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The article discusses the 1913 paper "Sur les fonctionnelles continues et les fonctionnelles analytiques," by René Gâteaux. Focus is given on the modern concept of functional analysis based from the ideas of Gâteaux in the paper. Also provided is information on Gâteaux derivative and influence function in estimation theory.

This paper introduces a topology optimization approach based on the virtual element method (VEM), incorporating uncertainties. The objective of this optimization process is to design an optimal material layout for problems governed by linear elasticity equations to minimize the volume while satisfying probabilistic compliance constraints. The VEM is used to solve the boundary value problem in reliability-based topology optimization (RBTO). In the comparison between the VEM and the standard finite element method (FEM), a key difference emerges in the absence of explicitly defined shape functions tied to discrete degrees of freedom in VEM. Unlike FEM, VEM directly constructs the discrete bilinear form and load linear form without the need for computing shape function derivatives within the elements. This flexibility accommodates meshes with intricate geometries and arbitrarily shaped elements. The paper discusses the computational efficiency of VEM RBTO and explores the geometric impact of tessellations on converged topologies, demonstrating reduced susceptibility to checkerboard patterns compared to conventional quadrilateral elements. Additionally, the single-loop approach is examined, showcasing comparable accuracy to the first-order/second-order reliability methods (FORM/SORM) of RBTO using VEM. Numerical results for several problems that demonstrate the feasibility of the proposed method are presented. [ABSTRACT FROM AUTHOR]

Users andedge servers arenot fullymutuallytrusted inmobile edge computing(MEC), andhenceblockchaincanbe introduced to provide trustable MEC. In blockchain-based MEC, each edge server functions as a node in bothMEC and blockchain, processing users' tasks and then uploading the task related information to the blockchain. That is, each edge server runs both users' offloaded tasks and blockchain tasks simultaneously.Note that there is a trade-off between the resource allocation for MEC and blockchain tasks. Therefore, the allocation of the resources of edge servers to the blockchain and the MEC is crucial for the processing delay of blockchain-based MEC. Most of the existing research tackles the problem of resource allocation in either blockchain or MEC,which leads to unfavorable performance of the blockchain-based MEC system. In this paper, we study how to allocate the computing resources of edge servers to the MEC and blockchain tasks with the aim to minimize the total system processing delay. For the problem, we propose a computing resource Allocation algorithm for Blockchain-based MEC (ABM) which utilizes the Slater's condition, Karush-Kuhn-Tucker (KKT) conditions, partial derivatives of the Lagrangian function and subgradient projectionmethod to obtain the solution. Simulation results show that ABM converges and effectively reduces the processing delay of blockchain-based MEC. [ABSTRACT FROM AUTHOR]

This paper proposes two data-driven models (including LSTM pricing model, WGAN pricing model) and an improved model of LSM based on GAN to analyze the pricing of convertible bonds. In addition, the LSM model with higher precision in traditional pricing model is selected for comparative study with other pricing models. It is found that the traditional LSM pricing model has a large error in the first-day pricing, and the pricing function needs to be further improved. Among the four pricing models, LSTM pricing model and WGAN pricing model have the best pricing effect. The WGAN pricing model is better than the LSTM pricing model (0.21%), and the LSM improved model (1.17%) is better than the traditional LSM model (2.26%). Applying the generative deep learning model GAN to the pricing of convertible bonds can circumvent the harsh preconditions of assumptions, and significantly improve the pricing effect of the traditional model. The scope of application of each model is different. Therefore, this paper proves the feasibility of the GAN model applied to the pricing of convertible bonds, and enriches the pricing function of derivatives in the financial field. [ABSTRACT FROM AUTHOR]

This paper addresses the uniqueness problem concerning the j-th derivative of a meromorphic function f(z) and the k-th derivative of its shift, f(z + c), where j, k are integers with 0 ≤ j < k. In this regard, our work surpasses the achievements of [2], as we have improved upon the existing results and provided a more refined understanding of this specific aspect. We give some illustrative examples to enhance the realism of the obtained outcomes. Denote by E(a, f) the set of all zeros of f-a, where each zero with multiplicity m is counted m times. In the paper proved, in particular, the following statement: Let f(z) be a non-constant meromorphic function of finite order, c be a non-zero finite complex number and j, k be integers such that 0 ≤ j < k. If f(j)(z) and f(k)(z + c) have the same a-points for a finite value a(̸= 0) and satisfy conditions... [ABSTRACT FROM AUTHOR]

There are numerous applications for finding zero of derivatives in function optimization. In this paper, a two-step fourth-order method was presented for finding a zero of the derivative. In the research process of iterative methods, determining the ball of convergence was one of the important issues. This paper discussed the radii of the convergence ball, uniqueness of the solution, and the measurable error distances. In particular, in contrast to Wang's method under hypotheses up to the fourth derivative, the local convergence of the new method was only analyzed under hypotheses up to the second derivative, and the convergence order of the new method was increased to four. Furthermore, different radii of the convergence ball was determined according to different weaker hypotheses. Finally, the convergence criteria was verified by three numerical examples and the new method was compared with Wang's method and the same order method by numerical experiments. The experimental results showed that the convergence order of the new method is four and the new method has higher accuracy at the same cost, so the new method is finer. [ABSTRACT FROM AUTHOR]

