Chain Rule Research Articles

An Information-Theoretic Proof of a Hypercontractive Inequality

The famous hypercontractive estimate discovered independently by Gross, Bonami and Beckner has had a great impact on combinatorics and theoretical computer science since it was first used in this setting in a seminal paper by Kahn, Kalai and Linial. The usual proofs of this inequality begin with two-point space, where some elementary calculus is used and then generalised immediately by introducing another dimension using submultiplicativity (Minkowski’s integral inequality). In this paper, we prove this inequality using information theory. We compare the entropy of a pair of correlated vectors in {0,1}n to their separate entropies, analysing them bit by bit (not as a figure of speech, but as the bits are revealed) using the chain rule of entropy.

Unified Generalizations of Hardy-Type Inequalities Through the Nabla Framework on Time Scales

This research investigates innovative extensions of Hardy-type inequalities through the use of nabla Hölder’s and nabla Jensen’s inequalities, combined with the nabla chain rule and the characteristics of convex and submultiplicative functions. We extend these inequalities within a cohesive framework that integrates elements of both continuous and discrete calculus. Furthermore, our study revisits specific integral inequalities from the existing literature, showcasing the wide-ranging relevance of our results.

Generalised Entropy Accumulation

Consider a sequential process in which each step outputs a system Ai\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_i$$\\end{document} and updates a side information register E. We prove that if this process satisfies a natural “non-signalling” condition between past outputs and future side information, the min-entropy of the outputs A1,⋯,An\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_1, \\dots , A_n$$\\end{document} conditioned on the side information E at the end of the process can be bounded from below by a sum of von Neumann entropies associated with the individual steps. This is a generalisation of the entropy accumulation theorem (EAT) (Dupuis et al. in Commun Math Phys 379: 867–913, 2020), which deals with a more restrictive model of side information: there, past side information cannot be updated in subsequent rounds, and newly generated side information has to satisfy a Markov condition. Due to its more general model of side-information, our generalised EAT can be applied more easily and to a broader range of cryptographic protocols. As examples, we give the first multi-round security proof for blind randomness expansion and a simplified analysis of the E91 QKD protocol. The proof of our generalised EAT relies on a new variant of Uhlmann’s theorem and new chain rules for the Rényi divergence and entropy, which might be of independent interest.

The Profound Nature of Derivatives and Integration: A Comprehensive Examination

This article delves into the fundamental concepts of derivatives and integration, two core principles of calculus that govern a wide range of mathematical and real-world applications. Derivatives, which measure the rate of change of a function, and integration, which calculates accumulation and areas under curves, are essential tools in fields such as physics, economics, engineering, and finance. The article offers a detailed exploration of their mathematical definitions, key properties—including linearity, the product and chain rules, and the Fundamental Theorem of Calculus—and the diverse techniques used in integration. The practical applications of these concepts are also examined, with special emphasis on their role in modeling dynamic systems, optimizing processes, and solving real-world problems such as motion analysis, financial modeling, and economic optimization. Through in-depth discussion and examples, this comprehensive article bridges the theoretical underpinnings of derivatives and integration with their vast and critical uses in various scientific and economic disciplines.

Misconceptions and Errors Among Grade 12 Students When Learning Differentiation Rules: A Case Study

t is an established fact that some grade 12 students learning differentiation in calculus are found struggling with the rule of differentiation and these rules include power rule, quotient rule, chain rule and product rule. Because of these, some students came up with some misconceptions which eventually resulted to the students having multiple errors when learning rules of differentiation. The reason associated with these common errors are not far from their failure to model some prerequisite knowledge in the laws of logarithm into their learning of the laws of differentiation. And this has resulted to the poor performance of some students in mathematics (calculus) since calculus carries about 40% in the overall grade in mathematics as a subject at grade 12 level. On this note, this paper presents an analysis of students’ errors and misconceptions in learning differentiation rule. The study was conducted among grade 12 students preparing for NSC examination in a high school in Limpopo province in South Africa. And the data was initially collected using 35 test scripts of grade 12 on the topic of differentiation and differentiation rule. A qualitative approach was considered, and the data collected was analysed, focusing on product rules, quotient rules and chain rules and the errors committed. The result of the study indicated that some grade 12 students make some errors and misconceptions in differentiation rules. And these were because of poor conceptual understanding, poor mathematics language understanding, and some other error. It was confirmed that the results of this study highlight the common mistakes and errors students make when learning differentiation rules, and these errors are; conceptual, systemic, language, and generalization errors.

