Derivative Of Function Research Articles

A constrained differential evolution algorithm for calculation of two-phase equilibria at given volume, temperature and moles

A variety of strategies including two types of direct solution methods based on Newton's method and indirect solution methods based on thermodynamic principles have been developed, which can find satisfactory solutions for the phase equilibrium calculation problem at given volume, temperature and moles (NVT-flash). However, these methods are sensitive to the initial values and depend on the objective function's derivative during calculation. To resolve these deficiencies, a constrained differential evolution algorithm (abbreviated as CDE) for calculating two-phase equilibria at given volume, temperature, and moles is presented in this paper. The total Helmholtz free energy is regarded as the objective function, and the moles vector and volume of a certain phase are taken as the decision variables. This paper deals with the constraints of the NVT-flash problem by using direct search method and exterior point method. The results of the investigations on pure substance, binary and ternary mixtures are in agreement with the published data, which proves the effectiveness in solving the NVT-flash problem and the energy attenuation characteristic of the algorithm. The proposed algorithm represents the first successful attempt to apply the meta-heuristic optimization algorithm to solve the phase equilibrium computation under NVT-flash, and it shows the promising application prospect of meta-heuristic optimization algorithms for solving the phase equilibrium calculation problems.

Elzaki residual power series method to solve fractional diffusion equation.

The time-fractional order differential equations are used in many different contexts to analyse the integrated scientific phenomenon. Hence these equations are the point of interest of the researchers. In this work, the diffusion equation for a one-dimensional time-fractional order is solved using a combination of residual power series method with Elzaki transforms. The residual power series approach is a useful technique for finding approximate analytical solutions of fractional differential equations that needs the residual function's (n-1)α derivative. Since it is challenging to determine a function's fractional-order derivative, the traditional residual power series method's application is somewhat constrained. The Elzaki transform with residual power series method is an attempt to get over the limitations of the residual power series method. The obtained numerical solutions are compared with the exact solution of this equation to discuss the method's applicability and efficiency. The results are also graphically displayed to show how the fractional derivative influences the behaviour of the solutions to the suggested method.

Computational analysis of time-fractional models in energy infrastructure applications

In this paper, we propose an effective numerical method to solve the one- and two-dimensional time-fractional convection-diffusion equations based on the Caputo derivative. The presented approach employs a hybrid method that combines Lucas and Fibonacci polynomials with the Caputo derivative definition. The main objective is to transform the problem into a time-discrete form utilizing the Caputo derivative technique and then approximate the function's derivative using Fibonacci polynomials. To evaluate the efficiency and accuracy of the proposed technique, we apply it to one- and two-dimensional problems and compare the results with the exact as well as with existing methods in recent literature. The comparison demonstrates that the proposed approach is highly efficient, accurate and ease to implement.

Two sharp inequalities for operators in a Hilbert space

In this paper we obtained generalisations of the L. V. Taikov’s and N. Ainulloev’s sharp inequalities, which estimate a norm of function's first-order derivative (L. V. Taikov) and a norm of function's second-order derivative (N. Ainulloev) via the modulus of continuity or the modulus of smoothness of the function itself and the modulus of continuity or the modulus of smoothness of the function's second-order derivative. The generalisations are obtained on the power of unbounded self-adjoint operators which act in a Hilbert space. The moduli of continuity or smoothness are defined by a strongly continuous group of unitary operators.

Distributed fuzzy fault-tolerant consensus of leader-follower multi-agent systems with mismatched uncertainties

In this paper, the distributed fuzzy fault-tolerant tracking consensus problem of leader-follower multi-agent systems (MASs) is studied. The objective system includes actuator faults, mismatched parameter uncertainties, nonlinear functions, and exogenous disturbances under switching communication topologies. To solve this problem, a distributed fuzzy fault-tolerant controller is proposed for each follower by adaptive mechanisms to track the state of the leader. Furthermore, the fuzzy logic system is utilized to approximate the unknown nonlinear dynamics. An error estimator is introduced between the mismatched parameter matrix and the input matrix. Then, a selective adaptive law with relative state information is adopted and applied. When calculating the Lyapunov function's derivative, the coupling terms related to consensus error and mismatched parameter uncertainties can be eliminated. Finally, a numerical simulation is given to validate the effectiveness of the proposed protocol.