The framework of this paper is behavioral finance and, more specifically, intertemporal choice when individuals exhibit decreasing impatience in their decision-making processes. After characterizing the two main types of decreasing impatience (moderately and strongly decreasing impatience), the main objective of this paper is to generalize these concepts when the criterion of time increase is given by an arbitrary function which describes such increments. In general, the methodology is mathematical calculus but particularly the concept of derivative according to the function which rules the increase of time. The main contribution of this paper is the characterization of this extension of the concept of decreasing impatience by using the aforementioned novel derivative and the well-known Prelec's index. [ABSTRACT FROM AUTHOR]

In this paper on a basis of functional analysis methods we justified theoretically the possibility of different orders root-polynomial functions application for interpolation and approximation of the boundary trajectory of an electron beam in case of its propagation in ionized gas with compensation of the space charge of the beam electrons. It is shown, that the root-polynomial functions satisfy to the second-order differential equation, describing the boundary trajectory of the beam electrons under such physical conditions. The results of interpolation and approximation of the boundary trajectory of the electron beam by root-polynomial functions from the second to the fifth order under the following physical conditions are presented. The interpolation results are compared with the corresponded results of the differential equation solution for the boundary trajectory of the electron beam using Runge-Kutta numerical method of the fourth order. These results are considered as reference ones for the interpolation task. To solve the approximation problem, in this paper an iterative algorithm based on the calculation of both values of the function and its derivatives at reference points is proposed. The approximation task is solved for a sample of numerical data obtained by experimental electron-beam equipment for real processes of current electron-beam technologies, which led to a rather large value of the experimental measurement error due to the effect of random factors associated with thermal treatment of products with electron beam. Test calculations show that the error of interpolation and approximation of numerical data, describing the boundary trajectory of electron beam in case of its propagation in ionized gas, does not exceed a few percent. The theoretical and practical results obtained in this paper are interesting for a wide range of specialists who are engaged in the physics of electron beams, the development of electron-beam technological equipment and implementation of current electron-beam technologies into industry. [ABSTRACT FROM AUTHOR]

This paper presents an investigation of the uniqueness problem of derivatives of a meromorphic function and its difference polynomial in view of a partially sharing. As a consequence of the main result, we improve the recent result of W. J. Chen and Z. G. Huang with the weaker hypotheses and also supplement several results in particular cases. [ABSTRACT FROM AUTHOR]

Wave propagation or acoustic emission waves caused by impact load can be simulated using the finite element (FE) method with a refined high-fidelity mesh near the impact location. This paper presents a method to refine a 3D finite element mesh by increasing the polynomial order near the impact location. Transition elements are required for such a refinement operation. Three protocols are defined to implement the transition elements within the low-order FE mesh. Due to the difficulty of formulating shape functions and verification, there are no transition elements beyond order two in the current literature for 3D elements. This paper develops a complete set of transition elements that facilitate the transition from first- to fourth-order Lagrangian elements, which facilitates mesh refinement following the protocols. The shape functions are computed and verified, and the interelement compatibility conditions are checked for each element case. The integration quadratures and shape function derivative matrices are also computed and made readily available for FE users. Finally, two examples are presented to illustrate the applicability of this method. [ABSTRACT FROM AUTHOR]

In this paper, we establish some upper bound estimates for the maximum modulus of the higher order derivative of a polynomial with restricted zeros on a circle of |z| = R, R ≥ 1, in terms of the maximum modulus of the same polynomial on the unit circle |z| = 1. These results generalize and sharpen some known results in this direction. [ABSTRACT FROM AUTHOR]

For µ ≥ −1/2, and under appropriate conditions on the sequence {Mp}∞p=0 of weights, the elements, the (weakly, weakly*, strongly) bounded subsets, and the (weakly, weakly*, strongly) convergent sequences in the dual of a space Kµ of type Hankel-K{Mp} can be represented by distributional derivatives of functions and measures in terms of iterated adjoints of the differential operator x−1Dx and the Bessel operator Sµ = x−µ−1/2 Dx x 2µ+1Dxx−µ−1/2. In this paper, such representations are compiled, and new ones involving adjoints of suitable iterations of the Zemanian differential operator Nµ = xµ+1/2Dxx−µ−1/2 are proved. Prior to this, new descriptions of the topology of the space Kµ are given in terms of the latter iterations. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive controller for a class of uncertain nonlinear systems is presented using a combination of Legendre polynomials and brain emotional learning-based intelligent controller (BELBIC). Recently, some versions of BELBIC have been presented with the aim of satisfying the universal approximation property using Gaussian basis function. However, the size of regressor vector is too large that imposes a heavy computational load to the processor. The novelty of this paper is presenting a new version of BELBIC with less computational burden using Legendre polynomials. Moreover, there are very few tuning parameters in Legendre polynomials. Another contribution of this paper is editing the stability analysis presented in recent related works. Due to the intrinsic non-differentiability of the adaptation rules of BELBIC, the second time derivative of Lyapunov function is undefined and thus, the Barbalat's lemma cannot be applied to verify the asymptotic convergence of the error function. Therefore, bounded-input-bounded-output (BIBO) stability can only be claimed for this controller. Simulation results on different case studies show that Legendre polynomials can improve the universal approximation property of BELBIC with less tuning parameters. Moreover, in the absence of the robust control term in the control law, the performance Legendre polynomials will not deteriorate, while the performance degrade in Gaussian basis function is quite considerable. [ABSTRACT FROM AUTHOR]

