Chain Rule Research Articles

On generalized and fractional derivatives and their applications to classical mechanics

A generalized differential operator on the real line is defined by means of a limiting process. When an additional fractional parameter is introduced, this process leads to a locally defined fractional derivative. The study of such generalized derivatives includes, as a special case, basic results involving the classical derivative and current research involving fractional differential operators. All our operators satisfy properties such as the sum, product/quotient rules, and the chain rule. We study a Sturm–Liouville eigenvalue problem with generalized derivatives and show that the general case is actually a consequence of standard Sturm–Liouville Theory. As an application of the developments herein we find the general solution of a generalized harmonic oscillator. We also consider the classical problem of a planar motion under a central force and show that the general solution of this problem is still generically an ellipse, and that this result is true independently of the choice of the generalized derivatives being used modulo a time shift. The generic nature of phase plane orbits modulo a time shift is extended to the classical gravitational n-body problem of Newton to show that the global nature of these orbits is independent of the choice of the generalized derivatives being used in defining the force law. Finally, restricting the generalized derivatives to a special class of fractional derivatives, we consider the question of motion under gravity with and without resistance and arrive at a new notion of time that depends on the fractional parameter. The results herein are meant to clarify and extend many known results in the literature and intended to show the limitations and use of generalized derivatives and corresponding fractional derivatives.

Fuzzy Fractional Quadratic Regulator Problem Under Granular Fuzzy Fractional Derivatives

In this paper, a class of uncertain linear dynamical systems called fuzzy fractional linear dynamical systems is investigated. The aim is to find control inputs to keep the states of the fuzzy fractional dynamical systems near the zero in an optimal manner. The optimality criterion is in a form of a granular fuzzy integral whose integrand is a quadratic function of the state variables and control inputs. The fuzzy fractional dynamical system is described using fuzzy fractional differential equations (FFDEs). In order to achieve the aim, an effective approach for solving FFDEs should be at disposal. Due to some restrictions imposed by the previous approaches dealing with FFDEs, a new approach is proposed. The proposed approach is based on the granular derivative and the so-called relative-distance-measure fuzzy interval arithmetic. New definitions of fuzzy fractional derivatives and integral called left and right granular Riemann–Liouville fuzzy fractional derivatives, left and right granular Caputo fuzzy fractional derivatives, and the left and right granular fuzzy fractional integral are also presented. In addition, the concepts of granular fuzzy partial derivative and granular fuzzy chain rule are introduced. By the approximations of the granular fuzzy fractional integral and the granular Caputo fuzzy fractional derivative, the approximation solution to the FFDEs is obtained. Consequently, based on the new concepts and theorems, the solution to the fuzzy fractional quadratic regulator problem is given by a theorem. This paper closes with an example of regulating the motion of Boeing 747 in longitudinal direction with the presence of uncertainty in the initial conditions and the coefficients.

Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions

We consider an integral transform introduced by Prabhakar, involving generalised multi-parameter Mittag-Leffler functions, which can be used to introduce and investigate several different models of fractional calculus. We derive a new series expression for this transform, in terms of classical Riemann–Liouville fractional integrals, and use it to obtain or verify series formulae in various specific cases corresponding to different fractional-calculus models. We demonstrate the power of our result by applying the series formula to derive analogues of the product and chain rules in more general fractional contexts. We also discuss how the Prabhakar model can be used to explore the idea of fractional iteration in connection with semigroup properties.

