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Exact solutions of nonlinear differential equations are of great importance to the theory and practice of complex systems. The main point of this review article is to discuss a specific methodology for obtaining such exact solutions. The methodology is called the SEsM, or the Simple Equations Method. The article begins with a short overview of the literature connected to the methodology for obtaining exact solutions of nonlinear differential equations. This overview includes research on nonlinear waves, research on the methodology of the Inverse Scattering Transform method, and the method of Hirota, as well as some of the nonlinear equations studied by these methods. The overview continues with articles devoted to the phenomena described by the exact solutions of the nonlinear differential equations and articles about mathematical results connected to the methodology for obtaining such exact solutions. Several articles devoted to the numerical study of nonlinear waves are mentioned. Then, the approach to the SEsM is described starting from the Hopf–Cole transformation, the research of Kudryashov on the Method of the Simplest Equation, the approach to the Modified Method of the Simplest Equation, and the development of this methodology towards the SEsM. The description of the algorithm of the SEsM begins with the transformations that convert the nonlinearity of the solved complicated equation into a treatable kind of nonlinearity. Next, we discuss the use of composite functions in the steps of the algorithms. Special attention is given to the role of the simple equation in the SEsM. The connection of the methodology with other methods for obtaining exact multisoliton solutions of nonlinear differential equations is discussed. These methods are the Inverse Scattering Transform method and the Hirota method. Numerous examples of the application of the SEsM for obtaining exact solutions of nonlinear differential equations are demonstrated. One of the examples is connected to the exact solution of an equation that occurs in the SIR model of epidemic spreading. The solution of this equation can be used for modeling epidemic waves, for example, COVID-19 epidemic waves. Other examples of the application of the SEsM methodology are connected to the use of the differential equation of Bernoulli and Riccati as simple equations for obtaining exact solutions of more complicated nonlinear differential equations. The SEsM leads to a definition of a specific special function through a simple equation containing polynomial nonlinearities. The special function contains specific cases of numerous well-known functions such as the trigonometric and hyperbolic functions and the elliptic functions of Jacobi, Weierstrass, etc. Among the examples are the solutions of the differential equations of Fisher, equation of Burgers–Huxley, generalized equation of Camassa–Holm, generalized equation of Swift–Hohenberg, generalized Rayleigh equation, etc. Finally, we discuss the connection between the SEsM and the other methods for obtaining exact solutions of nonintegrable nonlinear differential equations. We present a conjecture about the relationship of the SEsM with these methods.

Based on thermal-entransy theory, the multi-objective constructal design of quadrilateral heat generation body (HGB) with similar shapes of leaves is studied further. The relationship between the aspect ratio of quadrilateral HGB and average temperature difference based on thermal-entransy dissipation is compared with that between the aspect ratio of quadrilateral HGB and the maximum temperature difference (MTD). The relationship between a composite function, consisting of linear weighting sum of the average temperature difference and MTD, and aspect ratio is obtained, and the optimal aspect ratios under minimum composite function with different weighting coefficients are obtained. Using the NSGA-II algorithm, the Pareto frontier containing a series of compromise results of average temperature difference and MTD is obtained, and optimization results are compared using the deviation index. There is no aspect ratio to make both MTD and average temperature difference reach the minimum, and the optimal aspect ratio under the minimum MTD is smaller than that under the minimum average temperature difference. The optimal aspect ratio is obtained by making the composite function reach the minimum, and the optimal aspect ratios obtained by minimizing the composite function with different weighting coefficients are different. Compared with the construct of the initial design, the value of the composite function with optimal construct decreases by 1.9%, and the aspect ratio of the quadrilateral HGB decreases by 9.1%. The average temperature difference with the optimal construct increases by 2.1%, and the MTD with the optimal construct decreases by 5.6%. The deviation index under multi-objective optimization is smaller than that under single-objective optimization, and the obtained construct has better comprehensive thermal conductivity. Compared with TOPSIS and LINMAP decision-making methods, the average temperature difference with composite function optimization increases by 0.55% and 0.62% respectively, but the MTD with composite function optimization decreases by 0.84% and 0.96%.

It has been confirmed that a larger stability domain has more advantages for system operation. In this paper, a novel one-cycle control (OCC) embedded with a composite function is proposed. Its control principle is based on maintaining the symmetry of the volt-second value for the inductor in each cycle. Then, selecting the reference voltage as the study variable, the stability boundaries and identify stable parameter domains are studied by establishing a state-space average model. The results demonstrate that, compared with the conventional OCC, the proposed OCC with composite function u embedded achieves an expanded stable parameter domain, effectively delaying the occurrence of instability phenomena from uref=2uin to uref=3uin. Both the simulation and experimental results conclusively validate the theoretical analysis, confirming the effectiveness and superiority of the proposed control strategy.