The work proves that under conditions of instability of the second derivatives of the function in the minimization region, the estimate of the convergence rate of Newton's method is determined by the parameters of the irreducible part of the conditionality degree of the problem. These parameters represent the degree of difference between eigenvalues of the matrices of the second derivatives in the coordinate system, where this difference is minimal, and the resulting estimate of the convergence rate subsequently acts as a standard. The paper studies the convergence rate of the relaxation subgradient method (RSM) with optimization of the parameters of two-rank correction of metric matrices on smooth strongly convex functions with a Lipschitz gradient without assumptions about the existence of second derivatives of the function. The considered RSM is similar in structure to quasi-Newton minimization methods. Unlike the latter, its metric matrix is not an approximation of the inverse matrix of second derivatives but is adjusted in such a way that it enables one to find the descent direction that takes the method beyond a certain neighborhood of the current minimum as a result of one-dimensional minimization along it. This means that the metric matrix enables one to turn the current gradient into a direction that is gradient-consistent with the set of gradients of some neighborhood of the current minimum. Under broad assumptions on the parameters of transformations of metric matrices, an estimate of the convergence rate of the studied RSM and an estimate of its ability to exclude removable linear background are obtained. The obtained estimates turn out to be qualitatively similar to estimates for Newton's method. In this case, the assumption of the existence of second derivatives of the function is not required. A computational experiment was carried out in which the quasi-Newton BFGS method and the subgradient method under study were compared on various types of smooth functions. The testing results indicate the effectiveness of the subgradient method in minimizing smooth functions with a high degree of conditionality of the problem and its ability to eliminate the linear background that worsens the convergence. [ABSTRACT FROM AUTHOR]

K-essence theories are usually studied in the framework of a single scalar field ϕ. Namely, the Lagrangian of K-essence is the function of the single scalar field ϕ and its covariant derivative. However, in this paper, we explore a double-field pure K-essence, i.e., the corresponding Lagrangian is the function of covariant derivatives of double scalar fields without a dependency on scalar fields themselves. This is why we call it double-field pure K-essence. The novelty of this K-essence is that its Lagrangian contains the quotient term of the kinetic energies from the two scalar fields. This results in the presence of many interesting features; for example, the equation of state can be arbitrarily small and arbitrarily large. In comparison, the range of the equation of state for quintessence is − 1 to + 1 . Interestingly, this novel K-essence can play the role of an inflation field, dark matter, or dark energy by appropriately selecting the expressions of Lagrangian. [ABSTRACT FROM AUTHOR]

In this paper, combining B-spline function and Tikhonov regularization, we propose an online identification approach for reconstructing a smooth function and its derivative from scattered data with heteroscedasticity. Our methodology offers the unique advantage of enabling real-time updates based on new input data, eliminating the reliance on historical information. First, to address the challenge of heteroscedasticity and computation cost, we employ weight coefficients along with a judiciously chosen set of knots for interpolation. Second, a reasonable approach is provided to select weight coefficients and the regularization parameter in objective functional. Finally, We substantiate the efficacy of our approach through a numerical example and demonstrate its applicability in solving inverse problems. It is worth mentioning that the algorithm not only ensures the calculation efficiency, but also trades the data accuracy through the data volume. [ABSTRACT FROM AUTHOR]

This paper is devoted to the lifespan estimates of small classical solutions of the initial value problems for one dimensional wave equations with semilinear terms of the spatial derivative of the unknown function. It is natural that the result is same as the one for semilinear terms of the timederivative. But there are so many differences among their proofs. Moreover, it is meaningful to study this problem in the sense that it may help us to investigate its blow-up boundary in the near future. [ABSTRACT FROM AUTHOR]