Matrix function optimization under weighted boundary constraints and its applications in network control

The matrix function optimization under weighted boundary constraints on the matrix variables is investigated in this work. An “index-notation-arrangement based chain rule” (I-Chain rule) is introduced to obtain the gradient of a matrix function. By doing this, we propose the weighted trace-constraint-based projected gradient method (WTPGM) and weighted orthornormal-constraint-based projected gradient method (WOPGM) to locate a point of minimum of an objective/cost function of matrix variables iteratively subject to weighted trace constraint and weighted orthonormal constraint, respectively. New techniques are implemented to establish the convergence property of both algorithms. In addition, compared with the existing scheme termed “orthornormal-constraint-based projected gradient method” (OPGM) that requires the gradient has to be represented by the multiplication of a symmetrical matrix and the matrix variable itself, such a condition has been relaxed in WOPGM. Simulation results show the effectiveness of our methods not only in network control but also in other learning problems. We believe that the results reveal interesting physical insights in the field of network control and allow extensive applications of matrix function optimization problems in science and engineering.

Second-Order Karush–Kuhn–Tucker Optimality Conditions for Set-Valued Optimization Subject to Mixed Constraints

In this paper, we propose the concept of second-order composed adjacent contingent derivatives for set-valued maps and hence discuss the included relationship to the second-order composed contingent derivatives. Under the appropriate conditions, Lipschitz properties of the first order derivatives and a chain rule for such second-order composed contingent derivatives are demonstrated. By virtue of second-order composed adjacent contingent derivatives and second-order composed contingent derivatives, we establish second-order Karush–Kuhn–Tucker sufficient and necessary optimality conditions for a set-valued optimization problem subject to mixed constraints. Applying a separation theorem for convex sets and cone-Aubin properties, we address stronger necessary optimality conditions and extend the second-order Kurcyusz–Robinson–Zowe regularity assumption for a problem with mixed constraints, in which the derivatives of the objective and the constraint functions are considered in separated ways. When the results regress to vector optimization, we also extend and improve some recent existing results.

Responses to dominance challenge are a function of psychopathy level: A multimethod study.

This study assesses the relationship between psychopathy and dominance, both as a trait level and in 2 interpersonal contexts. Following the interpersonal circumplex model, we expected that interacting with a dominant interviewer would reduce dominance displays by low psychopathy participants (interpersonal complementarity) but might increase such displays in those with higher psychopathy (anticomplementarity). Psychopathic traits and dominance were assessed in a community sample (N = 91) using multiple dominance indicators: self-reported overall dominance level, job preference, dominance displays observed during identical interviews with a subordinate and dominant interviewer, and baseline and postinterview measures of personal space and testosterone level. Psychopathic traits were positively related to dominance on both trait measures and in the interviews. As a trait, higher self-reported psychopathy scores were associated with higher levels of self-reported dominance and preference for supervisory job positions. Higher Factor 1 psychopathy scores were associated with increased dominance display in interaction with the dominant interviewer. Higher Factor 2 scores were associated with allowing the dominant interviewer to approach more closely (reduced physical distancing). Psychopathy was partly related to a delayed increase in testosterone levels after interaction. Psychopathic traits may include a general tendency to dominate, a tendency which increases when interacting with a dominant other. The emotional/interpersonal psychopathy factor is specifically related to increased dominance displays when interacting with a dominant individual, while the behavioral factor of psychopathy is related to reduced physical distancing of a dominant partner. At higher psychopathy levels the general rule of dominant-subordinate reciprocity during social interaction is reversed. (PsycINFO Database Record

SOME FUNDAMENTAL RESULTS ON FUZZY CALCULUS

In this paper, we study fuzzy calculus in two main branches differential and integral. Some rules for finding limit and $gH$-derivative of $gH$-difference, constant multiple of two fuzzy-valued functions are obtained and we also present fuzzy chain rule for calculating $gH$-derivative of a composite function. Two techniques namely, Leibniz's rule and integration by parts are introduced for fuzzy integrals. Furthermore, we prove three essential theorems such as a fuzzy intermediate value theorem, fuzzy mean value theorem for integral and mean value theorem for $gH$-derivative. We derive a Bolzano's theorem, Rolle's theorem and some properties for $gH$-differentiable functions. To illustrate and explain these rules and theorems, we have provided several examples in details.

