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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 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]

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]

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]

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]

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]

The paper presents various derivatives of set-valued mappings, their main properties and how they are related to each other. Next, we consider Cauchy problems with linear homogeneous set-valued differential equations with different types of derivatives (Hukuhara derivative, PSderivative and BG-derivative). It is known that such initial value problems with PS-derivative and BG-derivative have infinitely many solutions. Two of these solutions are called basic. These are solutions such that the diameter function of the solution section is a monotonically increasing (the first basic solution) or monotonically decreasing (the second basic solution) function. However, the second basic solution does not always exist. We provide conditions for the existence of basic solutions of such initial value problems. It is shown that their existence depends on the type of derivative, the matrix of coefficients on the right-hand and the type of the initial set. Model examples are considered. [ABSTRACT FROM AUTHOR]

Modelling phenomena with interval differential equations (IDEs) is an effective way to consider the uncertainties that are unavoidable when collecting data. Similarly to the theory of ordinary differential equations, IDEs have been parallelly investigated with the interval difference equations from the beginning. These two branches can be regarded as one when unifying continuous and discrete solution domains. A conspicuous advantage when merging these areas is that the proof of several analogous properties in both theories need not be repeated. The paper provides a common and efficient tool for studying IDEs not only with continuous or discrete solution domains but also with more general ones. We propose the diamond-α derivative for interval-valued functions (IVFs) on time scales with respect to the generalized Hukuhara difference. Differently from most of the studies on the derivatives of functions on time scales, using the language of epsilon-delta, the novel concept is naturally studied according to the limit of IVFs on time scales as in classical mathematics. A particular class of IDEs on time scales is then considered with respect to the diamond-α derivative. Numerical problems are elaborated to illustrate the necessity and efficiency of the latter. [ABSTRACT FROM AUTHOR]

In this paper, mixed moduli of smoothness of functions of two variables are studied. We prove Ulyanov-type inequalities between mixed moduli of smoothness of positive orders in different metrics. Estimates for the mixed moduli of smoothness of the derivative of a function are also obtained in terms of the mixed moduli of smoothness of the function itself. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two novel wavelet approximations tailored for functions g$$ g $$ exhibiting a restricted second derivative and a bounded Tth$$ {T}&#x0005E;{th} $$ derivative. Employing the Hermite wavelet method, we derive these approximations to address the need for effective representations of such functions. Our findings reveal that these new wavelet approximations offer enhanced accuracy and efficiency in capturing the underlying structure of g$$ g $$, making them valuable tools in various applications requiring precise function approximations with limited derivative constraints. [ABSTRACT FROM AUTHOR]

This paper proposes a low-cost, penalty parameter-free, and pressure-robust Stokes solver based on the enriched Galerkin (EG) method with a discontinuous velocity enrichment function. The EG method employs the interior penalty discontinuous Galerkin (IPDG) formulation to weakly impose the continuity of the velocity function. However, despite its advantage of symmetry, the symmetric IPDG formulation requires a lot of computational effort to choose an optimal penalty parameter and compute different trace terms. To reduce such effort, we replace the derivatives of the velocity function with its weak derivatives computed by the geometric data of elements. Therefore, our modified EG (mEG) method is a penalty parameter-free numerical scheme that has reduced computational complexity and conserves the optimal convergence orders. Moreover, we achieve pressure robustness for the mEG method by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. The theoretical results are confirmed through numerical experiments with two- and three-dimensional examples. [ABSTRACT FROM AUTHOR]

In this paper, based on the generalized differentiability concept, first, some properties of continuity of the derivative function and the existence of switching points for a fuzzy-valued function are obtained. Next, the new sufficient conditions for existence of solutions to the periodic boundary problems for first-order linear fuzzy differential equations are presented, in detail. Finally, some examples are solved to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate optimality conditions of nonsmooth sparsity multiobjective optimization problem (shortly, SMOP) by the advanced variational analysis. We present the variational analysis characterizations, such as tangent cones, normal cones, dual cones and second-order tangent set, of the sparse set, and give the relationships among the sparse set and its tangent cones and second-order tangent set. The first-order necessary conditions for local weakly Pareto efficient solution of SMOP are established under some suitable conditions. We also obtain the equivalence between basic feasible point and stationary point defined by the Fréchet normal cone of SMOP. The sufficient optimality conditions of SMOP are derived under the pseudoconvexity. Moreover, the second-order necessary and sufficient optimality conditions of SMOP are established by the Dini directional derivatives of the objective function and the Bouligand tangent cone and second-order tangent set of the sparse set. [ABSTRACT FROM AUTHOR]