In the paper, we discuss the distribution of uniqueness and its elements over the extended complex plane from different polynomials of view. We obtain some new results regarding the structure and position of uniqueness. These new results have immense applications like classifying different expressions to be or not to be unique. The principal objective of the paper is to study the uniqueness of meromorphic functions when sharing a small function a(z) IM with restricted finite order and its nonlinear differential polynomials. The lemma on the logarithmic derivative by Halburb and Korhonen (Journal of Mathematical Analysis and Applications, 314 (2006), 477-87) is the starting point of this kind of research. In this direction, the current focus in this field involves exploring unique results for the differential-difference polynomials of meromorphic functions, covering both derivatives and differences. Liu et al. (Applied Mathematics A Journal of Chinese Universities, 27 (2012), 94-104) have notably contributed to this research. Their research establishes that when n ≤ k+2 for a finite-order transcendental entire function f the differential-difference polynomial [fnf(z + c)](k) - α(z) has infinitely many zeros. Here, α(z) is characterized by its smallness relatively to f. Additionally, for two distinct meromorphic functions f and g, both of finite order, if the differentialdifference polynomials [fnf(z + c)](k) and [gng(z + c)](k) share the value 1 in the same set, then f(z) = c1edz, g(z) = c2e-dz. We prove two results, which significantly generalize the results of Dyavanal and Mathai (Ukrainian Math. J., 71 (2019), 1032-1042), and Zhang and Xu (Comput. Math. Appl., 61 (2011), 722-730) and citing a proper example we have shown that the result is true only for a particular case. Finally, we present the compact version of the same result as an improvement. [ABSTRACT FROM AUTHOR]

This paper deals with a perturbed nonlinear system of fractional order differential equations (FrODEs) with Caputo derivative. The purpose of the paper is to discuss uniform stability (US), asymptotic stability (AS), Mittag-Leffer stability (MLS) of zero solution and boundedness at infinity of non-zero solutions of this perturbed nonlinear system of FrODEs with Caputo derivative. We obtain four new theorems on these mathematical concepts via a Lyapunov function (LF) and its Caputo derivative. For illustration, an example is provided which satisfies assumptions of the four new results and, in particular, shows their applications. The new results of this paper generalize and improve some recent ones in the literature and they have contributions to theory of FrODEs. [ABSTRACT FROM AUTHOR]

Apart from the fact that it was not shown that stretched exponential decay is the correct mechanism here, the GauLor shape is a non-physical shape glued together from two shapes which, individually, do have a physical basis. The argumentation that the new line shape named GauLor with Lorentz character in the central part and Gaussian wings is correct and is based on a comparison with the line shape resulting from a stretched exponential decay. This also brings us to Figure 9 in their paper where the authors argue that the fitting by a Gaussian or Lorentzian line shape requires many more components compared to their new line shape, 3-7 components are mentioned for the various spectra. [Extracted from the article]

Loss functions are an essential part in modern data-driven approaches, such as bi-level training scheme and machine learnings. In this paper, we propose a loss function consisting of a r -order (an)-isotropic total variation semi-norms TV r , r ∈ ℝ + , defined via the Riemann–Liouville (RL) fractional derivative. We focus on studying key theoretical properties, such as the lower semi-continuity and compactness with respect to both the function and the order of derivative r , of such loss functions. [ABSTRACT FROM AUTHOR]

We use the results from the very recent paper (Popa in Constr Approx, 2022. https://doi.org/10.1007/s00365-022-09584-3) to prove the complete asymptotic evaluation for general modified Mellin–Gauss–Weierstrass convolution operators. [ABSTRACT FROM AUTHOR]

In this paper, the fast algorithms of the derivatives of Bessel functions with respect to the parameter are obtained. Based on these fast algorithms, we discuss the calculations of the derivatives of the functions related to the heterogeneous Bessel differential equation, such as Anger, Weber, Struve and modified Struve functions. In addition, the fast calculation of some integrals related to these functions are obtained. At last, numerical examples show the algorithms given in this paper are fast and high precision. [ABSTRACT FROM AUTHOR]

This paper considers the estimation of SOC and SOH for lithium batteries using multi-innovation Levenberg–Marquardt and adaptive weighting unscented Kalman filter algorithms. For parameter identification, the second-order derivative of the objective function to optimize the traditional gradient descent algorithm is used. For SOC estimation, an adaptive weighting unscented Kalman filter algorithm is proposed to deal with the nonlinear update problem of the mean and covariance, which can substantially improve the estimation accuracy of the internal state of the lithium battery. Compared with fixed weights in the traditional unscented Kalman filtering algorithm, this algorithm adaptively adjusts the weights according to the state and measured values to improve the state estimation update accuracy. Finally, according to simulations, the errors of this algorithm are all lower than 1.63 %, which confirms the effectiveness of this algorithm. [ABSTRACT FROM AUTHOR]