A Computational Architecture for Coupling Heterogeneous Numerical Models and Computing Coupled Derivatives

One of the challenges in computational modeling is coupling models to solve multidisciplinary problems. Flow-based computational frameworks alleviate part of the challenge through a modular approach, where data flows from component to component. However, existing flow-based frameworks are inefficient when coupled derivatives are needed for optimization. To address this, we develop the modular analysis and unified derivatives (MAUD) architecture. MAUD formulates the multidisciplinary model as a nonlinear system of equations, which leads to a linear equation that unifies all methods for computing derivatives. This enables flow-based frameworks that use the MAUD architecture to provide a common interface for the chain rule, adjoint method, coupled adjoint method, and hybrid methods; MAUD automatically uses the appropriate method for the problem. A hierarchical, matrix-free approach enables modern solution techniques such as Newton--Krylov solvers to be used within this monolithic formulation without computational overhead. Two demonstration problems are solved using a Python implementation of MAUD: a nanosatellite optimization with more than 2 million unknowns and 25,000 design variables, and an aircraft optimization involving over 6,000 design variables and 23,000 constraints. MAUD is now implemented in the open source framework OpenMDAO, which has been used to solve aircraft, satellite, wind turbine, and turbofan engine design problems.

Opial and Pólya Type Inequalities Via Convexity

Abstract In this paper, we prove some new dynamic inequalities related to Opial and Pólya type inequalities on a time scale 𝕋. We will derive the integral and discrete inequalities of Pólya’s type as special cases and also derive several classical integral inequalities of Opial’s type that has been obtained in the literature as special cases. The main results will be proved by using the chain rule, Hölder’s inequality and Jensen’s inequality, Taylor formula on time scales.

Prediction of Physical-Chemical and Fire Hazard Characteristics by Carbon Chain Rules.1. Alkanals

Prediction of Physical-Chemical and Fire Hazard Characteristics by Carbon Chain Rules.1. Alkanals

MODULAR BUNDLE ADJUSTMENT FOR PHOTOGRAMMETRIC COMPUTATIONS

Abstract. In this paper we investigate how the residuals in bundle adjustment can be split into a composition of simple functions. According to the chain rule, the Jacobian (linearisation) of the residual can be formed as a product of the Jacobians of the individual steps. When implemented, this enables a modularisation of the computation of the bundle adjustment residuals and Jacobians where each component has limited responsibility. This enables simple replacement of components to e.g. implement different projection or rotation models by exchanging a module. The technique has previously been used to implement bundle adjustment in the open-source package DBAT (Börlin and Grussenmeyer, 2013) based on the Photogrammetric and Computer Vision interpretations of Brown (1971) lens distortion model. In this paper, we applied the technique to investigate how affine distortions can be used to model the projection of a tilt-shift lens. Two extended distortion models were implemented to test the hypothesis that the ordering of the affine and lens distortion steps can be changed to reduce the size of the residuals of a tilt-shift lens calibration. Results on synthetic data confirm that the ordering of the affine and lens distortion steps matter and is detectable by DBAT. However, when applied to a real camera calibration data set of a tilt-shift lens, no difference between the extended models was seen. This suggests that the tested hypothesis is false and that other effects need to be modelled to better explain the projection. The relatively low implementation effort that was needed to generate the models suggest that the technique can be used to investigate other novel projection models in photogrammetry, including modelling changes in the 3D geometry to better understand the tilt-shift lens.

Generalized Beta Regression to Elicit Conditional Distributions of Medical Variables

Univariate conditional models are of core importance in supporting medical reasoning, as they allow to decompose a joint probability distribution using the chain rule. Although several methods are available for the elicitation of the joint prior distribution of parameters when the response is a medical categorical variable, the case of a medical continuous response is typically difficult to address, because its sample space is often bounded to an interval and its relationship with explanatory variables may be not linear. In these situations, the elicitation of an informative prior distribution on parameters of a univariate conditional model is challenging, because some level of statistical training is required to a medical expert for interpreting parameters and for retrieving appropriate quantitative information about them. The task can be eased and made efficient by recognizing that physicians typically distinguish among values involving medically normal and pathological patient conditions on the grounds of their personal clinical experience. In this paper, we propose a Generalized Beta regression where parameter elicitation is performed by establishing a correspondence among measured values expressed as relative positions within intervals with a clinical interpretation, regardless the original scales of variables. Software implementing the elicitation procedure is freely available.