Full and reduced‐order H∞ filtering of Takagi–Sugeno fuzzy time‐varying delay systems: Input–output approach

SummaryThis article addresses the issue of delay‐dependent H∞ filtering design for TakagiŮSugeno fuzzy time‐varying delay systems using the input–output approach. A three‐term approximation model has been used to transform the original system into two interconnected subsystems. Since the nonquadratic Lyapunov–Krasovskii functional requires to deal with the membership function's (MF) time derivative, upper‐bound inequalities have been added to the obtained conditions. Based on the scaled small gain theorem, nonquadratic Lyapunov–Krasovskii functional approach and considering the bounds of the MF time‐derivative, the H∞ full‐ and reduced‐order filters are designed and then formulated in terms of linear matrix inequalities. Finally, illustrative examples are presented to demonstrate the validity of the proposed methods.

Constructing a subgradient from directional derivatives for functions of two variables

For any scalar-valued bivariate function that is locally Lipschitz continuous and directionally differentiable, it is shown that a subgradient may always be constructed from the function's directional derivatives in the four compass directions, arranged in a so-called "compass difference". When the original function is nonconvex, the obtained subgradient is an element of Clarke's generalized gradient, but the result appears to be novel even for convex functions. The function is not required to be represented in any particular form, and no further assumptions are required, though the result is strengthened when the function is additionally L-smooth in the sense of Nesterov. For certain optimal-value functions and certain parametric solutions of differential equation systems, these new results appear to provide the only known way to compute a subgradient. These results also imply that centered finite differences will converge to a subgradient for bivariate nonsmooth functions. As a dual result, we find that any compact convex set in two dimensions contains the midpoint of its interval hull. Examples are included for illustration, and it is demonstrated that these results do not extend directly to functions of more than two variables or sets in higher dimensions.

Nonlocal Transport Equations---Existence and Uniqueness of Solutions and Relation to the Corresponding Conservation Laws

In this contribution, we study the existence and uniqueness of nonlocal transport equations. The term “nonlocal” refers to the fact that the flux function's derivative will be integrated over a neighborhood of the corresponding space-time coordinate. We will demonstrate existence and uniqueness of weak solutions for $TV\cap L^{\infty}$ initial datum and provide stability estimates. Moreover, we investigate the convergence of the nonlocal transport equation to the corresponding local conservation law when the nonlocal reach tends to zero. For quadratic flux functions (including Burgers' equation and the Lighthill--Whitham--Richards traffic flow model), we establish convergence to a weak solution of the local conservation law for “symmetric” nonlocal terms. For specific quasi-convex and quasi-concave initial datum we even obtain convergence to the local entropy solution. We demonstrate that for “nonsymmetric” nonlocal approximations the solution cannot converge to the entropy solution or even a weak solution. We conclude with additional numerical examples showing that convergence appears to hold for more general initial datum.

Indefinite integrals of special functions from hybrid equations

ABSTRACTElementary linear first and second order differential equations can always be constructed for twice differentiable functions by explicitly including the function's derivatives in the definition of these equations. If the function also obeys a conventional differential equation, information from this equation can be introduced into the elementary equations to give blended linear equations which are here called hybrid equations. Integration theorems are derived for these hybrid equations and several universal integrals are also derived. The paper presents integrals derived with these methods for cylinder functions, associated Legendre functions, and the Gegenbauer, Chebyshev, Hermite, Jacobi and Laguerre orthogonal polynomials. All the results presented have been checked using Mathematica.

Implicit surface reconstruction based on generalized radial basis functions interpolant with distinct constraints

• We propose a generalized radial basis function interpolant with difference constraints. • Various types of constraints are derived to satisfy the interpolation conditions . • An adaptive iterative algorithm is used for approximating gradient and tangent constraints. • The numerical results reveal the improved performance of our method. In this work, we present a new algorithm for reconstruction of implicit surfaces from a set of cloud points with normals (Hermite data), based on the generalized radial basis functions interpolant with various types of constraints. Our key contribution is a novel construction of difference constraints in the interpolant to satisfy certain linear functional data, named as the domain constraints, the difference constraints of the gradient and the difference constraints of the tangent. To avoid ambiguous constraints, we apply an iterative algorithm to determine the adaptive distance in the difference constraints. The extended interpolant can not only provide an exact interpolation to the function values, but also approximate the function's derivatives by constructing a difference operator along the gradient or tangent direction. We interpolate the sampling points using our method by constructing a signed distance field in the geometry domain. Compared to the Hermite–Birkhoff interpolation with RBFs, the interpolant we use has faster solution efficiency. The experimental results of several data sets show the reliability and performance of our method in a variety of practical scenarios.