This paper presents a topology optimization (TopO) method for conjugate heat transfer (CHT), with turbulent flows. Topological changes are controlled by an artificial material distribution field (design variables), defined at the cells of a background grid and used to distinguish a fluid from a solid material. To effectively solve the CHT problem, it is crucial to impose exact boundary conditions at the computed fluid–solid interface (FSI); this is the purpose of introducing the cut-cell method. On the grid, including also cut cells, the incompressible Navier–Stokes equations, coupled with the Spalart–Allmaras turbulence model with wall functions, and the temperature equation are solved. The continuous adjoint method computes the derivatives of the objective function(s) and constraints with respect to the material distribution field, starting from the computation of derivatives with respect to the positions of nodes on the FSI and then applying the chain rule of differentiation. In this work, the continuous adjoint PDEs are discretized using schemes that are consistent with the primal discretization, and this will be referred to as the "Think Discrete–Do Continuous" (TDDC) adjoint. The accuracy of the gradient computed by the TDDC adjoint is verified and the proposed method is assessed in the optimization of two 2D cases, both in turbulent flow conditions. The performance of the TopO designs is investigated in terms of the number of required refinement steps per optimization cycle, the Reynolds number of the flow, and the maximum allowed power dissipation. To illustrate the benefits of the proposed method, the first case is also optimized using a density-based TopO that imposes Brinkman penalization terms in solid areas, and comparisons are made. [ABSTRACT FROM AUTHOR]

This paper deals with integral control of a class of switched affine systems characterised by time-varying differential equations that, along with the discontinuities caused by the switching, also depend periodically on an exogenous parameter, which is an affine function of time. The goal is to design a state-dependent switching function with integral action that ensures global asymptotic tracking of a trajectory of interest. The integral action is responsible for providing robustness to the system against uncertainties and load variations, whenever the discrepancy between the model and the real system is sufficiently bounded. Due to the integrator, all the convex combinations of the dynamic matrices are non-Hurwitz. In this case, the main difficulty is to guarantee that the time derivative of the Lyapunov function is strictly negative definite, which is done by means of LMIs and conditions on the affine terms. To the best of our knowledge, this is the first Lyapunov-based switching rule with integral action able to ensure global asymptotic tracking of a pre-specified profile. When applied to AC circuits the advantage is to design the switching rule based on the original system, without resorting to averaged models. The theory is illustrated by the control of a three-phase AC-DC converter. [ABSTRACT FROM AUTHOR]

This study shows the link between computer science and applied mathematics. It conducts a dynamics investigation of new root solvers using computer tools and develops a new family of single-step simple root-finding methods. The convergence order of the proposed family of iterative methods is two, according to the convergence analysis carried out using symbolic computation in the computer algebra system CAS-Maple 18. Without further evaluations of a given nonlinear function and its derivatives, a very rapid convergence rate is achieved, demonstrating the remarkable computing efficiency of the novel technique. To determine the simple roots of nonlinear equations, this paper discusses the dynamic analysis of one-parameter families using symbolic computation, computer animation, and multi-precision arithmetic. To choose the best parametric value used in iterative schemes, it implements the parametric and dynamical plane technique using CAS-MATLAB@R2011b. The dynamic evaluation of the methods is also presented utilizing basins of attraction to analyze their convergence behavior. Aside from visualizing iterative processes, this method illustrates not only iterative processes but also gives useful information regarding the convergence of the numerical scheme based on initial guessed values. Some nonlinear problems that arise in science and engineering are used to demonstrate the performance and efficiency of the newly developed method compared to the existing method in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce second derivative multistep collocation methods for the numerical integration of ordinary differential equations (ODEs). These methods combine the concepts of both multistep methods and collocation methods, using second derivative of the solution in the collocation points, to achieve an accurate and efficient solution with strong stability properties, that is, A-stability for ODEs. Using the second-order derivatives leads to high order of convergency in the proposed methods. These methods approximate the ODE solution by using the numerical solution in some points in the r previous steps and by matching the function values and its derivatives at a set of collocation methods. Also, these methods utilize information from the second derivative of the solution in the collocation methods. We present the construction of the technique and discuss the analysis of the order of accuracy and linear stability properties. Finally, some numerical results are provided to confirm the theoretical expectations. A stiff system of ODEs, the Robertson chemical kinetics problem, and the two-body Pleiades problem are the case studies for comparing the efficiency of the proposed methods with existing methods. [ABSTRACT FROM AUTHOR]

Oscillations in option price convergence have long been a problematic aspect of tree methods, inhibiting the use of repeated Richardson extrapolation that could otherwise greatly accelerate convergence, a feature integral to some of the most efficient modern methods. These oscillations are typically caused by the fluctuating positions of nodes around the discontinuities in the payoff function or its derivatives. Our paper addresses this crucial gap that typically prohibits the use of lattice methods when high efficiency is needed. Focusing on double barrier options, we develop a trinomial tree in which the positions of the nodes are precisely adjusted to align with these discontinuities throughout the option's lifespan and across various time steps. This alignment enables the use of repeated extrapolation to achieve high order convergence, including near barriers, a well-known challenge in many tree methods. Maintaining the inherent simplicity and adaptability of tree methods, our approach is easily applicable to other models and option types. [ABSTRACT FROM AUTHOR]

