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This paper is devoted to studying the regularity properties for the new maximal operator M φ {M_{\varphi}} and the fractional new maximal operator M φ , β {M_{\varphi,\beta}} in the local case. Some new pointwise gradient estimates of M φ , Ω {M_{\varphi,\Omega}} and M φ , β , Ω {M_{\varphi,\beta,\Omega}} are given. Moreover, the boundedness of M φ , Ω {M_{\varphi,\Omega}} and M φ , β , Ω {M_{\varphi,\beta,\Omega}} on Sobolev space is established. As applications, we also obtain the bounds of the above operators on Sobolev space with zero boundary values. [ABSTRACT FROM AUTHOR]

Multivariate uncertain calculus is a branch of mathematics that deals with differentiation and integration of uncertain fields based on uncertainty theory. This paper defines partial derivatives of uncertain fields for the first time by putting forward the concept of Liu field. Then the fundamental theorem, chain rule and integration by parts of multivariate uncertain calculus are derived. Finally, this paper presents an uncertain partial differential equation, and gives its integral form. [ABSTRACT FROM AUTHOR]

Active contour model (ACM) is considered as one of the most frequently employed models in image segmentation due to its effectiveness and efficiency. However, the segmentation results of images with intensity non-uniformity processed by the majority of existing ACMs are possibly inaccurate or even wrong in the forms of edge leakage, long convergence time and poor robustness. In addition, they usually become unstable with the existence of different initial contours and unevenly distributed intensity. To better solve these problems and improve segmentation results, this paper puts forward an ACM approach using adaptive local pre-fitting energy (ALPF) for image segmentation with intensity non-uniformity. Firstly, the pre-fitting functions generate fitted images inside and outside contour line ahead of iteration, which significantly reduces convergence time of level set function. Next, an adaptive regularization function is designed to normalize the energy range of data-driven term, which improves robustness and stability to different initial contours and intensity non-uniformity. Lastly, an improved length constraint term is utilized to continuously smooth and shorten zero level set, which reduces the chance of edge leakage and filters out irrelevant background noise. In contrast with newly constructed ACMs, ALPF model not only improves segmentation accuracy (Intersection over union(IOU)), but also significantly reduces computation cost (CPU operating time T), while handling three types of images. Experiments also indicate that it is not only more robust to different initial contours as well as different noise, but also more competent to process images with intensity non-uniformity. [ABSTRACT FROM AUTHOR]

Using recent estimates of maximum modulus for partial derivatives of the analytic functions with bounded L-index in joint variables we describe maximum modulus of these functions at the polydisc skeleton with given radii by the maximum modulus with lesser radii. Such a description is sufficient and necessary condition of boundedness of L-index in joint variables for functions which are analytic in a complete Reinhardt domain. The vector-valued function L is a positive and continuous function in the domain and its values at a point is greater than reciprocal of distance from the point to the boundary of the Reinhardt domain multiplied by some constant. [ABSTRACT FROM AUTHOR]

The manuscript is an initiative to construct a full and exhaustive theory of analytical multivariate functions in any complete Reinhardt domain by introducing the concept of L -index in joint variables for these functions for a given continuous, non-negative, non-vanishing, vector-valued mapping L defined in an interior of the domain with some behavior restrictions. The complete Reinhardt domain is an example of a domain having a circular symmetry in each complex dimension. Our results are based on the results obtained for such classes of holomorphic functions: entire multivariate functions, as well as functions which are analytical in the unit ball, in the unit polydisc, and in the Cartesian product of the complex plane and the unit disc. For a full exhaustion of the domain, polydiscs with some radii and centers are used. Estimates of the maximum modulus for partial derivatives of the functions belonging to the class are presented. The maximum is evaluated at the skeleton of some polydiscs with any center and with some radii depending on the center and the function L and, at most, it equals a some constant multiplied by the partial derivative modulus at the center of the polydisc. Other obtained statements are similar to the described one. [ABSTRACT FROM AUTHOR]

For derivative function estimation, conventional research only focuses on the derivative estimation of one-dimensional functions. This paper considers partial derivatives estimation of a multivariate variance function in a heteroscedastic model. A wavelet estimator of partial derivatives of a multivariate variance function is proposed. The convergence rates of a wavelet estimator under different estimation errors are discussed. It turns out that the strong convergence rate of the wavelet estimator is the same as the optimal uniform almost sure convergence rate of nonparametric function problems. [ABSTRACT FROM AUTHOR]