Optimization on matrix manifold based on gradient information and its applications in network control

Vector function optimization problems, in which one or more variables are multidimensional vectors or infinite-dimensional vectors, have been extensively studied and demonstrated in existing schemes. In various real life applications, a cost function to be optimized usually involves matrix variables subjected to certain constraints. Locating its minimum can be modeled as an optimization problem on matrix manifold which is investigated in this paper. We first present an index-notation-arrangement based chain rule (I-Chain rule) to obtain the gradient information of the cost function. Two iterative algorithms, namely, trace-constraint-based projected gradient method (TPGM) and orthonormal-constraint-based projected gradient method (OPGM) are proposed and their convergence properties are established. We find that the network control problems can be effectively solved by both TPGM and OPGM. Two important phenomena are observed. For controlling directed networks with selectable inputs, both TPGM and OPGM tend to locate the nodes that divides the elementary stem/circle/dilation equally for consuming less energy, with OPGM having a slightly higher chance than TPGM. For controlling directed networks by only evolving the connection strengths on a fixed network structure, we find that after a network adaptively changes its topology in such a way that many similar sub-networks are gradually evolved, the control cost attains its minimum. Our work takes a further step from understanding optimization problems on matrix manifold to extending their applications in science and engineering.

Automatic differentiation of a finite-volume-based transient heat conduction code for sensitivity analysis

A general method for computing derivatives of solution fields and other simulation outputs, with respect to arbitrary input quantities, is proposed. The method of automatic differentiation is used to carry out differentiation and propagate derivatives through the simulation code by chain rule, in forward order. An object-oriented approach using the operator overloading and templating features of the C++ programming language is presented. Verification results are given for a plane wall with surface convection, where the derivative of the dimensionless temperature field with respect to the Biot number is computed and compared to an analytical solution. Further results are given for conduction in a composite material with regions of different thermal conductivity. The derivative of the temperature field is computed with respect to the conductivity of one of the phases using the proposed method.

Electrical circuits described by fractional conformable derivative

SummaryIn the last 3 years, the fractional conformable derivative and its properties have been introduced. Unlike other definitions, this new fractional derivative is based on the basic limit definition of the derivative and satisfies the same formulas of derivation, such as product and quotient of 2 functions and the chain rule. Using this new derivative, we obtain a new class of linear ordinary differential equations with noninteger power variable coefficients for the Resistance Capacitance (RC), Inductance Capacitance (LC), and Resistance, Inductance Capacitance (RLC) electric circuits. The numerical solutions are solved through the Matlab software. Solutions depend on time and on the fractional order parameter 0 < γ ≤ 1. The computing using this new derivative is much easier than using other definitions of fractional derivative. It has been shown that in the particular case γ = 1, these solutions become the ordinary ones. Also, a comparison has been made with the Caputo fractional derivative for the case of the RC circuit with constant source.

Dimension of quantum mechanical path, chain rule, and extension of Landau’s energy straggling method using Fα-calculus

$F^\alpha$-calculus was recently presented for fractals. We show that the $F^\alpha$-derivative satisfies the chain rule and affirms that the dimension of a quantum mechanical path is two. $F^\alpha$-calculus allows us to extend quantum mechanics to fractal curves. To show the applicability of $F^\alpha$-calculus, we study the fractal model of a particle in a box. $F^\alpha$-calculus is suitable for describing the motion of particles with a fractal route through matter. In addition, we extend Landau's energy straggling method of charged particles to fractals.