In 2014, Sharma and Arora introduced two efficient Jarratt-like methods for solving systems of non-linear equations which are of convergence order four and six. To prove the respective convergence order, they used Taylor expansion which demands existence of derivative of the function up to order seven. In this paper, we obtain the respective convergence order for these methods using assumptions only on first three derivatives of the function. Other problems with this approach are: the lack of computable a priori estimates on the error distances involved as well as isolation of the solution results. These concerns constitute our motivation for this article. One extension of the fourth order method is presented which is of convergence order eight and the same is proved without any extra assumptions on the higher order derivatives. All the results are proved in a general Banach space setting. Numerical examples and dynamics of the methods are studied to analyse the performance of the method. [ABSTRACT FROM AUTHOR]

In the paper, with the aid of the Faà di Bruno formula, in terms of the central factorial numbers and the Stirling numbers of the second kinds, the authors derive several series expansions for any positive integer powers of the sinc and sinhc functions, discover several closed-form expressions for partial Bell polynomials of all derivatives of the sinc function, establish several series expansions for any real powers of the sinc and sinhc functions, and present several identities for central factorial numbers of the second kind and for the Stirling numbers of the second kind. [ABSTRACT FROM AUTHOR]

In this paper, a new approaching technique is offered to unravel multi-pantograph-type delay differential equations. The suggested new method is a collocation method based on integration and Boubaker polynomials. As the main idea of the method, the process starts by approaching the first derivative function in the equation in the form of truncated Boubaker series. Then this approximating form is composed in the matrix form. The unknown function is then obtained by integrating the approximate derivative function and expressing it as a matrix. Using the approximation, the matrix forms for the proportionally delayed terms in the equation are derived. In addition, operational matrix forms are constructed for convenience in the method. By using these matrix forms and matrix operations, the problem is reduced to a system of algebraic linear equations. The method is illustrated through numerical implementations and compared with existing techniques in the literature. The results demonstrate the effectiveness and reliability of the proposed approach, highlighting its superiority over other methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of interval-valued multiobjective optimization problems (in short, (IVMOP)) and formulate an associated multiobjective optimization problem, referred to as (MOP). We establish that the Pareto optimal solution of the associated (MOP) is an effective solution of (IVMOP). Using this characteristic of the associated (MOP), we introduce a variant of Newton's algorithm for the considered (IVMOP). The proposed algorithm exhibits superlinear convergence to a locally effective solution of (IVMOP), provided the objective function of (IVMOP) is twice generalized Hukuhara differentiable and locally strongly convex. Furthermore, if the second-order generalized Hukuhara partial derivatives of the objective function of (IVMOP) are generalized Hukuhara Lipschitz continuous, the rate of convergence is quadratic. We provide a suitable numerical example to illustrate the developed methodology. Moreover, we employ the proposed algorithm to solve a real-life portfolio optimization problem. [ABSTRACT FROM AUTHOR]

The cost function gradient vanishing or exploding problem and slow convergence speed are key issues when training deep neural networks (DNNs). In this paper, we investigate the forward and backward propagation processes of DNN training and explore the properties of the activation function and derivative function (ADF) employed. The outputs' distribution of ADF with near-zero mean is proposed to reduce gradient problems. Additionally, the constant energy transfer of propagating data in the training process is also proposed to speed up convergence further. Based on wavelet frame theory, we derive a novel ADF, i.e., tight frame wavelet activation function (TFWAF) and tight frame wavelet derivative function (TFWDF) of the Mexican hat wavelet, to stabilize and accelerate DNN training. The nonlinearity of wavelet functions can strengthen the learning capacity of DNN models, while the sparse property of wavelets derived can reduce the overfitting problem and enhance the robustness of models. Experiments demonstrate that the proposed method stabilizes the DNN training process and accelerates convergence. [ABSTRACT FROM AUTHOR]

