Chain Rule Research Articles

Constructing common height maps with various entropy-based similarity metrics and utilizing layering method for heterogeneous robot teams

PurposeHeterogeneous teams consisting of unmanned ground vehicles and unmanned aerial vehicles are being used for different types of missions such as surveillance, tracking and exploration. Exploration missions with heterogeneous robot teams (HeRTs) should acquire a common map for understanding the surroundings better. The purpose of this paper is to provide a unique approach with cooperative use of agents that provides a well-detailed observation over the environment where challenging details and complex structures are involved. Also, this method is suitable for real-time applications and autonomous path planning for exploration.Design/methodology/approachLidar odometry and mapping and various similarity metrics such as Shannon entropy, Kullback–Leibler divergence, Jeffrey divergence, K divergence, Topsoe divergence, Jensen–Shannon divergence and Jensen divergence are used to construct a common height map of the environment. Furthermore, the authors presented the layering method that provides more accuracy and a better understanding of the common map.FindingsIn summary, with the experiments, the authors observed features located beneath the trees or the roofed top areas and above them without any need for global positioning system signal. Additionally, a more effective common map that enables planning trajectories for both vehicles is obtained with the determined similarity metric and the layering method.Originality/valueIn this study, the authors present a unique solution that implements various entropy-based similarity metrics with the aim of constructing common maps of the environment with HeRTs. To create common maps, Shannon entropy–based similarity metrics can be used, as it is the only one that holds the chain rule of conditional probability precisely. Seven distinct similarity metrics are compared, and the most effective one is chosen for getting a more comprehensive and valid common map. Moreover, different from all the studies in literature, the layering method is used to compute the similarities of each local map obtained by a HeRT. This method also provides the accuracy of the merged common map, as robots’ sight of view prevents the same observations of the environment in features such as a roofed area or trees. This novel approach can also be used in global positioning system-denied and closed environments. The results are verified with experiments.

Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations

<p style='text-indent:20px;'>Structure-preserving finite-difference schemes for general nonlinear fourth-order parabolic equations on the one-dimensional torus are derived. Examples include the thin-film and the Derrida–Lebowitz–Speer–Spohn equations. The schemes conserve the mass and dissipate the entropy. The scheme associated to the logarithmic entropy also preserves the positivity. The idea of the derivation is to reformulate the equations in such a way that the chain rule is avoided. A central finite-difference discretization is then applied to the reformulation. In this way, the same dissipation rates as in the continuous case are recovered. The strategy can be extended to a multi-dimensional thin-film equation. Numerical examples in one and two space dimensions illustrate the dissipation properties.

Using nonlinear functions to approximate a new quasi-Newton method for unconstrained optimization problems

In order to get a higher order accuracy of approximating the Hessian matrix of the objective function, we use the chain rule and propose two modified secant equations. An interesting property of the proposed methods is that these utilize information from two most recent steps where the usual secant equation uses only the latest step information. The other point of interest to one of the proposed methods is that it makes use of both gradient and function value information. We show that the modified BFGS methods based on the new secant equations are globally convergent. The presented experimental results illustrate that the proposed methods are efficient.

Prospects of Chain Rule of Differentiations and Inverse of Multiple Functions

In this perspective paper, I tried to explain that what will be the possible prospect of multiple functions in one and another through the chain rule of differentiation? The chain rule is a formula to compute the derivative of the functional composition of two or more functions. The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. The chain rule in that what changes I noted, how it can be modified to reduce the differentiation process of multiple functions in one and another, and implementing that process in the inverse of multiple functions in one and another, further proceedings them through the chain rule of differentiation with all prospects. Here I claim that the differentiation’s chain rule is not changed but the sequences of finding the derivative of multiple functions are changed.

Tunability and switching of Fano and Lorentz resonances in PTX-symmetric electronic systems

Resonance is a ubiquitous phenomenon observed in a wide range of physical systems. Recently, with the Fano resonance exerting remarkable potential for optical, acoustic, atomic, and electronic applications, it is vital to control and even dynamically reconfigure the resonance line shape and bandwidth, in addition to its frequency. In this work, we introduce a parity-time-reciprocal scaling (PTX)-symmetric structure, which can offer a promising avenue for tailoring the resonance frequency and line shape of electronic circuits. We have theoretically studied the resonance behavior of such a PTX-symmetric electronic system, particularly for dependencies of resonant peaks and line-shapes on the non-Hermiticity, coupling coefficient, and the scaling coefficient introduced by the reciprocal scaling (X) transformation. Our results demonstrate that, at resonance frequencies, a transition between Fano and Lorentzian line-shapes is possible with a specific reciprocal scaling rule applied to lumped-element circuits.