We present a theory for general partial derivatives of matrix functions of the form f(A(x)), where A(x) is a matrix path of several variables (x = (x1, . . . ,xj)). Building on results by Mathias [SIAM J. Matrix Anal. Appl., 17 (1996), pp. 610--620] for the first order derivative, we develop a block upper triangular form for higher order partial derivatives. This block form is used to derive conditions for existence and a generalized Daleckiī--Kreī formula for higher order derivatives. We show that certain specializations of this formula lead to classical formulas of quantum perturbation theory. We show how our results are related to earlier results for higher order Frechet derivatives. Block forms of complex step approximations are introduced, and we show how those are related to evaluation of derivatives through the upper triangular form. These relations are illustrated with numerical examples. [ABSTRACT FROM AUTHOR]

We use Action-Process-Object-Schema (APOS) theory to study students' geometric understanding of partial derivatives of functions of two variables. This study contributes to research on the teaching and learning of differential multivariable calculus and its didactics. This is an important area due to its multiple applications in science, mathematics, engineering, and technology (STEM). The study tests a previously proposed model of mental constructions students may use to understand partial derivatives through a set of activities designed to help students make the conjectured constructions. The model is based on the local linearity of differentiable two-variable functions, and the model-based activities explore the relationship between partial derivatives and tangent plane in different representations. We used semi-structured interviews with eleven students whose teacher used the three-part cycle—Activities designed with the genetic decomposition; collaborative work in small groups and Class discussion; and Exercises for home (ACE)—as pedagogical strategy. The model-based activity set based on local linearity and the ACE strategy helped students construct a geometric understanding of partial derivatives. Results led to reconsider and further refine the model. This study also resulted in improving activity sets and obtaining information on students' construction of second-order and mixed partial derivatives. [ABSTRACT FROM AUTHOR]

In this paper, a novel concept for a double transform in two dimensions known as the double SEJI integral transform has been proposed. Its key characteristics, including a few of its properties and theorems, have been established. A few wellknown functions were also available in the Double SEJI integral transform. Later, we learn about brand-new research on partial fractional Caputo derivatives and partial differential derivatives. Finally, we apply this new transform to various first- and second-order partial differential equations. [ABSTRACT FROM AUTHOR]

There is introduced a concept of index for the Fueter regular function of the quaternionic variables. There are considered three approaches (Fueter, Sudbery and Mariconda) constructing the Fueter regular function from a holomorphic function of complex variable. Using Mariconda's approach there are constucted some analogs of such elementary functions as the exponent, the sine and the cosine. For the Mariconda analogs we proved that they have bounded index and their indices equal 1, 2, 2, respectively. Using recent results on sum of entire functions whose derivatives are of bounded index it is established that the Fueter regular function constructed by Mariconda's approach is of bounded index, if the derivatives of its addends have bounded index. Also there was examined a function of the form H(q) = f1(x0 + ix1) + jf2(x2 + ix3), where f1 and f2 are entire functions of complex variable. For the function H it is proved its Fueter regularity and index boundedness if the first order derivatives of f1 and f2 have bounded index. Moreover, the index of the function H does not exceed the maximum of indices of the functions f'1 and f'2 increased by 1. [ABSTRACT FROM AUTHOR]

The manuscript is an initiative to construct a full and exhaustive theory of analytical multivariate functions in any complete Reinhardt domain by introducing the concept of L-index in joint variables for these functions for a given continuous, non-negative, non-vanishing, vector-valued mapping L defined in an interior of the domain with some behavior restrictions. The complete Reinhardt domain is an example of a domain having a circular symmetry in each complex dimension. Our results are based on the results obtained for such classes of holomorphic functions: entire multivariate functions, as well as functions which are analytical in the unit ball, in the unit polydisc, and in the Cartesian product of the complex plane and the unit disc. For a full exhaustion of the domain, polydiscs with some radii and centers are used. Estimates of the maximum modulus for partial derivatives of the functions belonging to the class are presented. The maximum is evaluated at the skeleton of some polydiscs with any center and with some radii depending on the center and the function L and, at most, it equals a some constant multiplied by the partial derivative modulus at the center of the polydisc. Other obtained statements are similar to the described one.