MATHEMATICAL APPROACH OF THE BACKPROPAGATION METHOD FOR CONSTRUCTING ARTIFICIAL NEURAL NETWORKS

Backpropagation isthe core part of a neural network. This method is used to efficiently train a network using a chain rule that allows differentiation of complex functions. In other words, after each pass through the network, the backpropagation method performs a backward pass to adjust the model parameters, such as weights and biases. This article highlights the importance of using the backpropagation method from the point of view of mathematical formulas for neural networks. The importance of using the backpropaga-tion learning algorithm to calculate the gradient (gradient descent) and the need to use the activation function to minimize the loss function is mathematically described and calculated by formulas, and also proven by calculating the matrix products of vectors for each layer of parameters - weights and biases and applying complex differential equations.

Parasitic plants regulate C and N distribution among common mycorrhizal networks linking host and neighboring plants.

Common mycorrhizal networks (CMNs) can link multiple plants and distribute nutrients among them. However, how parasitic plants regulate the carbon and nutrient exchange between CMNs and the linked plants is unknown. Thus, we conducted a container experiment with two Trifolium pratense grown in two plastic cores and connected only by CMNs using a 25-μm nylon fabric in each container. Host T. pratense was parasitized or not parasitized by Cuscuta gronovii. CMNs were left intact or broken by rotating the cores with the host or neighboring T. pratense. The dual 15N and 13C labeling method was used to evaluate the N distributed by CMNs to the host and neighboring T. pratense and the recently fixed C from the host and neighboring T. pratense to CMNs. The results showed that CMNs distributed more 15N to unparasitized neighboring T. pratense than the parasitized host T. pratense. Moreover, the unparasitized neighboring T. pratense provides more recently fixed C to CMNs than the parasitized host T. pratense. These results revealed that the parasite regulated C and nutrient exchange between CMNs and the linked plants following the reciprocal rewards rule. Moreover, this study highlights the importance of parasitic plants in the regulation of mutualistic interactions in ecological webs.

Exploring solutions to the fractional Boiti–Leon–Manna–Pempinelli equation characterizing wave dynamics in incompressible fluids

In this study, the (3 + 1)-dimensional space-time fractional Boiti–Leon–Manna–Pempinelli (BLMP) equation is investigated utilizing the Kudryashov method (KM) and the modified Kudryashov method (MKM). These two efficient methods are implemented to acquire exact closed-form solutions to the considered fractional BLMP equation, and novel solitary wave solutions are constructed involving hyperbolic functions, and exponential rational functions with arbitrary constant parameters. With the aid of a unique wave transformation, the nonlinear fractional differential equation is transformed into an ordinary differential equation. In this unique wave transformation, the conformable fractional derivative is considered to which the chain rule is applied. The obtained solutions are indeed beneficial for analyzing the dynamic behavior of the fractional BLMP equation in describing fluid propagation. A few three-dimensional representations of the attained exact analytic solutions are offered by accounting for the proper selection of the appropriate parameters with the help of mathematical software. As a result, novel soliton solutions, such as the anti-kink type and singular kink type waves, are acquired from the attained solutions. This approach is relatively straightforward and efficient, making it possible to utilize it to obtain exact solutions for higher-order nonlinear partial differential equations.