A general method for N-order integral-form Gauss's variational equations under impulsive control

In this paper, an innovative derivatives-based method is proposed to derive any higher-order integral-form Gauss's Variational Equations (GVEs) of the orbital elements under impulsive control. In this new method, the Taylor series expansion is applied to express the variation of each orbital element as a sum of terms that are calculated from the values of the derivatives of orbital elements. The required N-order derivatives of orbital elements can be successively computed from the (N-1)-order differential-form GVEs by using the chain rule of calculus. Using this method, the final expressions of the second-order and third-order integral-form GVEs are provided in detail. To testify the accuracy of these new results, the Monte Carlo simulations are performed randomly and the histogram data are analyzed. It is demonstrated that the proposed higher-order GVEs are more accurate than the traditional lower-order ones. The method and the resultant higher-order GVEs can be potentially used on spacecraft impulsive orbital maneuver or formation flying.

Spatial density neural network force fields with first-principles level accuracy and application to thermal transport

Constrained by the fixed mathematical form of most empirical potentials used in classical molecular dynamics (MD) simulations, many properties of materials cannot be captured within experimental accuracy. On the other hand, accurate electronic structure calculations based on quantum theory, most notably density functional theory (DFT), are limited to several hundred atoms within a picosecond, which makes the method inadequate for modeling systems beyond the nanoscale. A combination of speed from classical MD and fidelity from DFT can be achieved through machine learning methods. Herewith, we developed an approach named spatial density neural network force fields (SDNNFFs) by training neural networks to ``learn'' and predict DFT-level forces. Our model focuses on the usage of a three-dimensional mesh of density functions, which together act as a mapping of the atomic environment and provides a physical representation of the forces acting on the central atom. Several notable advantages arise from the SDNNFF, including (1) the avoidance of the chain rule on the total energy and other variables by direct calculation of the forces from the neural network, (2) the ever large $N\ifmmode\times\else\texttimes\fi{}t$ scaling of the training data, where $N$ is the number of atoms in a supercell and $t$ is the number of evaluated structures by first-principles, and (3) the significant reduction in parameters and human effort needed to successfully train a force- and/or property-converged neural network force field. Overall, we focus on modeling DFT-level forces with minimal computational cost and parametrization for rapid prediction of phonon-based properties and future molecular dynamics of large-scale systems. To demonstrate the SDNNFF, we trained several models on diamond structures, including bulk silicon (Si), diamond, silicon carbide (SiC), and boron arsenide (BAs), and predicted their phonon dispersions and lattice thermal conductivities using the direct solution to the phonon Boltzmann transport equation. For phonon properties, we utilized a fitting method for obtaining the second- and third-order force constants, which outperforms the highly force-sensitive finite displacement method when employing neural network force fields. In comparison to DFT lattice thermal conductivity, we obtained high precision results from our SDNNFF within 0.7% for Si, 6.2% for diamond, 2.76% for SiC, and 7.46% for BAs, with further agreement with experiments. The phonon dispersions from the SDNNFF also matched those from direct DFT and experiments. The developed approach for accurately predicting phonon transport properties of crystalline materials would largely benefit the design of advanced materials with improved performance, such as complex thermoelectric devices and low thermal resistance interfaces for nanoelectronics. Future applications of our SDNNFF model could be extended toward including atomic energy into the algorithm and simulating large-scale heterogeneous systems for quasielectronic representations for various properties.

