doses in a combined medicinal product containing 2 pharmaceutical substances. A mathematical model is proposed, which is a multiple regression equation from two variables, which is a square polynomial. The proposed model makes it possible to calculate the optimal doses of pharmaceutical substances in a combined drug by further differentiating the regression equation in partial derivatives. Two examples of dose calculation are presented.

This paper presents the mathematical modeling of the oxygen transfer process using partial differential equations (PDEs). This process is crucial in various environmental and engineering applications, such as wastewater treatment, aeration systems, and natural water bodies, in order to maintain water quality. The authors solved the typical PDE for describing the change in oxygen concentration over time and present the developed model of the differential equation with the term "source", indicating that the model could be used to optimize oxygen transfer in various environmental and engineering applications, contributing to improved water quality and system efficiency. [ABSTRACT FROM AUTHOR]

Ni-based superalloys are widely used in aerospace applications. However, traditional constitutive equations often lack the necessary accuracy to predict their high-temperature behavior. A novel constitutive model, utilizing Taylor series expansions and partial derivatives, is proposed to predict the high-temperature flow behavior of a nickel-based superalloy. Hot compression tests were conducted at various strain rates (0.01 s−1, 0.1 s−1, 1 s−1, and 10 s−1) and temperatures (850 °C to 1200 °C) to gather comprehensive experimental data. The performance of the new model was evaluated against classical models, specifically the Arrhenius and Hensel–Spittel (HS) models, using metrics such as the correlation coefficient (R), root mean square error (RMSE), sum of squared errors (SSE), and sum of absolute errors (SAE). The key findings reveal that the new model achieves superior prediction accuracy with an R value of 0.9948 and significantly lower RMSE (22.5), SSE (16,356), and SAE (5561 MPa) compared to the Arrhenius and HS models. Additionally, the stability of the first-order partial derivative of logarithmic stress with respect to temperature ( ∂ l n σ / ∂ T ) indicates that the logarithmic stress–temperature relationship can be approximated by a linear function with minimal curvature, which is effectively described by a second-degree polynomial. Furthermore, the relationship between logarithmic stress and logarithmic strain rate ( ∂ l n σ / ∂ l n ε ˙ ) is more precisely captured using a third-degree polynomial. The accuracy of the new model provides an analytical basis for finite element simulation software. This helps better control and optimize processes, thus improving manufacturing efficiency and product quality. This study enables the optimization of high-temperature forming processes for current superalloy products, especially in aerospace engineering and materials science. It also provides a reference for future research on constitutive models and high-temperature material behavior in various industrial applications. [ABSTRACT FROM AUTHOR]

This paper deals with modified double Laplace transforms of partial derivatives and their applications. Starting with standard results on modified double Laplace transform, we developed theorems on modified double Laplace transform of first and second-order partial derivatives. Further, we applied our results to solve homogeneous and non-homogeneous partial differential equations, D' Alembert's wave equation, and Klein-Gordon equation. [ABSTRACT FROM AUTHOR]

This paper presents the mathematical modeling of the oxygen transfer process using partial differential equations (PDEs). This process is crucial in various environmental and engineering applications, such as wastewater treatment, aeration systems, and natural water bodies, in order to maintain water quality. The authors solved the typical PDE for describing the change in oxygen concentration over time and present the developed model of the differential equation with the term “source”, indicating that the model could be used to optimize oxygen transfer in various environmental and engineering applications, contributing to improved water quality and system efficiency.

The new Laplace variational iterative method is used in this research for solving the (2+1)-D and (3+1)-D Burgers equations. This technique relies on the modified variational iteration method and the Laplace transform. To apply this approach, the differential problem is first transformed into an algebraic form using the Laplace transform, and then the algebraic equations are iteratively solved using the modified variational iterative approach. By utilizing this technique, the Burgers equations can be solved both numerically and analytically. The study demonstrates the effectiveness of the new Laplace variational iterative approach through three specific examples. [ABSTRACT FROM AUTHOR]

The method for constructing the Chebyshev approximation of a discrete multivariable function with reproducing the values of the function and its partial derivatives at given points is proposed. The method is based on constructing a boundary mean-power approximation with appropriate interpolation conditions. The authors use an iterative scheme based on the least squares method with a variable weight function for constructing the mean-power approximation. The results of approximating the one-variable function confirm the fulfillment of the characteristic feature of the Chebyshev approximation with the reproduction of the function and its derivative values at given points. The test examples instantiate the fast convergence of the proposed method. [ABSTRACT FROM AUTHOR]