To study the hot deformation behavior of and obtain the optimal hot processing parameters for 2219 aluminum alloy, a new, precise constitutive model based on the partial derivative of flow data was constructed and hot processing maps were constructed based on the new model. First, isothermal compression experiments were conducted at strain rates of 0.01–10 s−1 and temperatures of 573–773 K, and the high-order differences of the logarithmic stress with respect to the temperature and logarithmic strain rate were calculated. Second, a new, precise constitutive model based on the high-order differences was constructed, and the predictive accuracies of the new model and the Arrhenius model were compared. Finally, the hot processing maps of 2219 aluminum alloy were constructed using the new model, and its optimal hot processing parameters were validated with metallographic experiments. The results showed that a first-order approximation between logarithmic stress and temperature and a third-order approximation between logarithmic stress and the logarithmic strain rate need to be considered to construct a high-precision constitutive model without significantly increasing material parameters. The new model exhibited a significantly higher prediction accuracy than the Arrhenius model at a high strain rate and low temperature levels. With an increase in temperature, the energy dissipation increased at a constant strain rate, and with an increase in the strain rate, the energy dissipation first increased and then decreased at constant temperature. The best region for hot processing was located in the temperature range of 673–773 K and the strain rate range of 0.1–1 s−1. The results of microstructure analysis were in good agreement with the prediction results of hot processing maps. Hot processing maps can be used to guide the hot working process formulation of 2219 aluminum alloy. [ABSTRACT FROM AUTHOR]

To solve the problem of insufficient predictability in the classical models for the Ti6242s alloy, a new constitutive model was proposed, based on the partial derivatives from experimental data and the Taylor series. Firstly, hot compression experiments on the Ti6242s alloy at different temperatures and different strain rates were carried out, and the Arrhenius model and Hensel–Spittel model were constructed. Secondly, the partial derivatives of logarithmic stress with respect to temperature and logarithmic strain rate at low, medium and high strain levels were analyzed. Thirdly, two new constitutive models with first- and second-order approximation were proposed to meet the requirements of high precision. In this new model, by analyzing the high-order differential data of experimental data and combining the Taylor series theory, the minimum number of terms that can accurately approximate the experimental rheological data was found, thereby achieving an accurate prediction of flow stress with minimal material parameters. In the new model, by analyzing the high-order differential of the experimental data and combining the theory of the Taylor series, the minimum number of terms that can accurately approximate the experimental rheological data was found, thereby achieving an accurate prediction of flow stress with minimal material parameters. Finally, the prediction accuracies for the classical model and the new model were compared, and the predictabilities for the classical models and the new model were proved by mathematical means. The results show that the prediction accuracies of the Arrhenius model and the Hensel–Spittel model are low in the single-phase region and high in the two-phase region. In addition, second-order approximation is required between the logarithmic stress and logarithmic strain rate, and first-order approximation is required between logarithmic stress and temperature to establish a high-precision model. The order of prediction accuracy of the four models from high to low is the quadratic model, Arrhenius model, linear model and HS model. The prediction accuracy of the quadratic model in all temperatures and strain rates had no significant difference, and was higher than the other models. The quadratic model can greatly improve prediction accuracy without significantly increasing the material parameters. [ABSTRACT FROM AUTHOR]

For derivative function estimation, conventional research only focuses on the derivative estimation of one-dimensional functions. This paper considers partial derivatives estimation of a multivariate variance function in a heteroscedastic model. A wavelet estimator of partial derivatives of a multivariate variance function is proposed. The convergence rates of a wavelet estimator under different estimation errors are discussed. It turns out that the strong convergence rate of the wavelet estimator is the same as the optimal uniform almost sure convergence rate of nonparametric function problems.