The non-profit sharing economy from a social exchange theory perspective: a case from World Wide Opportunities on Organic Farms in Taiwan

The non-profit form of the Sharing Economy (SE) is arguably better positioned to manifest the alternative nature of SE that supports sustainable production and consumption than its for-profit counterpart. However, interactions in non-profit SE are imbued with uncertainties that can discourage participants’ continuing engagement. Social exchange theory was adopted to explore the dynamism of non-profit SE exchange in the context of WWOOF (World Wide Opportunities on Organic Farms) as a niche area. The study was conducted primarily through semi-structured interviews with eighteen WWOOF farm hosts in Taiwan. The findings revealed that most hosts opened their farms to share their organic farming knowledge with WWOOF guests as a measure to attain social/environmental, cultural, and economic values that motivated their participation in WWOOF. Depending on the value(s) salient in driving WWOOF hosts’ participation, the exchange rules of reciprocity, rationality, and altruism were variously applied to manage uncertainties and limited resource capacities, and address issues arising from demands for emotional labor and social dilemmas. Positive exchange outcomes were experienced when exchange rules lead to reduced uncertainties, and enhanced equality, fairness, and joint responsibility. Implications of how the findings can be applied to promote sustainable production and consumption through non-profit SE are discussed.

Polyhedral DC Decomposition and DCA Optimization of Piecewise Linear Functions

For piecewise linear functions f : R n ↦ R we show how their abs-linear representation can be extended to yield simultaneously their decomposition into a convex f ˇ and a concave part f ^ , including a pair of generalized gradients g ˇ ∈ R n ∋ g ^ . The latter satisfy strict chain rules and can be computed in the reverse mode of algorithmic differentiation, at a small multiple of the cost of evaluating f itself. It is shown how f ˇ and f ^ can be expressed as a single maximum and a single minimum of affine functions, respectively. The two subgradients g ˇ and − g ^ are then used to drive DCA algorithms, where the (convex) inner problem can be solved in finitely many steps, e.g., by a Simplex variant or the true steepest descent method. Using a reflection technique to update the gradients of the concave part, one can ensure finite convergence to a local minimizer of f, provided the Linear Independence Kink Qualification holds. For piecewise smooth objectives the approach can be used as an inner method for successive piecewise linearization.

Sparse and low-rank regularized deep subspace clustering

Subspace clustering aims at discovering the intrinsic structure of data in unsupervised fashion. As ever in most of approaches, an affinity matrix is constructed by learning from original data or the corresponding hand-crafted feature with some constraints on the self-expressive matrix (SEM), which is then followed by spectral clustering algorithm. Based on successful applications of deep technologies, it has become popular to simultaneously accomplish deep feature and self-representation learning for subspace clustering. However, deep feature and SEM in previous deep methods are lack of precise constraints, which is sub-optimal to conform with the linear subspace model. To address this, we propose an approach, namely sparse and low-rank regularized deep subspace clustering (SLR-DSC). In the proposed SLR-DSC, an end-to-end framework is proposed by introducing sparse and low-rank constraints on deep feature and SEM respectively. The sparse deep feature and low-rank regularized SEM implemented via fully-connected layers are encouraged to facilitate a more informative affinity matrix. In order to solve the nuclear norm minimization problem, a sub-gradient computation strategy is utilized to cater to the chain rule. Experiments on the data sets demonstrate that our method significantly outperforms the competitive unsupervised subspace clustering approaches.

Decomposition rules for quantum Rényi mutual information with an application to information exclusion relations

We prove decomposition rules for quantum Rényi mutual information, generalizing the relation I(A : B) = H(A) − H(A|B) to inequalities between Rényi mutual information and Rényi entropy of different orders. The proof uses Beigi’s generalization of Reisz–Thorin interpolation to operator norms [J. Math. Phys. 54(12), 122202 (2013)] and a variation of the argument developed by Dupuis [J. Math. Phys. 56, 022203 (2015)], which was used to show chain rules for conditional Rényi entropies. The resulting decomposition rule is then applied to establish an information exclusion relation for Rényi mutual information, generalizing the original relation by Hall [Phys. Rev. Lett. 74, 3307 (1995)].

The New Travelling Wave Solutions of Time Fractional Fitzhugh-Nagumo Equation with Sine-Gordon Expansion Method

Authors aimed to employ the sine-Gordon expansion method to acquire the new exact solutions of fractional Fitzhugh-Nagumo equation which is a stripped type of the Hodgkin-Huxley model that expresses in extensive way activation and deactivation dynamics of neuron spiking. By using the wave transformations, by the practicality of chain rule and applicability of the conformable fractional derivative, the fractional nonlinear partial differential equation (FNPDE) changes to a nonlinear ordinary differential equation. So the exact solution of the considered equation can be obtained correctly with the aid of efficient and reliable analytical techniques.