Thematization of derivative schema in university students: nuances in constructing relations between a function's successive derivatives

ABSTRACTThis study is part of a more extensive research project that addresses the understanding of the derivative concept in university students with prior instruction in differential calculus. In particular, we focus on the analysis of students’ responses to a sequence of tasks that require a high level of understanding of the concept, and complement this information with clinical interviews. APOS (Action-Process-Object-Schema) theory and the configuration of the derivative concept that is characterized by: mathematical elements, logical relations and the representation modes that students use to solve a task were used in the analysis of students’ responses. The results obtained suggest that thematizing the derivative schema is difficult to achieve. In addition, nuances were observed in responses given by those students who succeeded, indicating differences in the construction of relations between the successive derivatives of a function.

Generalization performance of radial basis function networks.

This paper studies the generalization performance of radial basis function (RBF) networks using local Rademacher complexities. We propose a general result on controlling local Rademacher complexities with the L1 -metric capacity. We then apply this result to estimate the RBF networks' complexities, based on which a novel estimation error bound is obtained. An effective approximation error bound is also derived by carefully investigating the Hölder continuity of the lp loss function's derivative. Furthermore, it is demonstrated that the RBF network minimizing an appropriately constructed structural risk admits a significantly better learning rate when compared with the existing results. An empirical study is also performed to justify the application of our structural risk in model selection.

Secants, Tangents, Rotations, and Reflections

SummaryGiven the graph of a continuous function on an interval, if we know the slopes of all the secant lines, then we can determine which rotations and reflections of the graph are also graphs of functions and, for those that are, whether the new functions are one-to-one. Provided that the original function is differentiable on the interior of the interval, we can determine the slopes of all secant lines by calculating the range of the function's derivative, provided that we know any subintervals where the function is linear.

Foundational Certification of Code Transformations Using Automatic Differentiation

Automatic Dierentiation (AD) is concerned with the semantics augmentation of an input program representing a function to form a transformed program that computes the function's derivatives. To ensure the correctness of the AD transformed code (particularly for safety-critical applications), we aim at certifying the algebraic manipulations at the heart of the AD process. We have considered a WHILE-language, and have shown how such proofs can be constructed by using appropriate relational Hoare logic. In particular, we have shown how such inference rules can be constructed for both the forward- and reverse-mode AD by using an abductive logical reasoning.

Foundational Certification of Code Transformations Using Automatic Differentiation

Automatic Differentiation (AD) is concerned with the semantics augmentation of an input program representing a function to form a transformed program that computes the function's derivatives. To ensure the correctness of the AD transformed code, particularly for safety critical applications, we aim at certifying the algebraic manipulations at the heart of the AD process. We have considered a WHILE-language and have shown how such proofs can be constructed by using an appropriate relational Hoare logic.In particular, we have shown how such inference rules can be constructed for both the forward and reverse mode AD by using an abductive logical reasoning.

A Note on Downside Risk Aversion Measures

In this paper, we first present a unified characterization of downside risk aversion measures, using a popular approach in the literature. In the process, we make the following contribution: we characterize the two downside risk aversion measures cautiousness and the ratio of a utility function's third derivative to its first derivative, give a new characterization of prudence, and establish a new downside risk aversion measure, the negative relative rate of change in absolute risk aversion. We then show how these measures are related to the Arrow-Pratt risk aversion measure and how they are related to each other. We also explain the link between these downside risk aversion measures and skewness preference and show the effect of downside risk aversion on option prices.