We prove two main results on Denjoy–Carleman classes: (1) a composite function theorem which asserts that a function $f(x)$ in a quasianalytic Denjoy–Carleman class ${\\mathcal{Q}}_{M}$ , which is formally composite with a generically submersive mapping $y=\\unicode[STIX]{x1D711}(x)$ of class ${\\mathcal{Q}}_{M}$ , at a single given point in the source (or in the target) of $\\unicode[STIX]{x1D711}$ can be written locally as $f=g\\circ \\unicode[STIX]{x1D711}$ , where $g(y)$ belongs to a shifted Denjoy–Carleman class ${\\mathcal{Q}}_{M^{(p)}}$ ; (2) a statement on a similar loss of regularity for functions definable in the $o$ -minimal structure given by expansion of the real field by restricted functions of quasianalytic class ${\\mathcal{Q}}_{M}$ . Both results depend on an estimate for the regularity of a ${\\mathcal{C}}^{\\infty }$ solution $g$ of the equation $f=g\\circ \\unicode[STIX]{x1D711}$ , with $f$ and $\\unicode[STIX]{x1D711}$ as above. The composite function result depends also on a quasianalytic continuation theorem, which shows that the formal assumption at a given point in (1) propagates to a formal composition condition at every point in a neighbourhood.

This paper develops a dynamics-based nonsingular interval model and proposes a first-order composite function interval perturbation method (FCFIPM) for luffing angular response field analysis of the dual automobile cranes system (DACS) with narrowly bounded uncertainty. By using the nonsingular interval model to describe a structure parameter with bounded uncertainty, the reasonable lower and upper bounds can be obtained, which is quite different from the traditional interval model with approximate bounds only from a large number of samples. Firstly, for the DACS with deterministic information, the inverse kinematics is analyzed, and the dynamic model of the DACS is established based on the virtual work principle and the inverse kinematics. Secondly, considering the nonsingularity of the dynamic response curves, a dynamics-based nonsingular interval model is introduced. Based on the nonsingular interval model, the interval luffing angular response vector equilibrium equation of the DACS is established. Thirdly, a first-order composite function interval perturbation method is proposed. In the FCFIPM, the composite function vectors are expanded by using the first-order Taylor series expansion, based on the differential property of composite function and monotonic analysis technique, the lower and upper bounds of the interval luffing angular response vector of the crane 1 and crane 2 of the DACS are determined. The first case is to investigate the deterministic kinematics and dynamics of the DACS with a given trajectory. The second case is provided to illustrate the detailed implementation process of constructing a dynamics-based nonsingular interval model. Finally, some numerical examples are given to verify the feasibility and efficiency of the FCFIPM for solving the luffing angular response field problem with narrowly interval parameters.

The recurrence relation for the coefficients of higher order Bell polynomials, i.e. of the Bell polynomials relevant to 𝑛thderivative of a multiple composite function, is proved. Therefore, starting from this recurrence relation and by using the computer algebra program Mathematica©, some tables for complete higher order Bell polynomials and the relevant numbers are derived.

In certain critical infrastructures, correlations between cyber and physical components can be exploited to launch strategic attacks, so that disruptions to one component may affect others and possibly the entire infrastructure. Such correlations must be explicitly taken into account in ensuring the survival of the infrastructure. For large discrete infrastructures characterized by the number of cyber and physical components, we characterize the cyber-physical interactions at two levels: (i) the cyber-physical failure correlation function specifies the conditional survival probability of the cyber sub-infrastructure given that of the physical sub-infrastructure (both specified by their marginal probabilities), and (ii) individual survival probabilities of both sub-infrastructures are characterized by first-order differential conditions expressed in terms of their multiplier functions. We formulate an abstract problem of ensuring the survival probability of a cyber-physical infrastructure with discrete components as a game between the provider and attacker, whose utility functions are composed of infrastructure survival probability terms and cost terms, both expressed in terms of the number of components attacked and reinforced. We derive Nash equilibrium conditions and sensitivity functions that highlight the dependence of infrastructure survival probability on cost terms, correlation functions, multiplier functions, and sub-infrastructure survival probabilities. We apply these analytical results to characterize the defense postures of simplified models of metro systems, cloud computing infrastructures, and smart power grids.

To estimate the derivatives of a deterministic signal arriving in real time, a state observer is introduced, built as a replica of the virtual canonical model with an unknown bounded input. The results of a comparative analysis of observers with linear and piecewise linear corrective actions are presented, which are confirmed by the results of numerical modeling for piecewise differentiable composite function. Its derivatives have jump discontinuity points at the moments when the form of the function changes. It is shown that at these moments the observer variables with linear corrective actions with high gains have large surges, which increase by an order of magnitude with an increase in the order of the estimated derivative. Observers with bounded piecewise linear corrective actions do not have these problems and are recommended for use in practical applications.

We recover a recurrence relation for representing in an easy form the coefficients of the Bell polynomials, which are known in literature as the partial Bell polynomials. Several applications in the framework of classical calculus are derived, avoiding the use of operational techniques. Furthermore, we generalize this result to the coefficients of the second-order Bell polynomials, i.e. of the Bell polynomials relevant to nth derivative of a composite function of the type f(g(h(t))). The second-order Bell polynomials and the relevant Bell numbers are introduced. Further extension of the nth derivative of M-nested functions is also touched on.

The composition of functions is a key concept in advanced mathematics, with transversal applications in disciplines such as physics, economics, and biology. This study details a pedagogical experience conducted in an Algebra course for engineering, in which Python was integrated as a tool to model data and solve problems related to composite functions. The methodology included collaborative and contextualized activities that linked the theory of function composition to simulated real-world situations. The results indicate a significant improvement in the understanding of the concept of function composition, an increase in student motivation, and a positive perception of the relevance of applied mathematics.