Constructing faradaic electrode with superior desalination performance is important for expanding the applications of capacitive deionization (CDI). Herein, a simple one‐step alkalized treatment for in situ synthesis of 1D TiO2 nanowires on the surface of 2D Ti3C2 nanosheets, forming a Ti3C2‐MXene partially derived hierarchical 1D/2D TiO2/Ti3C2 heterostructure as the cathode electrode is reported. Cross‐linked TiO2 nanowires on the surface help avoid layer stacking while acting as the protective layer against contact of internal Ti3C2 with dissolved oxygen in water. The inner Ti3C2 MXene nanosheets cross over the TiO2 nanowires can provide abundant active adsorption sites and short ion/electron diffusion pathways.. Density functional theory calculations demonstrated that Ti3C2 can consecutively inject electrons into TiO2, indicating the high electrochemical activity of the TiO2/Ti3C2. Benefiting from the 1D/2D hierarchical structure and synergistic effect of TiO2 and Ti3C2, TiO2/Ti3C2 heterostructure presents a favorable hybrid CDI performance, with a superior desalination capacity (75.62 mg g−1), fast salt adsorption rate (1.3 mg g−1 min−1), and satisfactory cycling stability, which is better than that of most published MXene‐based electrodes. This study provides a feasible partial derivative strategy for construction of a hierarchical 1D/2D heterostructure to overcome the restrictions of 2D MXene nanosheets in CDI. [ABSTRACT FROM AUTHOR]

The differentiability of the one parameter family of Okomoto's functions as functions of $x$ has been analyzed extensively since their introduction in 2005. As an analogue to a similar investigation, in this paper, we consider the partial derivative of Okomoto's functions with respect to the parameter $a$. We place a significant focus on $a = 1/3$ to describe the properties of a nowhere differentiable function $K(x)$ for which the set of points of infinite derivative produces an example of a measure zero set with Hausdorff dimension $1$. [ABSTRACT FROM AUTHOR]

• Eliminate the need for tedious analysis and derivation of intermediate variables. • Simplify the establishment of constraint equations and require constructing fewer equations. • The proposed method is more straightforward to establish with explicit form than open dynamics Engine's method and requires less computational complexity. • Reduce the difficulty of engineering implementation via explicit, canonical, and symbolic characteristics, especially for high-DOF mobile multi-axis systems. The wheel-ground unilateral constraint is essential in establishing the complete mobile multi-axis system dynamics. To reduce the calculation complexity and improve the dynamic performance, an innovative joint-space dynamic theory for mobile multi-axis systems with unilateral constraints is proposed. This present study builds on our existing explicit dynamics studies of tree-chain rigid multi-axis systems. By analyzing the formulation of the unilateral constraints, the complexity of establishing the constraint equations is reduced while the physical implications are clear. The constraint equations are derived and established based on the explicit partial derivative equations, where the expression of the constraint equations is greatly simplified. Then, based on the process of backward force iteration and in combination with the derived dynamic and unilateral constraint equations, the solution and analytical procedure for unilateral constraints are presented. The accuracy of the proposed method is proved by the three-wheeled multi-axis system and Mars rover examples. The proposed method is explicit, canonical, and symbolic and has the advantages of simple modeling and low computational complexity, which is analyzed by comparing it with the Open Dynamic Engine's approach. [ABSTRACT FROM AUTHOR]

To study the hot deformation behavior of and obtain the optimal hot processing parameters for 2219 aluminum alloy, a new, precise constitutive model based on the partial derivative of flow data was constructed and hot processing maps were constructed based on the new model. First, isothermal compression experiments were conducted at strain rates of 0.01–10 s−1 and temperatures of 573–773 K, and the high-order differences of the logarithmic stress with respect to the temperature and logarithmic strain rate were calculated. Second, a new, precise constitutive model based on the high-order differences was constructed, and the predictive accuracies of the new model and the Arrhenius model were compared. Finally, the hot processing maps of 2219 aluminum alloy were constructed using the new model, and its optimal hot processing parameters were validated with metallographic experiments. The results showed that a first-order approximation between logarithmic stress and temperature and a third-order approximation between logarithmic stress and the logarithmic strain rate need to be considered to construct a high-precision constitutive model without significantly increasing material parameters. The new model exhibited a significantly higher prediction accuracy than the Arrhenius model at a high strain rate and low temperature levels. With an increase in temperature, the energy dissipation increased at a constant strain rate, and with an increase in the strain rate, the energy dissipation first increased and then decreased at constant temperature. The best region for hot processing was located in the temperature range of 673–773 K and the strain rate range of 0.1–1 s−1. The results of microstructure analysis were in good agreement with the prediction results of hot processing maps. Hot processing maps can be used to guide the hot working process formulation of 2219 aluminum alloy.