Spectral density estimation through a regularized inverse problem

In the study of stationary stochastic processes on the real line, the co- variance function and the spectral density function are parameters of considerable interest. They are equivalent ways of expressing the temporal dependence in the process. In this article, we consider the spectral density function and propose a new estimator that is not based on the periodogram; the estimator is derived through a regularized inverse problem. A further feature of the estimator is that the data are not required to be observed on a grid. When the regularization condition is based on the function's first derivative, we give the estimator in closed form as well as a bound on its mean squared error. Our numerical studies compare our new estima- tor of the spectral density to several well known estimators, and we demonstrate its increased statistical efficiency and much faster computation time.

What Can the Option-Implied Risk Aversion Really Tell Us?

In this paper we raise a question on the reliability of option implied risk aversion. We prove that given any number of options, there exist numerous risk aversion functions which are piecewise constant, have only two values, either a lower bound or an upper bound on the true risk aversion function, and are consistent with the prices of all these options. Similar results are proved with respect to the true risk aversion function's derivatives. These results show that the errors in the estimation of risk aversion and its derivatives caused purely by market incompleteness can be infinitely large and it is extremely difficult to ensure that the risk aversion function we extract from option prices is the true one.

TaylUR 3, a multivariate arbitrary-order automatic differentiation package for Fortran 95

This new version of TaylUR is based on a completely new core, which now is able to compute the numerical values of all of a complex-valued function's partial derivatives up to an arbitrary order, including mixed partial derivatives.

New version announcement for TaylUR, an arbitrary-order diagonal automatic differentiation package for Fortran 95

We present a new version of TaylUR, a Fortran 95 module to automatically compute the numerical values of a complex-valued function's derivatives with respect to several variables up to an arbitrary order in each variable, but excluding mixed derivatives. The new version fixes a potentially serious bug in the code for exponential-related functions that could corrupt the imaginary parts of derivatives, as well as being compatible with a wider range of compilers. Program summary Title of program: TaylUR Catalogue identifier: ADXR_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADXR_v2_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions: none Programming language used: Fortran 95 Computer: Any computer with a conforming Fortran 95 compiler Operating system: Any system with a conforming Fortran 95 compiler No. of lines in distributed program, including test data, etc.: 6548 No. of bytes in distributed program, including test data, etc.: 17 468 Catalogue identifier of previous version: ADXR_v1_0 Journal reference of previous version: Comput. Phys. Comm. 174 (2006) 569–576 Does the new version supersede the previous version?: yes Distribution format:tar.gz Nature of problem: Problems that require potentially high orders of derivatives with respect to some variables or derivatives of complex-valued functions, such as, e.g., expansions of Feynman diagrams in particle masses in perturbative Quantum Field Theory. Solution method: Arithmetic operators and Fortran intrinsics are overloaded to act correctly on objects of a defined type taylor, which encodes a function along with its first few derivatives with respect to the user-defined independent variables. Derivatives of products and composite functions are computed using Leibniz's rule and Fàa di Bruno's formula. Reasons for the new version: The previous version [G.M. von Hippel, TaylUR, an arbitrary-order diagonal automatic differentiation package for Fortran 95, Comput. Phys. Comm. 174 (2006) 569–576] contained a potentially serious bug in the functions overloading the exponential-related intrinsics ( EXP, LOG, SIN, COS, TAN, SINH, COSH, TANH), which could corrupt the imaginary parts of derivatives. It also contained some features which caused it to crash when compiled with certain compilers (notably the NAG and Lahey/Fujitsu compilers). Summary of revisions: The bug in the exponential-related intrinsics has been corrected. A number of additional changes have been made to the code to enable better compatibility with a greater range of compilers, including the NAG and Lahey/Fujitsu compilers. Users of some of these compilers may have to define useintrinsic as a preprocessor symbol when compiling TaylUR. Restrictions: Memory and CPU time constraints may restrict the number of variables and Taylor expansion order that can be achieved. Loss of numerical accuracy due to cancellation may become an issue at very high orders. Unusual features: No mixed higher-order derivatives are computed. The complex conjugation operation assumes all independent variables to be real. Running time: The running time of TaylUR operations depends linearly on the number of variables. Its dependence on the Taylor expansion order varies from linear (for linear operations) through quadratic (for multiplication) to exponential (for elementary function calls).