A method for constructing the Chebyshev approximation of a discrete multivariable function with reproduction of function’s values and partial derivatives at given points is proposed. The idea of the method is based on the construction of the boundary mean-power approximation with the appropriate interpolation conditions. An iterative scheme based on the least squares method with a variable weight function is used to construct the mean-power approximation. The results of the approximation of the function of one variable confirm the fulfillment of the characteristic property of the Chebyshev approximation with the reproduction of the values of the function and the values of its derivatives at the given points. The test examples confirm the fast convergence of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the concept of single conformable Sehu transformation (CSHT) to double conformable transformation (CDFSHT). Moreover, we are able to prove some theorems and properties related to this work. We apply the double conformable Sehu transform to solve the initial and boundary problems of linear and (non)homogenous conformable fractional partial differential equations (PDEs). The validity and the applicability of the proposed technique are shown by three numerical examples. Mathematica software is used for Euclidean division of polynomials and drawing graphs. [ABSTRACT FROM AUTHOR]

A new way of developing a mathematical model of the dynamics of the factors forming the considered social, political, economic, ecological or other space of life activity of society, depending on local changes in parameters affecting these factors, is presented. A feature of the proposed approach is the use of a matrix of marginal values included in the study that make up the Jacobi matrix of noted factors. The dependence of the factors describing the socio-economic system on the parameters of the model is obtained in explicit form. Under certain conditions, the described relations have the form of mappings close to extension. A generalized assessment of these transformations is proposed. Accounting for this assessment is important for preventing crisis phenomena. The model is intended to be used for informational, forecasting, management and governance purposes in the presence of a sufficient digitalization’s degree of public structures, without which it is problematic to receive and transmit data for building the model, and perform related calculations.

This paper extends work in the areas of quantitative reasoning and covariational reasoning at the undergraduate level. Task-based interviews were used to examine third-semester calculus students' reasoning about partial derivatives in five tasks, two of which are situated in a mathematics context. The other three tasks are situated in real-world contexts (i.e. weight, revenue, and mortgage). There are three main findings from this study. First, interpreting quantities representing partial derivatives in real-world contexts was problematic for most of the students. Second, students' interpretations of partial derivatives were not consistent across tasks. Third, all the students engaged in several levels of covariational reasoning when interpreting quantities representing partial derivatives in the context of weight, revenue, and mortgage. Implications for calculus instruction and directions for future research are included. [ABSTRACT FROM AUTHOR]

From literature, we know that making the connections between mathematics and physics is not trivial for most students, even at the advanced level. In the specific context of partial derivatives in thermodynamics, research suggests that making explicit connections between the mathematics and the physics is necessary to foster student understanding. In this paper, we investigate how graphical reasoning can help undergraduate students in making connections between the partial derivatives of temperature with respect to position and to time and their respective physical meaning in the context of one-dimensional systems modeled by the heat equation. We conducted task-based, think aloud interviews with four pairs of undergraduate students majoring in physics or mathematics and use dynamic conceptual blending diagrams to analyze their reasoning processes and the role of the graphs therein. The results in this paper indicate that stimulating graphical reasoning is a promising way to promote the blending of mathematical and physical knowledge. Specifically, our data shows that constructing graphs can be a catalyst which helps students to give physical meaning to the partial derivatives ∂ T / ∂ t and/or ∂ T / ∂ x . Therefore, in teaching/learning activities, actively promoting graph construction and reasoning based on those graphs offers perspectives to support the blending of mathematics and physics. [ABSTRACT FROM AUTHOR]

In this work, we introduce the SEE (Sadiq, Emad, Eman) integral transform to find the general solution of the first order complex differential equations with constant coefficients. The usage of SEE integral transform has proven its efficiency in solving a wide range of linear operator equations. [ABSTRACT FROM AUTHOR]

In the traditional approach to differentiability, found in almost all university textbooks, this notion is considered only for interior points of the domain of function or for functions with an open domain. This approach leads to the fact that differentiability has usually been considered only for functions with an open domain in R n , which severely limits the possibility of applying the potential techniques and tools of differential calculus to a broader class of functions. Although there is a great need for generalization of the notion of differentiability of a function in various problems of mathematical analysis and other mathematical branches, the notion of differentiability of a function at the non-interior points of its domain has almost not been considered or successfully defined. In this paper, we have generalized the differentiability of scalar and vector functions of several variables by defining it at non-interior points of the domain of the function, which include not only boundary points but also all points at which the notion of linearization is meaningful (points admitting nbd rays). This generalization allows applications in all areas where standard differentiability can be applied. With this generalized approach to differentiability, some unexpected phenomena may occur, such as a function discontinuity at a point where a function is differentiable, the non-uniqueness of differentials... However, if one reduces this theory only to points with some special properties (points admitting a linearization space with dimension equal to the dimension of the ambient Euclidean space of the domain and admitting a raylike neighborhood, which includes the interior points of a domain), then all properties and theorems belonging to the known theory of differentiability remain valid in this extended theory. For generalized differentiability, the corresponding calculus (differentiation techniques) is also provided by matrices—representatives of differentials at points. In this calculus the role of partial derivatives (which in general cannot exist for differentiable functions at some points) is taken by directional derivatives. [ABSTRACT FROM AUTHOR]