A second-order shape optimization algorithm for solving the exterior Bernoulli free boundary problem using a new boundary cost functional

The exterior Bernoulli problem is rephrased into a shape optimization problem using a new type of objective function called the Dirichlet-data-gap cost function which measures the $$L^2$$ -distance between the Dirichlet data of two state functions. The first-order shape derivative of the cost function is explicitly determined via the chain rule approach. Using the same technique, the second-order shape derivative of the cost function at the solution of the free boundary problem is also computed. The gradient and Hessian informations are then used to formulate an efficient second-order gradient-based descent algorithm to numerically solve the minimization problem. The feasibility of the proposed method is illustrated through various numerical examples.

A novel mathematical method to analyze the free vibration of eccentric annular plates

Vibration analysis of nominally axisymmetric plate structures with either imperfections or geometric asymmetries due to practical motivations is of interest in designing and developing some mechanical structures. Semi-analytical methods to model such structures suffer from either choosing inappropriate admissible functions or both plausible convergence issues and additional computations owing to employing the addition theorem of Bessel functions. Therefore, the present study aims at developing a new mathematical method to analyze the vibrational behavior of circular plates with geometric asymmetries. The suggested approach makes use of the separation of variables to determine general solutions of the partial differential equation of the plate transverse displacement while defining multiple polar coordinate systems each of which offers a formulation of the plate deformation. Moreover, closed-form geometric equations and the chain rule for determining derivatives are implemented to move from one coordinate system to the other to satisfy boundary conditions without any need for the cumbersome transformation involved in using the addition theorem. A finite element model is also constructed to evaluate the validity of the proposed method before studying the effects of the cutout location and size on natural frequencies and mode shapes of eccentric annular plates.

A neurodynamic optimization approach for complex-variables programming problem

A neural network model upon differential inclusion is designed for solving the complex-variables convex programming, and the chain rule for real-valued function with the complex-variables is established in this paper. The model does not need to choose penalty parameters when applied to practical problems, which makes it easier to design. The result is obtained that its state reaches the feasible region in finite time. Furthermore, the convergence for its state to an optimal solution is proved. Some typical examples are shown for the effectiveness of the designed model.

Post‐stack impedance blocky inversion based on analytic solution of viscous acoustic wave equation

ABSTRACTThe conventional impedance inversion method ignores the attenuation effect, transmission loss and inter‐layer multiple waves; the smooth‐like regularization approach makes the corresponding impedance solution excessively smooth. Both fundamentally limit the resolution of impedance result and lead to the inadequate ability of boundary characterization. Therefore, a post‐stack impedance blocky inversion method based on the analytic solution of viscous acoustic equation is proposed. Based on the derived recursive formula of reflections, the 1D viscous acoustic wave equation is solved analytically to obtain zero‐offset full‐wave field response. Applying chain rule, the analytical expression of the Fréchet derivative is derived for gradient‐descent non‐linear inversion. Combined with smooth constraints, the blocky constraints can be introduced into the Bayesian inference framework to obtain stable and well‐defined inversion results. According to the above theory, we firstly use model data to analyse the influence of incompleteness of forward method on seismic response, and further verify the effectiveness of the proposed method. Then the Q‐value sensitivity analysis of seismic trace is carried out to reduce the difficulty of Q‐value estimation. Finally, the real data from Lower Congo Basin in West Africa indicate that the proposed approach provide the high‐resolution and well‐defined impedance result. As a supplement and development of linear impedance inversion method, the non‐linear viscous inversion could recover more realistic and reliable impedance profiles.

