Chain Rule Research Articles

Microdisplays as a versatile tool for the optical simulation of crystal diffraction in the classroom

Presented here is a flexible and low-cost setup for demonstrating X-ray, electron or neutron diffraction methods in the classroom. Using programmable spatial light modulators extracted from a commercial video projector, physical diffraction patterns are generated, in real time, of any two-dimensional structure which can be displayed on a computer screen. This concept enables hands-on experience beyond simplest-case scenarios of scattering phenomena, and for the students, the transfer to the regime of visible light greatly enhances intuitive understanding of diffraction in real and reciprocal space. The idea and working principle of the modified video projector are explained, technical advice is given for its choice and successful modification, and a Python-based open-source program code is provided. Program features include the design and interactive manipulation of two-dimensional crystal structures to allow a straightforward demonstration of concepts such as reciprocal space, structure factors, selection rules, symmetry and symmetry violation, as well as structural disorder. This approach has already proved helpful in teaching crystal diffraction to undergraduate students in materials science.

Calculus rules of the generalized contingent derivative and applications to set-valued optimization

In the paper, we develop sum and chain rules of the generalized contingent derivative for set-valued mappings. Then, their applications to sensitivity analysis and optimality conditions for some particular optimization problems are given. Our results extend some recent existing ones in the literature.

A chain rule formula for higher derivations and inverses of polynomial maps

The multidimensional chain rule formula for analytic functions and its generalization to higher derivatives perfectly work in the algebraic setting in characteristic zero. In positive characteristic one runs into problems due to denominators in these formulas. In this article we show a direct analog of these formulas using higher derivations which are defined in any characteristic. We also use these formulas to show how higher derivations to different coordinate systems are related to each other. Finally, we apply this to polynomial automorphisms in arbitrary characteristic and obtain a formula for the inverse of such a polynomial automorphism.

On fractional calculus with general analytic kernels

Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann–Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel functions. We demonstrate, under some assumptions, how all of these modifications can be considered as special cases of a single, unifying, model of fractional calculus. We provide a fundamental connection with classical fractional calculus by writing these general fractional operators in terms of the original Riemann–Liouville fractional integral operator. We also consider inversion properties of the new operators, prove analogues of the Leibniz and chain rules in this model of fractional calculus, and solve some fractional differential equations using the new operators.

Single variable differential calculus under q ‐rung orthopair fuzzy environment: Limit, derivative, chain rules, and its application

The q-rung orthopair fuzzy set ( q-ROFS) that the sum of the qth power of the membership degree and the qth power of the nonmembership degree is restricted to one is a generalization of fuzzy set (FS). Recently, many researchers have given a series of aggregation operators to fuse q-rung orthopair fuzzy discrete information. Subsequently, although some scholars have also focused on studying q-rung orthopair fuzzy continuous information and give its continuity, derivative, differential, and integral, those studies are only considered from the perspective of multivariable fuzzy functions. Thus, the main aim of the paper is to study the q-rung orthopair fuzzy continuous single variable information. In this paper, we first define the concept of q-rung orthopair single variable fuzzy function ( q-ROSVFF) to describe the fuzzy continuous information, and give its domain to make sure that this kind of function is meaningful. Afterward, we propose the limits, continuities, and infinitesimal of q-ROSVFFs, and offer the relationship between the limit of q-ROSVFF and that of q-ROSVFF infinitesimal. On the basis of the definition of derivative in mathematical analysis, we define the subtraction and division derivatives and basic operational rules, and offer the simpler proofs for the derivatives of q-ROSVFFs. What is more, we propose the subtraction and division differential invariances, and give the approximate calculation formulas of q-ROSVFFs when the value of independent variable is changed small enough. In the real situation, fundamental functions cannot be used to express more complicated functions, thus we define the compound q-ROSVFFs and give their chain rules of subtraction and division derivatives. Finally, we use numerical examples by simulation to verify the feasibility and veracity of the approximate calculation on q-ROSVFFs.