In this article, we consider the correction of metric matrices in quasi-Newton methods (QNM) from the perspective of machine learning theory. Based on training information for estimating the matrix of the second derivatives of a function, we formulate a quality functional and minimize it by using gradient machine learning algorithms. We demonstrate that this approach leads us to the well-known ways of updating metric matrices used in QNM. The learning algorithm for finding metric matrices performs minimization along a system of directions, the orthogonality of which determines the convergence rate of the learning process. The degree of learning vectors' orthogonality can be increased both by choosing a QNM and by using additional orthogonalization methods. It has been shown theoretically that the orthogonality degree of learning vectors in the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method is higher than in the Davidon–Fletcher–Powell (DFP) method, which determines the advantage of the BFGS method. In our paper, we discuss some orthogonalization techniques. One of them is to include iterations with orthogonalization or an exact one-dimensional descent. As a result, it is theoretically possible to detect the cumulative effect of reducing the optimization space on quadratic functions. Another way to increase the orthogonality degree of learning vectors at the initial stages of the QNM is a special choice of initial metric matrices. Our computational experiments on problems with a high degree of conditionality have confirmed the stated theoretical assumptions. [ABSTRACT FROM AUTHOR]

In this paper we extend the family of test proposed by Majumder and Mitra [Detecting trend change in failure functions-an L-statistic approach. Stat Pap. 2019;62:31–52. doi: ] for testing exponentiality against bathtub (BT) failure rate (upside down bathtub (UBT)) to randomly right censored case. We derive the asymptotic distribution of the test statistic and using Monte Carlo simulation we obtain the empirical powers for certain alternative distributions. As a special case the asymptotic distribution of the test statistic proposed by Park [Testing whether failure rate changes its trend. IEEE Trans Reliab. 1988;37(4):375–378. doi: ] is obtained. Finally some examples of real life data are given for illustrative purpose. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose an enhancement to the primary frequency control (PFC) of the San Cristobal Island hybrid wind–diesel power system (WDPS). Naturally, variable speed wind turbines (VSWT) provide negligible inertia. Therefore, various control strategies, i.e., modified synthetic inertial control, droop control and traditional inertial control, if introduced into VSWT, enable them to release hidden inertia. Based on these strategies, a WDPS has been simulated under seven different control strategies, to evaluate the power system performance for frequency regulation (FR). Furthermore, the student psychology-based algorithm (SBPA) methodology is used to optimize the WDPS control. The results show that modified synthetic inertial control is the most suitable approach to provide FR. However, further exhaustive research validates that droop control is a better alternative than modified synthetic inertial control due to the negligible system performance differences. In addition, droop control does not require a frequency derivative function in the control system. Therefore, the hybrid system is more robust. Moreover, it reduces the steady state error, which makes the power system more stable. In addition, a pitch compensation control is introduced in blade pitch angle control (BPAC) to enhance the pitch angle smoothness and to help the power system to return to normal after perturbations. Moreover, to justify the performance of hybrid WDPS, it is tested under certain real-world contingency events, i.e., loss of a wind generator, increased wind speed, fluctuating wind speed, and simultaneously fluctuating load demand and wind speed. The simulation results validate the proposed WDPS control strategy performance. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to introduce the fourth and sixth-order compact finite difference methods for solving anti-periodic boundary value problems. Compact finite difference formulas can approximate the derivatives of a function more accurately than the standard finite difference formulas for the same number of grid points. The convergence analysis of the proposed method is also investigated. This analysis shows how the error between the approximate and exact solutions decreases as the grid space is reduced. To validate the proposed method's accuracy and efficiency, some computational experiments are provided. Moreover, a comparison is performed between the standard and compact finite difference methods. The experiments indicate that the compact finite difference method is more accurate and efficient than the standard one. [ABSTRACT FROM AUTHOR]

In this paper, a Newton-type iterative scheme for solving nonlinear systems is designed. In the process of proving the convergence order, we use the higher derivatives of the function and show that the convergence order of this iterative method is six. In order to avoid the influence of the existence of higher derivatives on the proof of convergence, we mainly discuss the convergence of this iterative method under weak conditions. In Banach space, the local convergence of the iterative scheme is established by using the ω -continuity condition of the first-order Fréchet derivative, and the application range of the iterative method is extended. In addition, we also give the radius of a convergence sphere and the uniqueness of its solution. Finally, the superiority of the new iterative method is illustrated by drawing attractive basins and comparing them with the average iterative times of other same-order iterative methods. Additionally, we utilize this iterative method to solve both nonlinear systems and nonlinear matrix sign functions. The applicability of this study is demonstrated by solving practical chemical problems. [ABSTRACT FROM AUTHOR]