A new modified BFGS method for unconstrained optimization problems

Using chain rule, we propose a modified secant equation to get a more accurate approximation of the second curvature of the objective function. Then, based on this modified secant equation we present a new BFGS method for solving unconstrained optimization problems. The proposed method makes use of both gradient and function values, and utilizes information from two most recent steps, while the usual secant relation uses only the latest step information. Under appropriate conditions, we show that the proposed method is globally convergent without convexity assumption on the objective function. Comparative numerical results show computational efficiency of the proposed method in the sense of the Dolan–More performance profiles.

Pointwise Partial Information Decomposition Using the Specificity and Ambiguity Lattices.

What are the distinct ways in which a set of predictor variables can provide information about a target variable? When does a variable provide unique information, when do variables share redundant information, and when do variables combine synergistically to provide complementary information? The redundancy lattice from the partial information decomposition of Williams and Beer provided a promising glimpse at the answer to these questions. However, this structure was constructed using a much criticised measure of redundant information, and despite sustained research, no completely satisfactory replacement measure has been proposed. In this paper, we take a different approach, applying the axiomatic derivation of the redundancy lattice to a single realisation from a set of discrete variables. To overcome the difficulty associated with signed pointwise mutual information, we apply this decomposition separately to the unsigned entropic components of pointwise mutual information which we refer to as the specificity and ambiguity. This yields a separate redundancy lattice for each component. Then based upon an operational interpretation of redundancy, we define measures of redundant specificity and ambiguity enabling us to evaluate the partial information atoms in each lattice. These atoms can be recombined to yield the sought-after multivariate information decomposition. We apply this framework to canonical examples from the literature and discuss the results and the various properties of the decomposition. In particular, the pointwise decomposition using specificity and ambiguity satisfies a chain rule over target variables, which provides new insights into the so-called two-bit-copy example.

A direct Primitive Variable Recovery Scheme for hyperbolic conservative equations: The case of relativistic hydrodynamics.

In this article we develop a Primitive Variable Recovery Scheme (PVRS) to solve any system of coupled differential conservative equations. This method obtains directly the primitive variables applying the chain rule to the time term of the conservative equations. With this, a traditional finite volume method for the flux is applied in order avoid violation of both, the entropy and “Rankine-Hugoniot” jump conditions. The time evolution is then computed using a forward finite difference scheme. This numerical technique evades the recovery of the primitive vector by solving an algebraic system of equations as it is often used and so, it generalises standard techniques to solve these kind of coupled systems. The article is presented bearing in mind special relativistic hydrodynamic numerical schemes with an added pedagogical view in the appendix section in order to easily comprehend the PVRS. We present the convergence of the method for standard shock-tube problems of special relativistic hydrodynamics and a graphical visualisation of the errors using the fluctuations of the numerical values with respect to exact analytic solutions. The PVRS circumvents the sometimes arduous computation that arises from standard numerical methods techniques, which obtain the desired primitive vector solution through an algebraic polynomial of the charges.

Periodic Dynamics for Memristor-based Bidirectional Associative Memory Neural Networks with Leakage Delays and Time-varying Delays

This paper deals with a class of memristor-based bidirectional associative memory (BAM) neural networks with leakage delays and time-varying delays. With the aid of the framework of Filippov solutions, Chain rule and some inequality techniques, a sufficient condition which ensures the boundedness and ultimate boundedness of solutions of memristor-based BAM neural networks with leakage delays and time-varying delays is established. Applying a new approach involving Yoshizawa-like theorem, we prove the existence of periodic solution of the memristor-based BAM neural networks. By using the theory of set-valued maps and functional differential inclusions, Lyapunov functional, a set of sufficient conditions which guarantee the uniqueness and global exponential stability of periodic solution of memristor-based BAM neural networks are derived. An example is given to illustrate the applicability and effectiveness of the theoretical predictions. The results obtained in this paper are completely new and complement the previously known studies of Li et al. [Existence and global exponential stability of periodic solution of memristor-based BAM neural networks with time-varying delays, Neural networks 75 (2016) 97-109.]