Chain Rule Research Articles

ANALISIS KESULITAN MAHASISWA DALAM MENYELESAIKAN SOAL DIFERENSIAL PADA MATA KULIAH MATEMATIKA DASAR

Its research is a descriptive qualitative research. The sampling method in this study used purposive sampling with 10 Informatics Management students in the first semester of the 2022/2023 academic year as subjects, who took Basic Mathematics courses. These students have difficulty solving differential problems, especially material on chain rules and high-order derivatives. The results of the Mid Semester Examination (UTS) obtained were only able to be answered by 20% of students. The purpose of this study was to analyze students' difficulties in answering differential questions, done by looking at learning outcomes and by collecting information obtained during interviews, based on four aspects of expertise, namely: 1) Information skills, 2) Arithmatics skills, 3) Language skills, and 4) Number fact skills. The method of information analysis is carried out by: collecting information, reducing, presenting information, and drawing conclusions. Data collection in this study used a triangulation procedure, namely by equating the information on the test results with the results of the interviews. The results of this study are that students have difficulty solving differential problems when viewed from the four aspects of expertise. Factors causing difficulties in this ability are 1) students are not able to analyze the correct method that will be used to solve the differential function reduction, 2) students do not understand mathematical operations well, and 3) students make mistakes reading and interpreting questions.

Eigensensitivity of damped system with distinct and repeated eigenvalues by chain rule

AbstractIn this paper, a novel algorithm for computing the derivatives of eigensolutions of asymmetric damped systems with distinct and repeated eigenvalues is developed without using second‐order derivatives of the eigenequations, which has a significant benefit over the existing published methods. To achieve this, the algorithm is proposed by (1) imposing consistent normalization conditions for both left and right eigenvectors throughout the computational system, and (2) solving sensitivity problems using the chain rule. These contributions add significant value to the proposed method. Additionally, the numerical difficulty is overcome by suggesting justifying Nelson (1976)'s technique based on the normalization conditions. Even though the proposed algorithm is simple and easy to implement, the employment of chain rule with new normalization in this note has never been reported in the literature, especially for the problem of eigensensitivity. Because this new algorithm does not require the second‐order derivatives of the eigenequations, the analytical and numerical solutions for eigensensitivity offered by this method are less complicated than existing available solutions. The validity and applicability of the method are demonstrated through five numerical examples, showcasing its implementation in displacement‐ and state‐space equations (rotordynamic system). Also, verifications are found to be in reasonable agreement with the approximated method.

Chain rules for a mutual information based on Rényi zero-relative entropy

Chain rules for a mutual information based on Rényi zero-relative entropy

Quantifying Causal Path-Specific Importance in Structural Causal Model

Path-specific effect analysis is a powerful tool in causal inference. This paper provides a definition of causal counterfactual path-specific importance score for the structural causal model (SCM). Different from existing path-specific effect definitions, which focus on the population level, the score defined in this paper can quantify the impact of a decision variable on an outcome variable along a specific pathway at the individual level. Moreover, the score has many desirable properties, including following the chain rule and being consistent. Finally, this paper presents an algorithm that can leverage these properties and find the k-most important paths with the highest importance scores in a causal graph effectively.

Gohar Fractional Derivative: Theory and Applications

The local fractional derivatives marked the beginning of a new era in fractional calculus. Due to their that have never been observed before in the field, they are able to fill in the gaps left by the nonlocal fractional derivatives and substantially increase the field’s theoretical and applied potential. In this article, we introduce a new local fractional derivative that possesses some classical properties of the integer-order calculus, such as the product rule, the quotient rule, the linearity, and the chain rule. It meets the fractional extensions of Rolle’s theorem and the mean value theorem and has more properties beyond those of previously defined local fractional derivatives. We reveal its geometric interpretation and physical meaning. We prove that a function can be differentiable in its sense without being classically differentiable. Moreover, we apply it to solve the Riccati fractional differential equations to demonstrate that it provides more accurate results with less error in comparison with the previously defined local fractional derivatives when applied to solve fractional differential equations. The numerical results obtained in this work by our local fractional derivative are shown to be in excellent agreement with those produced by other analytical and numerical methods such as the enhanced homotopy perturbation method (EHPM), the improved Adams-Bashforth-Moulton method(IABMM), the modified homotopy perturbation method (MHPM), the Bernstein polynomial method (BPM), the fractional Taylor basis method (FTBM), and the reproducing kernel method (RKM).