Minimax problems have gained tremendous attentions across the optimization and machine learning community recently. In this paper, we introduce a new quasi-Newton method for the minimax problems, which we call J-symmetric quasi-Newton method. The method is obtained by exploiting the J-symmetric structure of the second-order derivative of the objective function in minimax problem. We show that the Hessian estimation (as well as its inverse) can be updated by a rank-2 operation, and it turns out that the update rule is a natural generalization of the classic Powell symmetric Broyden method from minimization problems to minimax problems. In theory, we show that our proposed quasi-Newton algorithm enjoys local Q-superlinear convergence to a desirable solution under standard regularity conditions. Furthermore, we introduce a trust-region variant of the algorithm that enjoys global R-superlinear convergence. Finally, we present numerical experiments that verify our theory and show the effectiveness of our proposed algorithms compared to Broyden's method and the extragradient method on three classes of minimax problems. [ABSTRACT FROM AUTHOR]

Differential calculus represents a very important field of mathematics, primarily because of the possibility of its application. The derivative of a function, a key element of differential calculus, finds its application not only in mathematics but also in most other natural sciences. By its very nature, differential calculus is an abstract and, to understand, rather complicated and demanding theory for students. Due to the presence of highly abstract concepts, students have difficulties understanding the true meaning of differential calculus, and especially seeing a little further, i.e. recognizing situations where that theory can be applied. In higher education, differential calculus, primarily the derivative of a function, is an unavoidable moment in the teaching of mathematics, and it is very important how it will be presented to students. In teaching mathematics, it is very important that students have properly formed ideas about mathematical concepts. This is especially related to the concept of the derivative of a function, which is necessary for students not only as a concept that occurs in the teaching of mathematics but as a necessary tool for many other fields and sciences. Successful organization of the process of formation of mathematical concepts and successful management of their adoption can be achieved through the modernization of the teaching process. Special attention in this paper is devoted to the improvement of the teaching process through the application of modern computer technologies. The GeoGebra software, which was used for the purpose of preparing the material for this work, is described and shown. The GeoGebra software was chosen for its ability to create animations and simulations. The dynamic nature of the GeoGebra software answer the demands of modern teaching and fits perfectly into the students’ habits for the active learning process. For the purposes of this work, a dynamical material containing moving images, illustrations, and graphs which through its interesting story illustrates the features of the derivative of the function in a very interesting way was designed by the authors of the paper. Presenting the basic content related to the concept of the derivative of a function in this way enables students to understand and acquire these concepts more quickly. The material presented in the paper was realized using GeoGebra software. All the elements of the used GeoGebra material were explained in detail. The impressions of teachers and students about the effects of applying such materials in teaching mathematics and science were more than positive. Our mission for the future is to continue with the development and improvement of teaching materials and techniques by using modern technologies. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to define a new additive function related to generalized divisors of an integer and to study some of its properties. [ABSTRACT FROM AUTHOR]

This paper addresses discontinuities in the solutions of mathematical physics that describe actual processes and are not observed in experiments. The appearance of discontinuities is associated in this paper with the classical differential calculus based on the analysis of infinitesimal quantities. Nonlocal functions and nonlocal derivatives, which are not specified, in contrast to the traditional approach to a point, but are the results of averaging over small but finite intervals of the independent variable are introduced. Classical equations of mathematical physics preserve the traditional form but include nonlocal functions. These equations are supplemented with additional equations that link nonlocal and traditional functions. The proposed approach results in continuous solutions of the classical singular problems of mathematical physics. The problems of a string and a circular membrane loaded with concentrated forces are used to demonstrate the procedure. Analytical results are supported with experimental data. [ABSTRACT FROM AUTHOR]

Presented in this paper is a trigonometrically fitted scheme based on a class of improved hybrid method for the numerical integration of oscillatory problems. The trigonometric conditions are constructed through which a third algebraic order scheme is derived. Numerical properties of the scheme are analysed. A numerical experiment is conducted to validate the scheme. Results obtained reveal the superiority of the scheme over its equals in the literature. [ABSTRACT FROM AUTHOR]

This paper studies nonparametric estimations of the derivatives of the variance function in a heteroscedastic model. Using a wavelet method, a linear estimator and an adaptive nonlinear estimator are constructed. The convergence rates under risk of those two wavelet estimators are considered with some mild assumptions. A simulation study is presented to validate the performances of the wavelet estimators. [ABSTRACT FROM AUTHOR]

In this paper, the functional differential equation (D C a + α x) (t) + ∑ i = 0 m (T i x (i)) (t) = f (t) , t ∈ [ a , b ] , with Caputo fractional derivative D C a + α is studied. The operators Ti act from the space of continuous to the space of essentially bounded functions. They can be operators with deviations (delayed and advanced), integral operators and their various linear combinations and superpositions. Such equations could appear in various applications and in the study of systems of, for example, two fractional differential equations, when one of the components can be presented from the first equation and substituted then to another. For two-point problems with this equation, assertions about negativity of Green's functions and their derivatives with respect to t are obtained. Our technique is based on an analog of the Vallée-Poussin theorem for differential inequalities, which is proven in our paper and gives necessary and sufficient conditions of negativity of Green's functions and their derivatives for two-point problems: there exists a positive function v satisfying corresponding boundary conditions and the inequality (D C a + α ν) (t) + ∑ i = 0 m (T i v (i)) (t) < 0 , t ∈ [ a , b ] . Choosing the function v, we obtain explicit sufficient tests of sign-constancy of Green's functions and its derivatives. It is demonstrated that these tests cannot be improved in a general case. Influences of delays on these sufficient conditions are analyzed. It is demonstrated that the tests can be essentially improved for "small" deviations. [ABSTRACT FROM AUTHOR]