Sewer damage detection from imbalanced CCTV inspection data using deep convolutional neural networks with hierarchical classification

Abstract Accurate infrastructure condition assessment is critical for optimized maintenance and rehabilitation plan. Closed Circuit Television (CCTV) inspection has been widely applied in the internal inspection of sewerage systems. However, the manual approach adopted under current practice is expertise intensive and time-consuming. Previous research has attempted to apply specialized image processing techniques for the detection of specific defects with engineered features, such as cracks and joint offset. However, these engineered features are less generalizable than the state-of-the-art deep learning methods. Another crucial problem in defect classification is the imbalance between defects and non-defects due to high volume of normal images and the imbalance between different defects due to varying defect occurrence rates. This raises a big challenge for both traditional methods and deep learning methods. In this paper, a method based on the deep convolutional neural network is proposed to detect and classify defects from CCTV inspections. To improve the performance on imbalanced datasets, a hierarchical classification approach is introduced to supervise the learning process at different levels. The high-level detection task tries to discriminate images with defects from normal images. The low-level classification calculates the probability of each defect assuming the image has defects. The final defect classification is then derived from the chain rule of conditional probability. The network was trained and tested using inspection images collected from 24.7 km sewer lines. The high-level defect detection accuracy was improved from 78.4% to 83.2% with a hierarchical classification approach. Due to the difficulty to discriminate the defects, the low-level defect classification accuracy still needs improvements, but the proposed network with hierarchical classification also demonstrated superior performance over traditional approaches.

Fractional derivatives of composite functions and the Cauchy problem for the nonlinear half wave equation

We show new results of well-posedness for the Cauchy problem for the half wave equation with power-type nonlinear terms. For the purpose, we propose two approaches on the basis of the contraction-mapping argument. One of them relies upon the $$L_t^q L_x^\infty $$ Strichartz-type estimate together with the Ginibre–Ozawa–Velo type chain rule of fairly general fractional orders. This chain rule has a significance of its own. Furthermore, in addition to the weighted fractional chain rule established in Hidano et al. (Weighted fractional chain rule and nonlinear wave equations with minimal regularity. Preprint, arXiv:1605.06748v3 [math.AP], 2018), the other approach uses weighted space-time $$L^2$$ estimates for the inhomogeneous equation which are recovered from those for the second-order wave equation. In particular, by the latter approach we settle the problem left open in Bellazzini et al. (Math Ann 371(1–2):707–740, 2018) concerning the local well-posedness in $$H^{s}_{\mathrm{rad}}({\mathbb R}^n)$$ with $$s>1/2$$ .

Shortened Regenerating Codes

For general exact repair regenerating codes, the optimal trade-offs between the storage size and repair bandwidth remain undetermined. Various outer bounds and partial results have been proposed. Using a simple chain rule argument, we identify nonnegative differences between the functional repair and the exact repair outer bounds, and one of the differences is then bounded from below by the repair data of a shortened subcode. Our main result is a new outer bound for an exact repair regenerating code in terms of its shortened subcodes. In general, the new outer bound is implicit and depends on the choice of shortened subcodes. For the linear case, we obtain explicit bounds.

Prediction of Physical-Chemical and Fire Hazard Characteristics by Carbon Chain Rules 2. Carboxylic Acids

Prediction of Physical-Chemical and Fire Hazard Characteristics by Carbon Chain Rules 2. Carboxylic Acids

Differential algebras in codifferential categories

Differential categories were introduced by Blute, Cockett, and Seely as categorical models of differential linear logic and have since led to abstract formulations of many notions involving differentiation such as the directional derivative, differential forms, smooth manifolds, De Rham cohomology, etc. In this paper we study the generalization of differential algebras to the context of differential categories by introducing T-differential algebras, which can be seen as special cases of Blute, Lucyshyn-Wright, and O'Neill's notion of T-derivations. As such, T-differential algebras are axiomatized by the chain rule and as a consequence we obtain both the higher-order Leibniz rule and the Faà di Bruno formula for the higher-order chain rule. We also construct both free and cofree T-differential algebras for suitable codifferential categories and discuss power series of T-algebras.