Dynamics Behaviours of Kink Solitons in Conformable Kolmogorov–Petrovskii–Piskunov Equation

The current study introduces the generalised New Extended Direct Algebraic Method (gNEDAM) for producing and examining propagation of kink soliton solutions within the framework of the Conformable Kolmogorov–Petrovskii–Piskunov Equation (CKPPE), which entails conformable fractional derivatives into account. The primary justification around employing conformable derivatives in this study is their special ability to comply with the chain rule, allowing for in the solution of aimed nonlinear model. The CKPPE is a crucial model for a number of disciplines, such as mathematical biology, reaction-diffusion mechanisms, and population increase. CKPPE is transformed into a Nonlinear Ordinary Differential Equation by the proposed gNEDAM, and many kink soliton solutions are found by applying the series form solution. These kink soliton solutions shed light on propagation mechanisms within the framework of the CKPPE model. Furthermore, our research offers multiple graphical depictions that facilitate the examination and analysis of the propagation patterns of the identified kink soliton solutions. Through the integration of mathematical biology and reaction-diffusion principles, our research broadens our comprehension of intricate occurrences in various academic domains.

A new fractional derivative extending classical concepts: Theory and applications

In this paper, a novel general definition for the fractional derivative and fractional integral based on an undefined kernel function is introduced. For 0<α≤1, this definition aligns with classical interpretations and is applicable for calculating the derivative in an open negative interval I⊆[a,+∞),a∈R. Additionally, when α=1, the definition coincides with the classical derivative. Fundamental properties of the fractional integral and derivative, including the product rule, quotient rule, chain rule, Rolle’s theorem, and the mean value theorem, are derived. These properties are illustrated through various applications to demonstrate their applicability. Furthermore, some applications of solving fractional nonlinear systems of integro-differential equations using framelets are presented.

Results of existence and uniqueness for the Cauchy problem of semilinear heat equations on stratified Lie groups

The aim of this paper is to give existence and uniqueness results for solutions of the Cauchy problem for semilinear heat equations on stratified Lie groups G with the homogeneous dimension N. We consider the nonlinear function behaves like |u|α or |u|α−1u(α>1) and the initial data u0 belongs to the Sobolev spaces Lsp(G) for 1<p<∞ and 0<s<N/p. Since stratified Lie groups G include the Euclidean space Rn as an example, our results are an extension of the existence and uniqueness results obtained by F. Ribaud on Rn to G. It should be noted that our proof is very different from it given by Ribaud on Rn. We adopt the generalized fractional chain rule on G to obtain the estimate for the nonlinear term, which is very different from the paracomposition technique adopted by Ribaud on Rn. By using the generalized fractional chain rule on G, we can avoid the discussion of Fourier analysis on G and make the proof more simple.

Fast computation of function composition derivatives for flatness-based control of diffusion problems

The chain rule is a standard tool in differential calculus to find derivatives of composite functions. Faà di Bruno’s formula is a generalization of the chain rule and states a method to find high-order derivatives. In this contribution, we propose an algorithm based on Faà di Bruno’s formula and Bell polynomials (Bell in Ann Math 29:38–46, 1927; Parks and Krantz in A primer of real analytic functions, 2012) to compute the structure of derivatives of function compositions. The application of our method is showcased using trajectory planning for the heat equation (Laroche et al. in Int J Robust Nonlinear Control 10(8):629–643, 2000).

Stability criteria and calculus rules via conic contingent coderivatives in Banach spaces

This paper addresses the study of novel constructions of variational analysis and generalized differentiation that are appropriate for characterizing robust stability properties of constrained set-valued mappings/multifunctions between Banach spaces important in optimization theory and its applications. Our tools of generalized differentiation revolve around the newly introduced concept of ε-regular normal cone to sets and associated coderivative notions for set-valued mappings. Based on these constructions, we establish several characterizations of the central stability notion known as the relative Lipschitz-like property of set-valued mappings in infinite dimensions. Applying a new version of the constrained extremal principle of variational analysis, we develop comprehensive sum and chain rules for our major constructions of conic contingent coderivatives for multifunctions between appropriate classes of Banach spaces.