This paper intends to estimate the box dimension of the Weyl–Marchaud fractional derivative (Weyl–M derivative) for various choices of continuous functions on a compact subset of ℝ. We show that the Weyl–M derivative of order γ of a continuous function satisfying Hölder condition of order μ also satisfies Hölder condition of order μ − γ and the upper box dimension of the Weyl–M derivative increases at most linearly with the order γ. Moreover, the upper box dimension of the Weyl–M derivative of a continuous function satisfying the Lipschitz condition is not more than the sum of the box dimension of the function itself and order γ. Furthermore, we prove that the box dimension of the Weyl–M derivative of a certain continuous function which is of bounded variation is one. [ABSTRACT FROM AUTHOR]

In interval-valued three-way decision, the reflection of decision-makers' preference under the full consideration of interval-valued characteristics is particularly important. In this paper, we propose an interval-valued three-way decision model based on the cumulative prospect theory. First, by means of the interval distance measurement method, the loss function and the gain function are constructed to reflect the differences of interval radius and expectation simultaneously. Second, combined with the reference point, the prospect value function is utilized to reflect decision-makers' different risk preferences for gains and losses. Third, the calculation method of cumulative prospect value for taking action is given through the transformation of the prospect value function and cumulative weight function. Then, the new decision rules are deduced based on the principle of maximizing the cumulative prospect value. Finally, in order to verify the effectiveness and feasibility of the algorithm, the prospect value for decision-making and threshold changes are analyzed under different risk attitudes and different radii of the interval-valued decision model. In addition, compared with the interval-valued decision rough set model, our method in this paper has better decision prospects. [ABSTRACT FROM AUTHOR]

Abstract: Structured compartmental models in mathematical biology track age classes, stage classes, or size classes of a population. Structured modeling becomes important when mechanistic formulations or intraspecific interactions are class‐dependent. The classic derivation of such models from partial differential equations produces time delays in the transition rates between classes. In particular, the transition from juvenile to adult has a delay equal to the maturation period of the organism. In the literature, many structured compartmental models, posed as ordinary differential equations, omit this delay. We reviewed occurrences of continuous‐time compartmental models for age‐ and stage‐structured populations in the recent literature (2000–2016) to discover which papers did so. About half of the 249 papers we reviewed used a maturation delay. Papers with ecological models were more likely to have the delay than papers with disease models, and mathematically focused papers were more likely to have the delay than biologically focused papers. Recommendations for Resource Managers: Interacting populations often are modeled with systems of ordinary differential equations in which the state variables are numbers of individuals of each species and interaction terms depend only on the current state of the system. Single‐population continuous‐time models with age‐ or stage‐structure, in which state variables represent numbers of individuals in classes such as juveniles and adults, often but not always contain maturation time delays in the transition rates between classes. The exclusion of the delay typically changes the model dynamics. Managers should be aware of the maturation delay issue when considering the results of continuous‐time models of structured populations. Discrete‐time models have an inherent time delay, set by the census time step chosen by the modeler, and for that reason are convenient for modeling maturation and other biological delays. [ABSTRACT FROM AUTHOR]

Quantum calculus can provide new insights into the nonlinear behaviour of functions and equations, addressing problems that may be difficult to tackle by classical calculus due to high nonlinearity. Iterative methods for solving nonlinear equations can benefit greatly from the mathematical theory and tools provided by quantum calculus, e.g., using the concept of q-derivatives, which extends beyond classical derivatives. In this paper, we develop parallel numerical root-finding algorithms that approximate all distinct roots of nonlinear equations by utilizing q-analogies of the function derivative. Furthermore, we utilize neural networks to accelerate the convergence rate by providing accurate initial guesses for our parallel schemes. The global convergence of the q-parallel numerical techniques is demonstrated using random initial approximations on selected biomedical applications, and the efficiency, stability, and consistency of the proposed hybrid numerical schemes are analyzed. [ABSTRACT FROM AUTHOR]

In this paper, on the basis of zero set of reverse indexed polynomial, first introduced by us, we have investigated the uniqueness of meromorphic functions under weighted sharing in wider sense criteria [4], which in turn extend some earlier results in different directions. We have succeeded to identify a subclass of meromorphic functions for which uniqueness property exists for higher reverse indexed polynomial in literature. In the last section, we have presented the application of our results in case of derivatives of the concerned functions accompanied by series of examples. [ABSTRACT FROM AUTHOR]

In this paper, we study wavelet approximation of the Chebyshev polynomials of the first, second, third, and fourth kinds. We estimate the wavelet approximation of a function f having bounded first derivatives. [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]