Robust stability of mixed Cohen–Grossberg neural networks with discontinuous activation functions

PurposeThe purpose of this paper is to develop a method for the existence, uniqueness and globally robust stability of the equilibrium point for Cohen–Grossberg neural networks with time-varying delays, continuous distributed delays and a kind of discontinuous activation functions.Design/methodology/approachBased on the Leray–Schauder alternative theorem and chain rule, by using a novel integral inequality dealing with monotone non-decreasing function, the authors obtain a delay-dependent sufficient condition with less conservativeness for robust stability of considered neural networks.FindingsIt turns out that the authors’ delay-dependent sufficient condition can be formed in terms of linear matrix inequalities conditions. Two examples show the effectiveness of the obtained results.Originality/valueThe novelty of the proposed approach lies in dealing with a new kind of discontinuous activation functions by using the Leray–Schauder alternative theorem, chain rule and a novel integral inequality on monotone non-decreasing function.

Flexible Photogrammetric Computations Using Modular Bundle Adjustment: The Chain Rule and the Collinearity Equations

The main purpose of this article is to show that photogram-metric bundle-adjustment computations can be sequentially organized into modules. Furthermore, the chain rule can be used to simplify the computation of the analytical Jacobians needed for the adjustment. Novel projection models can be flexibly evaluated by inserting, modifying, or swapping the order of selected modules. As a proof of concept, two variants of the pinhole projection model with Brown lens distortion were implemented in the open-source Damped Bundle Adjustment Toolbox and applied to simulated and calibration data for a nonconventional lens system. The results show a significant difference for the simulated, error-free, data but not for the real calibration data. The current flexible implementation incurs a performance loss. However, in cases where flexibility is more important, the modular formulation should be a useful tool to investigate novel sensors, data-processing techniques, and refractive models.

Quaternion Filtering Based on Quaternion Involutions and its Application in Signal Processing

The quaternion gradient plays an important role in quaternion signal processing, and has undergone several modifications. Recently, three methods for obtaining the quaternion gradient have been proposed based on generalized HR calculus, the quaternion product, and quaternion involutions, respectively. The first method introduces the quaternion rotation, which is difficult to calculate and often relies on lookup tables; the second is cumbersome because it depends on all the real and imaginary parts of the variables and parameters; the third transforms the gradient from the quaternion domain to the real domain, and uses real derivatives and the real chain rule. In this paper, we generalize the quaternion involutions method to calculate the quaternion gradient of the quaternion matrix function. Several examples are presented in which the proposed gradient is applied to the adaptive filter, Kalman filter, and kernel filters to solve the associated optimization problems.

New exact solitary wave solutions to the space-time fractional differential equations with conformable derivative

The exact wave solutions to the space-time fractional modified Benjamin-Bona-Mahony (mBBM) and space time fractional Zakharov-Kuznetsov Benjamin-Bona-Mahony (ZKBBM) equations are studied in the sense of conformable derivative. The existence of chain rule and the derivative of composite functions permit the nonlinear fractional differential equations (NLFDEs) to convert into the ordinary differential equation using wave transformation. The wave solutions of these equations are examined by means of the expanding and effective two variable ( G' / G ,1/ G )-expansion method. The solutions are obtained in the form of hyperbolic, trigonometric and rational functions containing parameters. The method is efficient, convenient, accessible and is the generalization of the original ( G' / G )-expansion method.

New Travelling Wave Solutions for Time-Space Fractional Liouville and Sine-Gordon Equations

In this paper, the authors discussed the new wave solutions of time-space fractional Liouville andSine-Gordon equations by using a reliable analytical method called sub-equation method. The fractional derivativesof considered equations are handled in conformable sense. Conformable derivative which is an easy and applicabletype of fractional derivative, satisfies basic properties of known derivative with integer order such as Leibnizrule, quotient rule, chain rule. These properties of conformable derivative superior to other popular definitions onobtaining analytical solutions of fractional equations.

