Mathematical Examples Research Articles

An improved sufficient dimension reduction-based Kriging modeling method for high-dimensional evaluation-expensive problems

Kriging is a powerful surrogate modeling method for the analysis and optimization of computationally expensive problems. However, the efficient construction of high-precision models for high-dimensional expensive problems is an important challenge for Kriging method and has received much attention recently. To address this challenge, we propose an improved sufficient dimension reduction-based Kriging modeling method (KISDR) for high-dimensional evaluation-expensive problems. First, the martingale difference divergence is introduced into the sufficient dimension reduction to obtain a more accurate and stable estimate of the projection matrix, which can project the high-dimensional inputs into a low-dimensional latent space while maintaining sufficient information for response prediction. Second, we propose to utilize the ladle estimator to identify the dimension of latent space. The ladle estimator combines both eigenvalues and eigenvectors of the matrix and can identify the latent dimensionality more precisely. After that, a new Kriging correlation function is constructed by integrating the information of dimension reduction into the correlation structure, which significantly reduces the number of hyperparameters to be estimated. Finally, we devise a local optimization approach to fine-tune the Kriging hyperparameters to further improve the modeling accuracy. In this study, six mathematical examples and three engineering examples with dimensions varying from 30-D to 100-D are employed for performance analysis and comparison. The results indicate that the proposed KISDR can precisely identify and exploit the low-dimensional linear structure in the data to improve modeling accuracy and efficiency for high-dimensional evaluation-expensive problems.

The computation of the air conditioning loads life loss and its application to demand response

AbstractAir conditioning loads (ACLs) represent an increasing proportion of power system loads, offering significant potential for optimised scheduling and active participation in demand response (DR) programs. While many studies have focused on ON/OFF control schemes that satisfy system requirements, few have addressed quantifying the life loss of ACLs from the user perspective. To address this gap, a quantitative model of ACL life loss is established and an optimal scheduling model is developed for ACLs participating in DR that incorporates the cost of life loss. The relationship between life loss and refrigeration power is a complex non‐linear high‐order fractional function that cannot be solved by commercial solvers. Therefore, a bi‐objective multi‐weight optimisation algorithm is proposed with a complex non‐linear fraction based on the Dinkelbach algorithm and its feasibility through mathematical examples is verified. Finally, a numerical example based on the IEEE 39‐bus test system is provided to demonstrate the feasibility of the model and the effectiveness of the proposed solution method.

A possibility-based solution framework for interval uncertainty-based design optimization

A possibility-based solution framework called the equivalent deterministic method (EDM) is proposed to solve optimization problems with interval uncertainty, which provides an efficient tool for interval uncertainty-based design optimization. First, an interval uncertainty-based design optimization model is established based on the possibility theory and current or reference design from experienced designers. Second, the equivalent deterministic method is proposed to transform the interval optimization problem into a series of interval analyses and deterministic optimizations that are alternately solved, which decouples the double-loop nested optimization into an iterative solving process. The novelties and advantages of the proposed method lie in two aspects. First, the shift scalar is deducted in the objective function and constraints in the possibility degree space, rather than the uncertain design variables and parameters, through which the violated constraints rapidly move toward the flexible region and the design points are optimized progressively. Second, interval analysis methods that calculate the interval responses and possibility, rather than the maximum and minimum points, can be integrated into the proposed method, which enhances the application scope for the existing interval optimization methods. Finally, a mathematical example and two engineering applications are utilized to demonstrate the performance of accuracy and efficiency.

Validation metric of multi-output model based on energy distance

Model validation metrics have been developed to provide a quantitative measure that characterizes the agreement between predictions and observations. The validation metric of multi-output model has the problem of solving the joint probability density fuction (PDF). The traditional method is to reduce the dimension, but this may lead to the loss of some information. To avoid solving the joint PDF, this paper gives a validation metric method of multi-output model based on energy distance. At the same time, it avoids the problem of information loss caused by dimensionality reduction, and greatly reduces the difficulty of calculation. Then using mathematical examples, a set of numerical studies are designed to verify the correctness and stability of the method. Finally, we apply the metric to the Sandia validation challenge problem.