To solve the problem of insufficient predictability in the classical models for the Ti6242s alloy, a new constitutive model was proposed, based on the partial derivatives from experimental data and the Taylor series. Firstly, hot compression experiments on the Ti6242s alloy at different temperatures and different strain rates were carried out, and the Arrhenius model and Hensel–Spittel model were constructed. Secondly, the partial derivatives of logarithmic stress with respect to temperature and logarithmic strain rate at low, medium and high strain levels were analyzed. Thirdly, two new constitutive models with first- and second-order approximation were proposed to meet the requirements of high precision. In this new model, by analyzing the high-order differential data of experimental data and combining the Taylor series theory, the minimum number of terms that can accurately approximate the experimental rheological data was found, thereby achieving an accurate prediction of flow stress with minimal material parameters. In the new model, by analyzing the high-order differential of the experimental data and combining the theory of the Taylor series, the minimum number of terms that can accurately approximate the experimental rheological data was found, thereby achieving an accurate prediction of flow stress with minimal material parameters. Finally, the prediction accuracies for the classical model and the new model were compared, and the predictabilities for the classical models and the new model were proved by mathematical means. The results show that the prediction accuracies of the Arrhenius model and the Hensel–Spittel model are low in the single-phase region and high in the two-phase region. In addition, second-order approximation is required between the logarithmic stress and logarithmic strain rate, and first-order approximation is required between logarithmic stress and temperature to establish a high-precision model. The order of prediction accuracy of the four models from high to low is the quadratic model, Arrhenius model, linear model and HS model. The prediction accuracy of the quadratic model in all temperatures and strain rates had no significant difference, and was higher than the other models. The quadratic model can greatly improve prediction accuracy without significantly increasing the material parameters.

In this paper, a new general double integral transform is introduced. We present its essential properties and proved some useful results such as the double convolution theorem and derivative properties. Furthermore, we apply the proposed double general integral transform to solve some partial differential equations such as telegraph and Klein–Gordon equations. [ABSTRACT FROM AUTHOR]

The multivariable chain rule is often challenging to students because it is usually presented with ambiguities and other defects that hamper systematic and reliable application. A very simple formulation combines the derivation operators for functions and for expressions in a manner not found elsewhere due to common confusion between them. Some issues are rooted more deeply than others and are discussed in a broader perspective, starting with the function concept. The approach is illustrated using various applications including the transport equation, partial derivatives of a definite integral, and the distortionless (but not lossless) transmission line. This note is suitable for a lecture in any first-year course covering partial derivatives, as a complement to the other course material. [ABSTRACT FROM AUTHOR]

In the paper, we introduce a new partial derivative call it β-partial derivatives as the most natural extensions of the limit definitions of the partial derivative and the β-derivative, which obeys classical properties including: continuity, linearity, product rule, quotient rule, power rule, chain rule and vanishing derivatives for constant functions. As applications, we establish some new Opial-Traple type inequalities for the β-partial derivatives.

Abstract Introduction When the serum [creatinine] is changing, creatinine kinetics can still gauge the kidney function, and knowing the kinetic glomerular filtration rate (GFR) helps doctors take care of patients with renal failure. We wondered how the serum [creatinine] would respond if the kinetic GFR were tweaked. In every scenario, if the kinetic GFR decreased, the [creatinine] would increase, and vice versa. This opposing relationship was hypothesized to be universal. Methods Serum [creatinine] and kinetic GFR, along with other parameters, are described by a differential equation. We differentiated [creatinine] with respect to kinetic GFR to test if the two variables would change oppositely of each other, throughout the gamut of all allowable clinical values. To remove the discontinuities in the derivative, limits were solved. Results The derivative and its limits were comprehensively analyzed and proved to have a sign that is always negative, meaning that [creatinine] and kinetic GFR must indeed move in opposite directions. The derivative is bigger in absolute value at the higher end of the [creatinine] scale, where a small drop in the kinetic GFR can cause the [creatinine] to shoot upward, making acute kidney injury similar to chronic kidney disease in that regard. Conclusions All else being equal, a change in the kinetic GFR obligates the [creatinine] to change in the opposite direction. This does not negate the fact that an increasing [creatinine] can be compatible with a rising kinetic GFR, due to differences in how the time variable is treated.