Gravitational Apparent Motion-Based SINS Self-Alignment Method for Underwater Vehicles

To solve the self-alignment problem of strapdown inertial navigation system (SINS) for underwater vehicles, a novel gravitational apparent motion (GAM)-based method is proposed. Different from conventional GAM methods, the proposed GAM method can complete SINS self-alignment under swaying conditions without using the a priori local latitude information. First, we determine the gravity vector in the earth frame and the local latitude by using the gradient descent optimization and certain geometry constraints. Then, the self-alignment process is formulated as an optimization-based alignment quaternion determination problem by constructing an objective function with the estimated gravity vector. We employ gradient descent optimization to achieve the least square solution of the objective function. Thus, the attitude quaternion can be determined according to the quaternion product chain rule. The simulation and experiments results demonstrate the proposed GAM method without using the local latitude achieves an alignment accuracy close to conventional GAM methods during the coarse alignment process.

An estimate on the Bedrosian commutator in Sobolev space

While the Bedrosian identity for the Hilbert transform of a product does not hold for general Sobolev class functions, we show that the defect of this identity is more regular than would be naively expected. We use this result to give a stronger-than-expected estimate on the chain rule defect of the square root of the Laplacian.

Students’ ability to correctly apply differentiation rules to structurally different functions

The derivative concept is studied in first-year university mathematics. In this study, we focused on students’ ability to correctly apply the rules for derivatives of functions with the different structures that they encounter in their university studies. This was done by investigating the online responses of first-year students at the University of KwaZulu-Natal to online quizzes that contributed to their assessment. Based on this investigation, we then interviewed eight students to gain an insight into the thinking behind their responses. We report on the analysis of students’ responses to five items on the online quizzes based on the derivative concept. The categories in which those items were based are: condition for existence of derivative at a point; rules for derivatives of standard functions; application of chain rule to different function structures; the application of multiple rules; and application of derivatives to optimise a function. Our findings indicate that students had difficulty in detecting that multiple rules for derivatives were required to differentiate certain types of functions represented in symbolic form. Furthermore, students had difficulty in finding the derivative of a function when more than one application of the chain rule was required. However, there were students who had the ability to apply the rules for derivatives of functions without difficulty. In particular, most of the students were able to correctly recall the differentiation rules for functions with standard structures f(x)=xn, h(x)=ekx and y=[g(x)]n, n ≠ 0 and k is a non-zero constant. Students were also able to correctly apply the chain rule to an exponential function with base e, raised to 4x. The majority of students were able to correctly apply the chain rule together with differentiation rules for logarithmic and exponential (with bases a >1) function structures, and function structures that required the application of the product rule together with the chain rule. Most of the students were able to apply derivatives to optimise a function.Significance:
 
 A significant percentage of students who took online quizzes experienced difficulties with applying multiple differentiation rules in the context of a single function. The difficulties stemmed from their inability to detect from the structure of the function which rules should be applied and also the order in which those relevant rules should be applied.

Tsallis Entropy of Partitions in Quantum Logics

The purpose of the present paper is to study the Tsallis entropy Sq,t(P) of a partition P, and the Tsallis conditional entropy Sq,t(P|Q) of finite partitions P,Q in a quantum logic (L,t), where L is a quantum logic and t is a Bayessian state. We extend some results of Furuichi (J. Math. Phys. 47: 023302, 2006) to the partitions in (L,t). We study the concavity and subadditivity properties for Tsallis entropy of partitions in (L,t). We show that the subadditivity property in the case of q 1,Sq,t(P|Q) ≤ Sq,t(P), but in the case of $q<1, S_{q,t} (P|Q) \nleq S_{q,t} (P)$ , in general. We also study the chain rules for the information measures in (L,t); and present a numerical example to illustrate the results. We study Tsallis entropy and Tsallis conditional entropy of t-independent partitions. Finally, we show that, Tsallis entropy and Tsallis conditional entropy of partitions of a couple (L,t), are invariant under the relation of $\circeq _{t}$ .

The DOTI Approach: An Alternative Way of Teaching and Learning the Chain Rule

The DOTI Approach: An Alternative Way of Teaching and Learning the Chain Rule