MIXTURE DESIGN AND OPTIMUM REGION VISUALIZATION WITH SPREADSHEETS AT EXCEL: A TUTORIAL. This tutorial shows how to perform the data treatment for mixture design with two and multiple responses. Two data sets from the literature were used to exemplify the calculations. A template in Excel was proposed to prepare contour plot and response surface to visualize the optimum region of the mixture, which usually is made with commercial computational programs. The desirability function was used in example 2 (optimization of 5 responses simultaneously), and a code also available for download was proposed. Besides, 21 videos were prepared to show all the details for the readers and, it is available on YouTube. With this tutorial, it is possible to learn in a practical fashion how to handle the data from mixture design and, the authors hope to contribute with the researchers in this area.

The new Laplace variational iterative method is used in this research for solving the (2+1)-D and (3+1)-D Burgers equations. This technique relies on the modified variational iteration method and the Laplace transform. To apply this approach, the differential problem is first transformed into an algebraic form using the Laplace transform, and then the algebraic equations are iteratively solved using the modified variational iterative approach. By utilizing this technique, the Burgers equations can be solved both numerically and analytically. The study demonstrates the effectiveness of the new Laplace variational iterative approach through three specific examples.

Abstract Evaluating the exact first derivative of a feedforward neural network (FFNN) output with respect to the input feature is pivotal for performing the sensitivity analysis of the trained neural network with respect to the inputs. In this paper, a novel method is presented that computes the analytical quality first derivative of a trained feedforward neural network output with respect to the input features without the need for backpropagation. To this end, the complex step derivative approximation is illustrated, and its implementation in the framework of the feedforward neural network is described. Artificial datasets are generated, and the efficacy of the proposed method for both regression and classification tasks is demonstrated. The results obtained for the regression task indicated that the proposed method is capable of obtaining analytical quality derivatives, and in the case of the classification task, the least relevant features could be identified.

The generalization of the formula of a priori accuracy estimation for the case of implicitly given functions is performed. This is based on the classical definition of the root mean square error, which is given as the sum of the squares of the products of partial derivatives of the arguments and the errors of their definition. Differentiation of functions is carried out with the involvement of the theory of implicit functions of many variables, which does not require explicit assignment of the function by an analytical expression. The corresponding derivatives are determined by the differentiation of the equations in which the investigated function appears, according to the corresponding variables, including the function itself. Only the values of the function and arguments for which the accuracy is evaluated are required for the calculations. These values are found in different ways, including approximate methods for solving nonlinear equations (for example, Newton's method, half-division method). This approach is generalized to the case of several functions, which are already determined by a set of nonlinear equations. Their differentiation gives a linear system, which solutions are elements in the formulas for estimating the accuracy of each function. The solution of this system is determined by Cramer's method. Since the matrix of coefficients is the same for all linear systems, it is advisable to use the inverse matrix method to solve it. This significantly reduces the calculations. The values of the functions for which errors are determined are obtained from the set of equations that connect them. Finding them is much more problematic than for one variable. Thus, a strict priori estimate of accuracy is obtained without any restrictions on the studied functions, for example, in the form of their approximate representation by Taylor series or approximate estimates when solving equations. Proposed method is tested on test examples, which include the assessment of accuracy for both one and two variables, and is considered in the first case. The results of the calculations confirm the feasibility of using this technique. Therefore, with the traditional approach, the above algorithm can be used in more complex cases, i.e. for the case of implicit definition of the function.