In this paper, our purpose is to study the quaternary continuous classical boundary optimal control vector problem dominated by a quaternary linear parabolic boundary value problem. Under suitable assumptions and with a given quaternary continuous classical boundary control vector, the existence theorem for a unique quaternary state vector solution to the weak form is stated and demonstrated via the method of Galerkin. Furthermore, the continuity of the Lipschitz operator between the state vector solution to the weak form for the dominating equations and the corresponding are proved. The existence of a quaternary continuous classical boundary optimal control vector is stated and demonstrated under suitable Assumptions. The mathematical formulation for the quaternary adjoint boundary value problem associated with each considered boundary value problem is obtained and the Frѐchet derivative for the objective function is derived. Finally, the necessary conditions for the optimality theorem of the problem are stated and demonstrated. [ABSTRACT FROM AUTHOR]

New upper bounds are found in the problem of approximation to the th derivatives of a function of variables defined on a simplex by the derivatives of an algebraic polynomial of degree at most () interpolating the values of the function at equidistant nodes of the simplex. The estimates are obtained in terms of the diameter of the simplex, the angular characteristic introduced in the paper, the dimension , the degree of the polynomial, and the order of the derivative to be estimated and do not contain unknown parameters. These estimates are compared with those most frequently used in the literature. [ABSTRACT FROM AUTHOR]

A nitrogen-vacancy center based scanning magnetic microscope can be used to characterize magnetics at the nanoscale with high sensitivity. This paper reports a field-programmable-gate-array based hardware system that is designed to realize control and signal readout for fast scanning magnetic imaging with a nitrogen-vacancy center. A 10-channel 1 Msps @ 20 bit analog signal generator, a 12-channel 50 ps resolution pulse generator, a 300 Msps @ 16 bit lock-in amplifier with proportional integral derivative control function, and a 4-channel 200 Msps counter are integrated on the platform. A customized acceleration algorithm is realized with the re-configurable field-programmable-gate-array chip to accelerate the imaging speed of the nitrogen-vacancy system, and the experimental results prove that the imaging efficiency can be accelerated by five times compared to the system without the acceleration algorithm. The platform has considerable potential for future applications of fast scanning magnetic imaging. [ABSTRACT FROM AUTHOR]

Hermite polynomials arise when dealing with functions of normally distributed variables, and are commonly thought of as the analog of the simple polynomials on functions of regular variables. Therefore the Hermite expansion should be an analog of the Taylor expansion. Indeed there is a strong connection between the two - the general coefficient in the Hermite expansion is the weighted integral of the nth derivative, as compared to the nth derivative evaluated at zero in the case of Taylor. This fact can be used to derive the Hermite expansion for the integral and the derivative of a function. Furthermore, it provides a method of providing simple proofs of many of the Hermite identities. This connection is used to derive the Hermite expansions of the normal probability distribution function, the normal cumulative distribution function and the indicator function. Finally, an algorithm to numerically perform a Hermite expansion is presented, which is efficient in the sense that is only requires a single call to a quadrature method. [ABSTRACT FROM AUTHOR]

The Sharpe ratio function is a commonly used risk/return measure in financial econometrics. To estimate this function, most existing methods take a two-step procedure that first estimates the mean and volatility functions separately and then applies the plug-in method. In this paper, we propose a direct method via local maximum likelihood to simultaneously estimate the Sharpe ratio function and the negative log-volatility function as well as their derivatives. We establish the joint limiting distribution of the proposed estimators, and moreover extend the proposed method to estimate the multivariate Sharpe ratio function. We also evaluate the numerical performance of the proposed estimators through simulation studies, and compare them with existing methods. Finally, we apply the proposed method to the three-month US Treasury bill data and that captures a well-known covariate-dependent effect on the Sharpe ratio. [ABSTRACT FROM AUTHOR]

In this paper, we firstly establish an identity by using the notions of quantum derivatives and integrals. Using this quantum identity, quantum Newton-type inequalities associated with convex functions are proved. We also show that the newly established inequalities can be recaptured into some existing inequalities by taking q → 1 −. Finally, we give mathematical examples of convex functions to verify the newly established inequalities. [ABSTRACT FROM AUTHOR]