Modified Backpropagation Algorithm with Multiplicative Calculus in Neural Networks

Backpropagation is one of the most widely used algorithms for training feedforward deep neural networks. The algorithm requires a differentiable activation function and it performs computations of the gradient proceeding backwards through the feedforward deep neural network from the last layer through to the first layer. In order to calculate the gradient at a specific layer, the gradients of all layers are combined via the chain rule of calculus. One of the biggest disadvantages of the backpropagation is that it requires a large amount of training time. To overcome this issue, this paper proposes a modified backpropagation algorithm with multiplicative calculus. Multiplicative calculus provides an alternative to the classical calculus and it defines new kinds of derivative and integral forms in multiplicative form rather than addition and subtraction forms. The performance analyzes are discussed in various case studies and the results are given comparatively with classical backpropagation algorithm. It is found that the proposed modified backpropagation algorithm converges in less time to the solution and thus provides fast training in the given case studies. It is also shown that the proposed algorithm avoids the local minima problem.

Novel Integral Inequalities on Nabla Time Scales with C-Monotonic Functions

Through the paper, we present several inequalities involving C-monotonic functions with C≥1, on nabla calculus via time scales. It is known that dynamic inequalities generate many different inequalities in different calculus. The main results will be proved by applying the chain rule formula on nabla calculus. As a special case for our results, when T=R, we obtain the continuous analouges of inequalities that had previously been proved in the literature. When T=N, the results, to the best of the authors’ knowledge, are essentially new. Symmetrical properties of C-monotonic functions are critical in determining the best way to solve inequalities.

Neural network guided sinogram-domain iterative algorithm for artifact reduction.

Artifact reduction or removal is a challenging task when the artifact creation physics are not well modeled mathematically. One of such situations is metal artifacts in x-ray CT when the metallic material is unknown, and the x-ray spectrum is wide. A neural network is used to act as the objective function for iterative artifact reduction when the artifact model is unknown. A hypothetical unpredictable projection data distortion model is used to illustrate the proposed approach. The model is unpredictable, because it is controlled by a random variable. A convolutional neural network is trained to recognize the artifacts. The trained network is then used to compute the objective function for an iterative algorithm, which tries to reduce the artifacts in a computed tomography (CT) task. The objective function is evaluated in the image domain. The iterative algorithm for artifact reduction is in the projection domain. A gradient descent algorithm is used for the objective function optimization. The associated gradient is calculated with the chain rule. The learning curves illustrate the decreasing treads of the objective function as the number of iterations increases. The images after the iterative treatment show the reduction of artifacts. A quantitative metric, the Sum Square Difference (SSD), also indicates the effectiveness of the proposed method. The methodology of using a neural network as an objective function has potential value for situations where a human developed model is difficult to describe the underlying physics. Real-world applications are expected to be benefit from this methodology.

Novel Exact Traveling Wave Solutions for Nonlinear Wave Equations with Beta-Derivatives via the sine-Gordon Expansion Method

The main objectives of this research are to use the sine-Gordon expansion method (SGEM) along with the use of appropriate traveling transformations to extract new exact solitary wave solutions of the (2 + 1)- dimensional breaking soliton equation and the generalized Hirota-Satsuma coupled Korteweg de Vries (KdV) system equipped with beta partial derivatives. Using the chain rule, we convert the proposed nonlinear problems into nonlinear ordinary differential equations with integer orders. There is then no further demand for any normalization or discretization in the calculation process. The exact explicit solutions to the problems obtained with the SGEM are written in terms of hyperbolic functions. The exact solutions are new and published here for the first time. The effects of varying the fractional order of the beta-derivatives are studied through numerical simulations. 3D, 2D, and contour plots of solutions are shown for a range of values of fractional orders. As parameter values are changed, we can identify a kink-type solution, a bell-shaped solitary wave solution, and an anti-bell shaped soliton solution. All of the solutions have been carefully checked for correctness and could be very important for understanding nonlinear phenomena in beta partial differential equation models for systems involving the interaction of a Riemann wave with a long wave and interactions of two long waves with distinct dispersion relations.