The computationally hard protein–protein complex structure prediction problem is continuously fascinating to the scientific community due to its biological impact. The field has witnessed the application of geometric algorithms, randomized algorithms, and evolutionary algorithms to name a few. These techniques improve either the searching or scoring phase. An effective searching strategy does not generate a large conformation space that perhaps demands computational power. Another determining factor is the parameter chosen for score calculation. The proposed method is an attempt to curtail the conformations by limiting the search procedure to probable regions. In this method, partial derivatives are calculated on the coarse-grained representation of the surface residues to identify the optimal points on the protein surface. Contrary to the existing geometric-based algorithms that align the convex and concave regions of both proteins, this method aligns the concave regions of the receptor with convex regions of the ligand only and thus reduces the size of conformation space. The method's performance is evaluated using the 55 newly added targets in Protein–Protein Docking Benchmark v 5 and is found to be successful for around 47% of the targets. The proposed method explores the shape complementarity of interacting proteins to predict the complex structure. An invariant shape descriptor is crucial in shape based docking algorithm. Coarse-grained representation of proteins reduces the size of the conformational space. [ABSTRACT FROM AUTHOR]

In this article, we extend the concept of the Aboodh transform to the solution of partial differential equations of fractional order using Caputo's fractional derivative. The transformation concerned is applicable to solve many classes of partial differential equations with derivatives of order and integrals. Consequently, the fractional Delay Differential Equations (DDEs) which we are going to study in this work. The resulting numerical proofs show that the method converges favorably towards the analytical solution.

Wetlands are important ecosystems and constitute one of the major carbon reservoirs on Earth. Therefore, accurate estimation of wetland carbon stocks, quantitative analysis of spatial and temporal changes in wetland carbon stocks, and maintenance and enhancement of wetland carbon sink functions play a pivotal role in achieving China’s “carbon neutral” strategy. Vegetation net primary productivity (NPP) is one of the core components of wetland ecosystem carbon budgets. In this study, an improved CASA model and the spatiotemporal fusion algorithm were employed to obtain NPP data with good accuracy and high spatiotemporal resolution for the Dongting Lake wetland area for the last two decades. Quantitative methods based on partial derivatives were used to assess the contribution of climate and anthropogenic factors to NPP changes. The results demonstrate that the NPP of wetland vegetation in Dongting Lake showed a significant increasing trend from 2000 to 2019 (0.86 g C/m2-yr, P

The linear dynamical system in partial derivatives is considered. The system must be transferred from an arbitrary initial state to an arbitrary final state through an arbitrary number of control points. The problem is to find the control for this system. The criterion for the complete controllability of such a system is derived using the method of cascade decomposition. Control and state functions in analytical form are constructed (with any given degree of smoothness), using the method of cascade decomposition. [ABSTRACT FROM AUTHOR]

To better describe and understand the time dynamics in functional data analysis, it is often desirable to recover the partial derivatives of the random surface. A novel approach is proposed based on marginal functional principal component analysis to derive the representation for partial derivatives. To obtain the Karhunen-Loève expansion of the partial derivatives, an adaptive estimation is explored. Asymptotic results of the proposed estimates are established. Simulation studies show that the proposed methods perform well in finite samples. Application to the human mortality data reveals informative time dynamics in mortality rates. [ABSTRACT FROM AUTHOR]

Evaluating the exact first derivative of a feedforward neural network (FFNN) output with respect to the input feature is pivotal for performing the sensitivity analysis of the trained neural network with respect to the inputs. In this paper, a novel method is presented that computes the analytical quality first derivative of a trained feedforward neural network output with respect to the input features without the need for backpropagation. To this end, the complex step derivative approximation is illustrated, and its implementation in the framework of the feedforward neural network is described. Artificial datasets are generated, and the efficacy of the proposed method for both regression and classification tasks is demonstrated. The results obtained for the regression task indicated that the proposed method is capable of obtaining analytical quality derivatives, and in the case of the classification task, the least relevant features could be identified. [ABSTRACT FROM AUTHOR]

In the traditional approach to differentiability, found in almost all university textbooks, this notion is considered only for interior points of the domain of function or for functions with an open domain. This approach leads to the fact that differentiability has usually been considered only for functions with an open domain in Rn, which severely limits the possibility of applying the potential techniques and tools of differential calculus to a broader class of functions. Although there is a great need for generalization of the notion of differentiability of a function in various problems of mathematical analysis and other mathematical branches, the notion of differentiability of a function at the non-interior points of its domain has almost not been considered or successfully defined. In this paper, we have generalized the differentiability of scalar and vector functions of several variables by defining it at non-interior points of the domain of the function, which include not only boundary points but also all points at which the notion of linearization is meaningful (points admitting nbd rays). This generalization allows applications in all areas where standard differentiability can be applied. With this generalized approach to differentiability, some unexpected phenomena may occur, such as a function discontinuity at a point where a function is differentiable, the non-uniqueness of differentials… However, if one reduces this theory only to points with some special properties (points admitting a linearization space with dimension equal to the dimension of the ambient Euclidean space of the domain and admitting a raylike neighborhood, which includes the interior points of a domain), then all properties and theorems belonging to the known theory of differentiability remain valid in this extended theory. For generalized differentiability, the corresponding calculus (differentiation techniques) is also provided by matrices—representatives of differentials at points. In this calculus the role of partial derivatives (which in general cannot exist for differentiable functions at some points) is taken by directional derivatives.