A modular Bayesian approach to model calibration of low‐fidelity simulation models

AbstractLow‐fidelity computational models have been widely used for the computation of complex engineered systems to greatly save the computational time while sacrificing the computational precision. Model calibration can be implemented for low‐fidelity simulation models to remedy the errors of simulation results. This article proposes a modular Bayesian updating framework that considers epistemic uncertainty to achieve model calibration of low‐fidelity simulation models. The proposed framework mainly contains two steps: (1) A model calibration framework based on Bayesian theory is proposed that can simultaneously quantify the model form uncertainty, model parameter uncertainty, and the experimental measurement uncertainty, (2) the proposed method updates a low‐fidelity surrogate model via a Metropolis–Hastings (M–H) algorithm by using high‐fidelity experimental data. The proposed method greatly improves the prediction accuracy of the simulation model and enhances the computational efficiency. A mathematical example is used to testify the accuracy of the developed method, while the aerodynamics simulation of the NACA0012 airfoil is leveraged to demonstrate the engineering application. The results show that the posterior prediction mean is in better agreement with the experimental mean than the prior prediction mean, suggesting that the modular Bayesian updating method improves the prediction accuracy of low‐fidelity simulation models.

Gröbner Basis Approach for Solving Fuzzy Complex System of Linear Equations

The foundation of Gröbner’s work is a potent approach for separating polynomial systems of equations. In this study, we solve complex fuzzy systems of linear equations utilizing Gröbner’s basic technique. First, a system of explicit polynomial equations is generated by applying fuzzy computations and fuzzy complex computations to the fundamental system. Taking into account the lexicographical order, Gröbner’s basis is then determined for the ideal given by a definite equivalent polynomial system. The Gröbnerbase that was obtained has a high triangular structure. Thus, the system solution may be readily implemented by proceeding. Hence, all of the system’s answers are readily attainable. Next, a criterion is established to identify whether or not a solution exists for the primary system. The advantage of the proposed approach is that it acquires all of the problem’s solutions simultaneously. Using this technique, a mathematical example and an application issue based on an electrical circuit are resolved. Comparing the generated findings to the known outcomes reveals a high level of congruence.

Merchant Energy Storage Investment Analysis Considering Multi-Energy Integration

In this paper, a two-stage model of an integrated energy demand response is proposed, and the quantitative relationship between the two main concerns of investors, i.e., investment return and investment cycle and demand response, is verified by the experimental data. Energy storage technology is a key means through which to deal with the instability of modern energy sources. One of the key development paths in the electricity market is the development by energy merchants of energy storage power plants in the distribution network to engage in a grid demand response. This research proposes a two-stage energy storage configuration approach for a cold-heat-power multi-energy complementary multi-microgrid system. Considering the future bulk connections of distributed power generation, the two most critical points of energy storage station construction are the power generation equipment and specific scenarios for serving the community, as well as the purchase and sale price of electricity for serving the community microgrid (which directly affects the investment revenue). Therefore, this paper focuses on analyzing the different impacts caused by these two issues; namely, the two most important concerns for the construction of energy storage configurations. First, the basic model enabling wholesale electricity traders to construct energy storage power plants is presented. Second, for a multi-microgrid system with a complementary cold-heat-power multi-energy scenario, a two-stage optimum allocation model is constructed, whereby the upper model calculates the energy storage allocation problem and the lower model calculates the optimal dispatch problem. The lower model’s dispatch computation validates the upper configuration model’s reasonableness. Finally, the two-layer model is converted to a single-layer model by the KKT condition, and the nonlinear problem is converted to a linear problem with the big-M method. The validity is proved via mathematical examples, and it is demonstrated that the planned energy storage plants by merchants may accomplish resource savings and mutual advantages for both users and wholesale power traders.