Strengthening Gradient Descent by Sequential Motion Optimization for Deep Neural Networks

In this paper, we explore the advantages of heuristic mechanisms and devise a new optimization framework named Sequential Motion Optimization (SMO) to strengthen gradientbased methods. The key idea of SMO is inspired from a movement mechanism in a recent metaheuristic method called Balancing Composite Motion Optimization (BCMO). Specifically, SMO establishes a sequential motion chain of two gradientguided individuals including a leader and a follower to enhance the effectiveness of parameter updates in each iteration. A surrogate gradient model with low computation cost is theoretically established to estimate the gradient of the follower by that of the leader through chain rule during training process. Experimental results in terms of training quality on both fully-connected multilayer perceptrons (MLPs) and convolutional neural networks (CNNs) with respect to three popular benchmark datasets including MNIST, Fashion-MNIST and CIFAR-10 demonstrate the superior performance of the proposed framework in comparison with the vanilla stochastic gradient descent (SGD) implemented via back-propagation (BP) algorithm. Although this study only introduces the vanilla gradient descent (GD) as a main gradientguided factor in SMO for deep neural networks (DNNs) training application, it is greatly potential to combine with other gradientbased variants to improve its effectiveness and solve other largescale optimization problems in practice.

On the failure of the chain rule for the divergence of Sobolev vector fields

We construct a large class of incompressible vector fields with Sobolev regularity, in dimension [Formula: see text], for which the chain rule problem has a negative answer. In particular, for any renormalization map [Formula: see text] (satisfying suitable assumptions) and any (distributional) renormalization defect [Formula: see text] of the form [Formula: see text], where [Formula: see text] is an [Formula: see text] vector field, we can construct an incompressible Sobolev vector field [Formula: see text] and a density [Formula: see text] for which [Formula: see text] but [Formula: see text], provided [Formula: see text].

Wars over a piece of land or a land of peace? Kosovo and Ukraine

Understanding the Ukrainian situation through the description of the Kosovo war is the aim of this study. The analysis will revolve around three pairs of con-cepts: identity and territory, emotions and revenge, memory and forgiveness. It will be discussed whether, in a war, it is possible to speak of personal and social forgiveness in Ricoeur's terms. In this sense, the analysis will focus on the relation-ship between forgetting and memory, in particular on what birth (of new genera-tions) "could not erase", making an accumulation of memory of the facts that re-generate in an increasingly destructive collective revenge. If, in a war, breaking the rule of reciprocity is the way to forgiveness, this means taking away the power of guilt to oblige the future. Liberation from negative feelings, through loslassen or "letting go", opens up a new perspective that changes every kind of relationship, both between individuals and between Countries.

The Influence of Agrarian Reform, Social Capital, Farmer Spirituality and Personal Tribute on Farmers' Welfare in Kabupaten Bogor, West Java

Residents in various villages to have agricultural land. The distribution of soil types in the agrarian reform program in Kabupaten Bogor is divided into two categories, namely paddy fields and dry land. The number of family members beneficiaries of agrarian reform has an average of around 3-4 people with a standard deviation of 1.3. Agricultural land ownership status affects the welfare of farmers. Social capital is related to cooperation in a society to achieve a better quality of life supported by values ​​and norms as the main elements including trust, reciprocity and collective rules. The influence of spirituality on the welfare of farmers in Bogor Regency is very important to have. Some farmers think that having a high level of education is not important to work as a farmer. Personal attributes influence the welfare of farmers in Bogor Regency. The main objective of this study was to determine the effect of the agrarian reform program, social capital, farmer spirituality and personal attributes on the welfare of farmers in Kabupaten Bogor, West Java. This study was designed using a descriptive method with a quantitative approach. Moreover, To find out how much influence these personal attributes have, this study uses 5 (five) conceptual points in the hypothesis. The five points that influence personal attributes include extraversion, agreeableness, conscientiousness, neuroticism and openness. Most of the farmers feel less confident in carrying out their agricultural tasks, but they have good communication skills and are full of enthusiasm.

Deep neural network method for solving the fractional Burgers-type equations with conformable derivative

In this article, we introduce the modified physics-informed neural network (PINN) method for finding data-driven solutions of three classes of time-fractional Burgers-type equations under the conformable sense. Since conformable derivative satisfies the chain rule, automatic differentiation can be applied to compute it directly to avoid truncation and other numerical discretization. In addition, the locally adaptive activation function and two effective weighting strategies are introduced to improve solution accuracy. As a result, three numerical examples indicate that the modified PINN method gives an efficient and reliable solution.