시골 양반 역병(疫病) 분투기 - 18세기 구상덕의 『승총명록』을 중심으로

In the 18th century, Joseon experienced a series of famine and plague. In short, it was a time of disaster. Gu Sang-deok, a local literati in Gyeongsang-do, detailed the horrors of epidemics and the lives of people overcoming it in his diary. When plague was prevalent people hid in temples or rang gongs and drums to chase away ghosts. Some prayed to spirits of epidemics while some drank beef soup to recover their strength. Those in better condition visited doctors and hospitals to take medicine. They couldn’t be helpless by an attack of epidemics. Gu Sang-deok was busy taking care of his family, protecting children and old people in his rural community from dying of plague and hunger. He went to rural Confucian school to discuss various issues of community, and delivered the public opinion(or dissatisfaction) to District magistrate.<BR>In the event of a plague, the state not only distributed prescriptions and medical books for the epidemic throughout the country, but also set up hut shelters at local authorities to isolate patients and hold rituals to heal the fear of the plague. Of course it was not enough until modern segregation systems and vaccines were developed people in Joseon had to live with plague and more epidemics. The members of the community provided shelters to each other and shared their resources to help each other. Well-regulated commune, an effort to keep local society safe by themselves were the driving force of overcoming epidemics. It was based on the learning to become a sage of Neo-Confucianism, which emphasized rules of reciprocity. Such projects are by nature voluntarism.

시골 양반 역병(疫病) 분투기 - 18세기 구상덕의 『승총명록』을 중심으로

In the 18th century, Joseon experienced a series of famine and plague. In short, it was a time of disaster. Gu Sang-deok, a local literati in Gyeongsang-do, detailed the horrors of epidemics and the lives of people overcoming it in his diary. When plague was prevalent people hid in temples or rang gongs and drums to chase away ghosts. Some prayed to spirits of epidemics while some drank beef soup to recover their strength. Those in better condition visited doctors and hospitals to take medicine. They couldn’t be helpless by an attack of epidemics. Gu Sang-deok was busy taking care of his family, protecting children and old people in his rural community from dying of plague and hunger. He went to rural Confucian school to discuss various issues of community, and delivered the public opinion(or dissatisfaction) to District magistrate.<BR>In the event of a plague, the state not only distributed prescriptions and medical books for the epidemic throughout the country, but also set up hut shelters at local authorities to isolate patients and hold rituals to heal the fear of the plague. Of course it was not enough until modern segregation systems and vaccines were developed people in Joseon had to live with plague and more epidemics. The members of the community provided shelters to each other and shared their resources to help each other. Well-regulated commune, an effort to keep local society safe by themselves were the driving force of overcoming epidemics. It was based on the learning to become a sage of Neo-Confucianism, which emphasized rules of reciprocity. Such projects are by nature voluntarism.

Acoustic impedance sensitivity approach for defining absorbing material layout using the wave based method

This study used the wave based method (WBM) to investigate the optimum layout of an absorbing material on a vibrating surface. The structural dynamic behavior was examined using the finite element model, and the acoustic response was computed using the WBM. The thickness of the absorbing material on each structural element was considered as the design variable. The sensitivity of the acoustic response to the acoustic impedance was derived using wave based system matrices, and a polynomial function was used to describe the impedance variation with respect to the thickness of the absorbing material. Thereafter, the final design sensitivity of a vibro-acoustic model was derived with respect to the thickness of the damping material using the chain rule. The design sensitivity results were found to be in good agreement with the results of the central finite difference methods. The design optimization of the absorbing material layout was applied to a vibrating surface using the design sensitivity results under various excitations, and the results of the optimization showed a layout similar to the active intensity plot of the vibrating surface. The proposed method can easily be applied to reduce the noise produced by a vibrating structure.

New higher-order weak lower inner epiderivatives and application to Karush–Kuhn–Tucker necessary optimality conditions in set-valued optimization

The purpose of the paper is to establish higher-order Karush–Kuhn–Tucker higher-order necessary optimality conditions for set-valued optimization where the derivatives of objective and constraint functions are separated. We first introduce concepts of higher-order weak lower inner epiderivatives for set-valued maps and discuss some useful properties about new epiderivatives, for instance, convexity, subadditivity and chain rule. With the help of the new concept and its properties, we establish higher-order Karush–Kuhn–Tucker necessary optimality conditions which is the classical type Karush–Kuhn–Tucker optimality conditions and improve and enhance some recent existing results in the literatures. Several examples are provided to illustrate our results. Finally, we provide weak and strong duality theorems in set-valued optimization.