On Some New Dynamic Hilbert-Type Inequalities Across Time Scales

In this article, we present some novel dynamic Hilbert-type inequalities within the framework of time scales T. We achieve this by utilizing Hölder’s inequality, the chain rule, and the mean inequality. As specific instances of our findings (when T=N and T=R), we obtain the discrete and continuous analogues of previously established inequalities. Additionally, we derive other inequalities for different time scales, such as T=qN0 for q&gt;1, which, to the best of the authors’ knowledge, is a largely novel conclusion.

Heat equation on the hypergraph containing vertices with given data

This paper is concerned with a multivalued ordinary differential equation governed by the hypergraph Laplacian, which describes the diffusion of “heat” or “particles” on the vertices of hypergraph. We consider the case where the heat on several vertices is manipulated internally by the observer, namely, are fixed by some given functions. This situation can be reduced to a nonlinear evolution equation associated with a time-dependent subdifferential operator, whose solvability has been investigated in numerous previous researches. In this paper, we give an alternative proof of the solvability in order to avoid some complicated calculations arising from the chain rule for the time-dependent subdifferential. As for results which cannot be assured by the known abstract theory, we also discuss the continuous dependence of solution on the given data and the time-global behavior of solution.

The “Stolen Revolution” and Hope: ‘Progressive’ and ‘Conservative’ Forces of the Contemporary Political Context of Kyrgyzstan

Abstract This article examines the role of the progressive forces in recent revolutions in Kyrgyzstan, and asks why their movements have not led to real political change in this Central Asian country. In Kyrgyzstan, the progressive forces are identified with such groups as young people, the educated middle class, lawyers, and journalists. Their discourses, such as demands for fair elections and the rule of law, have been central to the three revolutions that have taken place in Kyrgyzstan since 2005, as has their concept of Umut, or hope. The central thesis of this article is that, although these forces may have led recent revolutionary movements, they have proved unable to establish political dominance after the completion of revolutionary processes. Instead, after each revolutionary attempt, the rules of “local politics” have been consistently reasserted in Kyrgyzstan. These are rules based on the power of informal social and political networks governed by rules of reciprocity and the pursuit of self-interest. In spite of this recurring outcome, the progressive forces retain their commitment to hope, and it is the prism of hope which this paper uses to make sense of revolution’s failure. In this paper, a contribution to the anthropology of revolution, that idea of hope is deconstructed: this permits a greater focus on how and why the promises of revolution have, so far, been betrayed and unfulfilled in contemporary Kyrgyzstan, where the idea of hope plays a fundamental representational and political role.

Discrete energy balance equation via a symplectic second-order method for two-phase flow in porous media

We propose and analyze a second-order partitioned time-stepping method for a two-phase flow problem in porous media. The algorithm is a refactorization of Cauchy's one-leg θ-method: the implicit backward Euler method on [tn,tn+θ], and a linear extrapolation on [tn+θ,tn+1]. In the backward Euler step, the decoupled equations are solved iteratively, with the iterations converging linearly. In the absence of the chain rule for time-discrete setting, we approximate the change in the free energy by the product of a second-order accurate discrete gradient (chemical potential) and the one-step increment of the state variables. Similar to the continuous case, we also prove a discrete Helmholtz free energy balance equation, without numerical dissipation. In the numerical tests we compare this symplectic implicit midpoint method (θ=1/2) with the classic backward Euler method, and two implicit-explicit time-lagging schemes. The midpoint method outperforms the other schemes in terms of rates of convergence, long-time behavior and energy approximation, for both small and large values of the time step.