Eigenvalue problems for p-div-curl systems

We consider eigenvalue problems for a general p-div-curl system. In particular, we consider a nonlinear eigenvalue problem similar to the standard Dirichlet eigenvalue problem for the p-Laplacian, but due to the presence of the curl, we see that eigenfunctions of the general p-div-curl operator do not enjoy nice regularity as in the p-Laplacian case. We first show that weak solutions of a particular nonlinear p-div-curl eigenvalue problem split into a good and bad part, where the good part is not even divergence free. We then show that the general p-div-curl system admits a Ljusternik-Schnirelman sequence of eigenvalues. Finally, we show a general chain rule for the curl and divergence.

Pointwise decay of solutions to the energy critical nonlinear Schrödinger equations

In this note, we prove pointwise decay in time of solutions to the 3D energy-critical nonlinear Schrödinger equations assuming data in L1∩H3. The main ingredients are the boundness of the Schrödinger propagators in Hardy space due to Miyachi [9] and a fractional Leibniz rule in the Hardy space. We also extend the fractional chain rule to the Hardy space.

Shape feature controlled topology optimization of attached piezoelectric actuators for vibration control of thin-walled smart structures

This paper proposes a method for shape feature controlled topology optimization of attached piezoelectric actuators of thin-walled smart structures, aiming at both vibration control performance as well as possible manufacturing constraints. This paper first proposes a modified K-means clustering method to derive the shape feature control functions. Then, an optimization model with system performance index (SPI) as its objectiveness and shape feature control functions as its explicit constraints is organized. Based on the Lyapunov equation, the adjoint method and the chain rule, the sensitivity analysis of both objective and constraints in optimization model is performed. A mathematical programming method is applied to solve the optimization model once the sensitivities is calculated. Then the LQR control algorithm is used in the optimized structure for vibration control. Numerical examples as well as engineering application are applied to prove effectiveness of the proposed methods. The results show feasibility of the proposed methods.

Analysis of analytical solutions of fractional Date-Jimbo-Kashiwara-Miwa equation

In this study, the authors solved the (2+1)-dimensional fractional Date-Jimbo-Kashiwara-Miwa equation for nonlinear dispersive waves in an inhomogeneous medium, which is an extension of the Kadomtsev-Petviashvili hierarchy. A powerful analytical technique, the new auxiliary equation method, is employed to develop the class of soliton solutions to the considered equation which contains dark, bright, and periodic wave solutions. A comparative analysis is done by using two fractional derivatives, namely, β and Atangana-Baleanu. While using this technique, the nonlinear ordinary differential equation in integer order is formed using chain rule from the fractional Date-Jimbo-Kashiwara-Miwa equation. There is no more a requirement to perform any normalization or discretization because of the chain rule. These obtained results are graphically portrayed such as 3D and 2D using suitable parametric values. The exact solutions to this equation are necessary to comprehend wave behavior in physical models. Moreover, a sensitive analysis to the fractional dynamical system is performed and graphically shown the behavior on initial conditions.

State of charge estimation of lithium-ion battery based on improved Hausdorff gradient using wavelet neural networks

In this paper, a parameter optimization algorithm via the wavelet neural network (WNN) based on fractal derivative is proposed to estimate the state of charge (SOC) from the measured current and voltage data for lithium-ion batteries. The improved Hausdorff derivative is applied to the parameter tuning of a WNN to improve the performance of the SOC estimation. The local property of the improved Hausdorff derivative is used to obtain the chain rule which is consistent with the integer-order derivative, and the parameter tuning formulas in WNN based on the improved Hausdorff gradient are obtained. The adaptive order tuning method is proposed and applied to optimize the parameter tuning. Compared with the integer-order optimization method, this method speeds up the optimization of the parameters and improves the optimization accuracy of SOC estimation. The experiment results verify the effectiveness of the proposed SOC estimation method via the WNN based on fractal derivative.

A meta network pruning framework for remaining useful life prediction of rocket engine bearings with temporal distribution discrepancy

Accurate remaining useful life prediction (RUL) is important for the reliability and safety of liquid rocket engines. In this paper, a meta network pruning framework with attention augmented convolutions is proposed for RUL prediction. To address the problem of distribution discrepancy in engineering data under transient working conditions, a data-driven distribution matching strategy is designed. Besides, in view of the prediction accuracy and computation complexity of the model, an iterative meta network pruning algorithm, which automatically calculates the meta-gradients of each convolutional kernel according to the chain rule, is developed to identify, and then delete the unimportant connections in the framework. The method is verified on a high-precision cryogenic rocket engine bearing experiment platform under liquid nitrogen and received better performance than benchmark algorithms.