Accurate and real-time prediction of umbilical component layout optimization based on convolutional neural network

Umbilicals are an essential piece of control equipment in an underwater production system and are composed of cables, optical cables, steel pipes, and other components. Different component layouts result in umbilicals exhibiting different geometric characteristics and bearing capacities. Considering the geometric characteristics and mechanical properties of the umbilical components, a quantifiable optimization mathematical model for the layout of umbilical components is established in this work. The Differential Evolution-Generalized Lagrange Multiplier (DE-GLM) method is proposed to solve the mathematical model. We demonstrate, using mathematical examples, that the DE-GLM method can avoid the uncertainty of artificial experience layout design. Nevertheless, the increasing types, numbers, and layers of components lead to the time cost of DE-GLM calculation to be higher. To avoid the excessive consumption of time and artificial costs caused by traditional layout algorithms, a novel Efficient Pyramid-Scale Aggregation-U-Net (EPSA-U-Net) convolutional neural network is proposed to achieve real-time accurate prediction of umbilical component layout results. After discussing, in detail, the effects of hyperparameters such as optimization algorithms, learning rates, and loss functions, on the performance of the network, we demonstrate that the method not only automatically obtains an umbilical component layout design form that satisfies the performance requirements in real-time but also achieves an accuracy rate of 93.28% compared with other convolutional neural networks. As such, it could replace complex artificial operations, significantly improving the efficiency of umbilical component layout design within a negligible time frame. Simultaneously, it also provides an intelligent, reliable, and efficient reference for the layout design of other similar structures and equipment for marine engineering.

Entropy estimation via uniformization

Entropy estimation is of practical importance in information theory and statistical science. Many existing entropy estimators suffer from fast growing estimation bias with respect to dimensionality, rendering them unsuitable for high-dimensional problems. In this work we propose a transform-based method for high-dimensional entropy estimation, which consists of the following two main ingredients. Firstly, we provide a modified k-nearest neighbors (k-NN) entropy estimator that can reduce estimation bias for samples closely resembling a uniform distribution. Second we design a normalizing flow based mapping that pushes samples toward the uniform distribution, and the relation between the entropy of the original samples and the transformed ones is also derived. As a result the entropy of a given set of samples is estimated by first transforming them toward the uniform distribution and then applying the proposed estimator to the transformed samples. The performance of the proposed method is compared against several existing entropy estimators, with both mathematical examples and real-world applications.

An interval uncertainty optimization algorithm based on radial basis function network differentiation

A new interval uncertainty optimization algorithm is proposed to replace two-layer nested optimization, owing to the low efficiency of the latter. The radial basis function network is established to obtain the first-order differential, which is difficult to achieve in practical engineering problems. The results obtained by this network differential method are verified by a mathematical example. The network differential method is combined with the interval perturbation method to compute the bounds of uncertain objective functions and constraints, and the subinterval method is introduced to address the large level of uncertainty. The example of a compression spring shows the feasibility of this interval analysis method. The interval uncertain optimization problem is transformed into a deterministic one through the interval order relationship and probability model, and solved using the genetic algorithm or non-dominated sorting genetic algorithm-II. A numerical example and electromagnetic buffer model demonstrate the accuracy, efficiency and practicability of this new method.

Indeks Harmonik dan Indeks Gutman Graf Nilradikal pada Gelanggang Komutatif dengan Satuan

Graph theory is a topic that is still an important subject to discuss. This is because until now graph theory has many practical applications in various disciplines, for example in biology, computer science, economics, informatics engineering, linguistics, mathematics, health, and social sciences. This study discusses the Gutman index of nilradical graphs in the commutative ring with unity.A nilradical graph whose vertices are non-zero nilpotent elements, when the domain is a commutative ring with units, it forms a complete graph only if the commutative ring with units we use is limited to a positive integer modulo n (Z_n). Where n is the square of the prime number p which is less than equal to 3. It is known that the general pattern of harmonic indices and Gutman indices of nilradical graphs in the commutative ring with unity are H(N(Z_(n=p^2 ) ))=((p-2)^2+(p -2))/2(p-2) and Gut(N(Z_(n=p^2 ) ))=(1/2 ((p-2)^2+(p-2))) (p- 2)^2 respectively. In its application, this general form can be used as a numerical parameter of a graph in chemical graph theory, molecular topology, and mathematical chemistry.