Introduction: When the serum [creatinine] is changing, creatinine kinetics can still gauge the kidney function, and knowing the kinetic glomerular filtration rate (GFR) helps doctors take care of patients with renal failure. We wondered how the serum [creatinine] would respond if the kinetic GFR were tweaked. In every scenario, if the kinetic GFR decreased, the [creatinine] would increase, and vice versa. This opposing relationship was hypothesized to be universal. Methods: Serum [creatinine] and kinetic GFR, along with other parameters, are described by a differential equation. We differentiated [creatinine] with respect to kinetic GFR to test if the two variables would change oppositely of each other, throughout the gamut of all allowable clinical values. To remove the discontinuities in the derivative, limits were solved. Results: The derivative and its limits were comprehensively analyzed and proved to have a sign that is always negative, meaning that [creatinine] and kinetic GFR must indeed move in opposite directions. The derivative is bigger in absolute value at the higher end of the [creatinine] scale, where a small drop in the kinetic GFR can cause the [creatinine] to shoot upward, making acute kidney injury similar to chronic kidney disease in that regard. Conclusions: All else being equal, a change in the kinetic GFR obligates the [creatinine] to change in the opposite direction. This does not negate the fact that an increasing [creatinine] can be compatible with a rising kinetic GFR, due to differences in how the time variable is treated. [ABSTRACT FROM AUTHOR]

In this paper, we give estimates on partial derivatives and logarithmic partial derivatives for holomorphic functions on polydiscs. Estimates will also be utilized to characterize entire solutions of a class of partial differential equations, which gives a new form of Picard's theorem and its higher dimensional version. [ABSTRACT FROM AUTHOR]

We generalize some criteria of boundedness of $\mathbf{L}$-index in joint variables for in a bidisc analytic functions. Our propositions give an estimate the maximum modulus on a skeleton in a bidisc and an estimate of $(p+1)$th partial derivative by lower order partial derivatives (analogue of Hayman's theorem).

In this paper, we investigate partial integrals and partial derivatives of bivariate fractal interpolation functions (FIFs). We also prove that the mixed Riemann–Liouville fractional integral and derivative of order γ = (p , q) ; p > 0 , q > 0 , of bivariate FIFs are again bivariate interpolation functions corresponding to some iterated function system (IFS). Furthermore, we discuss the integral transforms and fractional order integral transforms of the bivariate FIFs. [ABSTRACT FROM AUTHOR]

Intuitionistic fuzzy set (IFS) can depict an object simultaneously from both the superiority and inferiority aspects, and its basic element is an intuitionistic fuzzy number (IFN). Based on four basic operational laws of IFNs, the intuitionistic fuzzy calculus (IFC) has been proposed recently. However, there is no research on the limit theory and differential calculus of intuitionistic fuzzy functions with several variables based on the four basic operational laws of IFNs. Hence, the aim of this article is to systematically investigate them. In this article, we first introduce the concepts of limits and iterated limits, and then discuss their properties, based on which, we reveal their relationships among the limits, the iterated limits, and continuities. Finally, we offer the notions of the differentials by the intuitionistic fuzzy infinitesimals, and especially, we obtain the concrete values of partial derivatives. In addition, we discuss in detail the relationships between the total differentials and partial derivatives, which are thoroughly different from the corresponding relationships in the IFC of one variable. [ABSTRACT FROM AUTHOR]

We consider linear evolutionary systems of partial differential equations with constant coefficients of general form. We suppose that the matrix of operators at the higher time derivative of the sought vector-function is degenerate. These systems are called partial differential-algebraic equations (DAEs). The index is the most important characteristic defining the structure complexity of these equations. We discuss the ways of approach to the definition of index for partial DAEs and some related questions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the absolute convergence of Fourier series of functions in several variables for an odd-dimensional space when these functions have continuous partial derivatives. It should be noted that similar properties for an even-dimensional space were given in [L. D. Gogoladze and V. S. Tsagareishvili, On absolute convergence of multiple Fourier series (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. (2015), no. 9, 12-21; translation in Russian Math. (Iz. VUZ) 59 (2015), no. 9, 9-17]. The obtained results are the best possible in a certain sense. [ABSTRACT FROM AUTHOR]

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