Estimation of complete mutual information exploiting nonlinear magnitude-phase dependence: Application to spatial FNC for complex-valued fMRI data

BackgroundReal-valued mutual information (MI) has been used in spatial functional network connectivity (FNC) to measure high-order and nonlinear dependence between spatial maps extracted from magnitude-only functional magnetic resonance imaging (fMRI). However, real-valued MI cannot fully capture the group differences in spatial FNC from complex-valued fMRI data with magnitude and phase dependence. MethodsWe propose a complete complex-valued MI method according to the chain rule of MI. We fully exploit the dependence among magnitudes and phases of two complex-valued signals using second and fourth-order joint entropies, and propose to use a Gaussian copula transformation with a lower bound property to avoid inaccurate estimation of joint probability density function when computing the joint entropies. ResultsThe proposed method achieves more accurate MI estimates than the two histogram-based (normal and symbolic approaches) and kernel density estimation methods for simulated signals, and enhances group differences in spatial functional network connectivity for experimental complex-valued fMRI data. Comparison with existing methodsCompared with the simplified complex-valued MI and real-valued MI, the proposed method yields higher MI estimation accuracy, leading to 17.4 % and 145.5 % wider MI ranges, and more significant connectivity differences between healthy controls and schizophrenia patients. A unique connection between executive control network (EC) and right frontal parietal areas, and three additional connections mainly related to EC are detected than the simplified complex-valued MI. ConclusionsWith capability in quantifying MI fully and accurately, the proposed complex-valued MI is promising in providing qualified FNC biomarkers for identifying mental disorders such as schizophrenia.

Chemical Potentials and the One-Electron Hamiltonian of the Second-Order Perturbation Theory from the Functional Derivative Approach.

We develop a functional derivative approach to calculate the chemical potentials of second-order perturbation theory (MP2). In the functional derivative approach, the correlation part of the MP2 chemical potential, which is the derivative of the MP2 correlation energy with respect to the occupation number of frontier orbitals, is obtained from the chain rule via the noninteracting Green's function. First, the MP2 correlation energy is expressed in terms of the noninteracting Green's function, and its functional derivative to the noninteracting Green's function is the second-order self-energy. Then, the derivative of the noninteracting Green's function to the occupation number is obtained by including the orbital relaxation effect. We show that the MP2 chemical potentials obtained from the functional derivative approach agree with that obtained from the finite difference approach. The one-electron Hamiltonian, defined as the derivative of the MP2 energy with respect to the one particle density matrix, is also derived using the functional derivative approach, which can be used in the self-consistent calculations of MP2 and double-hybrid density functionals. The developed functional derivative approach is promising for calculating the chemical potentials and the one-electron Hamiltonian of approximate functionals and many-body perturbation approaches dependent explicitly on the noninteracting Green's function.

Effective hydrogen supply chain management framework considering nonlinear multi-stage process uncertainties

The large amount of carbon emitted due to industrial development is posing serious climate change. Under these circumstances, hydrogen energy is being used as a major energy source and is attracting attention as an alternative energy to fossil fuels, characterized by high carbon emissions. While the use of hydrogen has the obvious benefit of a low carbon footprint, the hydrogen production process possesses a high carbon footprint difference depending on the source of utilized electricity. This study considers several situations arising in an electrolysis-based hydrogen supply chain network (HSCN). A multi-stage stochastic programming (SP) is modeled to consider the uncertainty of demand and transportation capacity generated in HSCN. Moreover, a typical stochastic model has a chain rule between the decision variables resulting from nonlinearity and uncertainty, and these features make it difficult to derive an effective solution. Thus, in this study, a new methodology, Weighted Scenario Sample Average Approximation (WSSAA), is proposed to derive an effective solution for the multistage SP model considering various scenarios under the hydrogen supply chain. While a general stochastic programming approach fails to generate feasible solutions with volatile HSCN uncertainties, the proposed WSSAA framework provides feasible decisions considering multiple scenarios. The effectiveness of the proposed model applied with WSSAA is demonstrated through comparative experiments over the models used in existing studies.