We are interested in regular expressions that represent word relations in an alphabet-invariant way—for example, the set of all word pairs (u , v) where v is a prefix of u independently of what the alphabet is. Current software systems of formal language objects do not have a mechanism to define such objects. Labelled graphs (transducers and automata) with alphabet-invariant and user-defined labels were considered in a recent paper. In this paper we study derivatives of regular expressions over labels (atomic objects) in some set B. These labels can be any strings as long as the strings represent subsets of a certain monoid. We show that the number of partial derivatives of any type B regular expression is linearly bounded, and that one can define partial derivative labelled graphs, whose transition labels can be elements of another label set X as long as X and B refer to the same monoid. We also show how to use derivatives directly to decide whether a given word is in the language of a regular expression over set specs. Set specs and pairing specs are label sets allowing one to express languages and relations over large alphabets in a natural and concise way such that many algorithms work directly on these labels without the need to expand these labels to linear or quadratic size expressions. [ABSTRACT FROM AUTHOR]

Agrawal and Vinay [2008], Koiran [2012], and Tavenas [2013] have recently shown that an exp (ω (√n log n)) lower bound for depth four homogeneous circuits computing the permanent with bottom layer of x gates having fanin bounded by √n translates to a superpolynomial lower bound for general arithmetic circuits computing the permanent. Motivated by this, we examine the complexity of computing the permanent and determinant via such homogeneous depth four circuits with bounded bottom fanin. We show here that any homogeneous depth four arithmetic circuit with bottom fanin bounded by √n computing the permanent (or the determinant) must be of size exp (Ω(√n)). [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of (word) derivatives and partial derivatives due to (respectively) Brzozowski and Antimirov to tree derivatives using already known inductive formulae of quotients. We define a new family of extended regular tree expressions (using negation or intersection operators), and we show how to compute a Brzozowski-like inductive tree automaton; the fixed point of this construction, when it exists, is the derivative tree automaton. Such a deterministic tree automaton can be used to solve the membership test efficiently: the whole structure is not necessarily computed, and the derivative computations can be performed in parallel. We also show how to solve the membership test using our (Bottom-Up) partial derivatives, without computing an automaton. [ABSTRACT FROM AUTHOR]

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This study proposes a method to improve the permanent magnet synchronous motor (PMSM) dq0 model for torque using incremental inductance. The torque model is essentially a partial differential equation considering flux linkage and magnetic energy. In practice, non‐ideal factors, such as saturation non‐linearity, the cogging structure, and the distortion current, increase the difficulty of solving this equation. In this study, two parameters related to flux linkage and magnetic energy are derived using incremental inductance. In the differential range, they are independent of the currents. Therefore, the partial derivatives of flux linkage and magnetic energy can be expressed in the form of derivatives under the saturation non‐linearity conditions. Thereby a dq0 model that describes the torque ripple of a PMSM is established. This model can be used to calculate and analyse the transient characteristics of torque regardless of steady or unsteady current excitation. The main parameters of the model are obtained from finite‐element analysis (FEA). To reduce the dependency on FEA in the application, a derived version of this model is obtained by extending the range of the linearisation assumption. Finally, the rationality and effectiveness of the models are proven by comparison with the FEA and experiments, respectively. [ABSTRACT FROM AUTHOR]

Various surface-wave exploration methods have become increasingly important tools in investigating the properties of subsurface structures. Inversion of the experimental dispersion curves is generally an indispensable component of these methods. Accurate and reliable calculation of partial derivatives of surface-wave dispersion curves with respect to parameters of subsurface layers is critical to the success of these approaches if the linearized inversion strategies are adopted. Here we present an open-source MATLAB package, named SWPD (Surface Wave Partial Derivative), for modeling surface-wave (both Rayleigh- and Love-wave) dispersion curves (both phase and group velocity) and particularly for computing their partial derivatives with high precision. The package is able to compute partial derivatives of phase velocity and of Love-wave group velocity analytically based on the combined use of the reduced delta matrix theory and the implicit function theorem. For partial derivatives of Rayleigh-wave group velocity, a hemi-analytical method is presented, which analytically calculates all the first-order partial differentiations and approximates the mixed second-order partial differentiation term with a central difference scheme. We provide examples to demonstrate the effectiveness of this package, and demo scripts are also provided for users to reproduce all results of this paper and thus to become familiar with the package as quickly as possible. [ABSTRACT FROM AUTHOR]