Hierarchy of Ecological Homeostasis as A Principle of Systemology

The broad concept of “homeostasis” in relation not only to the organism, but also to the population, ecosystem and socio–ecological and economic systems (in the context of sustainable development) is discussed. Sustainability is a major quality of systems. Discusses the mathematical formalization and examples of various definitions of sustainability: the sustainability of Lyapunov, Lagrange (steadiness), Poisson (periodic stability), Svirezhev (hierarchical), Holling (elasticity), Fleishman (vitality), Gnedenko (reliability).

A new efficient semi-numerical method with a convergence control parameter for Lane–Emden–Fowler boundary value problem

We propose a new efficient semi-numerical method for solving the Lane–Emden–Fowler type equations that arise in various scientific applications. To avoid singular behavior at t=0, we convert our problem into an integral equation before establishing the recursive procedure for resolving the problem. Unlike the existing methods, the current approach does not require the calculation of the transcendental equations for the unknown coefficients. The existence of a unique solution to the problem is discussed in detail. A convergence parameter ħ is introduced to control the convergence and the rate of convergence of the method. Moreover, the convergence analysis of the current method is discussed. Several mathematical and physical examples are selected from the open literature whose exact solutions are unknown to check the new scheme’s accuracy and applicability. The present method gives a convergent series solution, whereas the ADM (Singh and Kumar, 2014) fails to provide a convergent solution. The new approach resolves the issue of the existing method, which allows only capturing the solutions for the smaller domain of the solution.

Some new parameterized Newton-type inequalities for differentiable functions via fractional integrals

The main goal of the current study is to establish some new parameterized Newton-type inequalities for differentiable convex functions in the setting of fractional calculus. For this, first we prove a parameterized integral identity involving fractional integrals and then prove Newton-type inequalities for differentiable convex functions. It is also shown that the newly established parameterized inequalities are refinements of the already proved inequalities in the literature for different choices of parameters. Finally, we discuss a mathematical example along with a plot to show the validity of the newly established inequalities.

A Trivial Source of Wonder: Some Mathematical Examples in Plato’s Dialogues

Abstract The purpose of this paper is to reassess some mathematical examples in Plato’s dialogues which at a first glance may appear to be nothing more than trivial puzzles. In order to provide the necessary background for this analysis, I shall begin by sketching a brief overview of Plato’s mathematical passages and discuss the criteria for aptly selecting them. Second, I shall explain what I mean by ‘mathematical examples,’ and reflect on their function in light of the discussion on παραδείγματα outlined in the Sophist and the Statesman. Against this framework, I shall move to a close examination of specific examples drawn from the Theaetetus (154c–55d), the Republic (523c–24d), and the Phaedo (96d–97b, 100e–101d). By placing these examples in the broader context of pre-Euclidean mathematics, I shall show that their mathematical content is often less elementary than might appear at first sight. Moreover, by placing emphasis on the specific philosophical concerns that motivated their introduction, I shall argue that the examples are not merely nonsensical jokes. Even if their illustrative purpose is not always immediately clear, and even if they can sound playful and bizarre, they in fact fulfil an important function: by virtue of their power to generate wonder or confusion, they serve as exercises and act as a trigger with respect to the difficult philosophical issues they are intended to clarify.

FRACTIONAL HERMITE–HADAMARD INEQUALITY, SIMPSON’S AND OSTROWSKI’S TYPE INEQUALITIES FOR CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF FUNCTIONS

We consider the convexity with respect to a pair of functions and establish a Hermite–Hadamard type inequality for Riemann–Liouville fractional integrals. Moreover, we derive some new Simpson’s and Ostrowski’s type inequalities for differentiable convex mapping with respect to a pair of functions. We also show that the newly established inequalities are the extension of some existing inequalities. Finally, we consider some mathematical examples and graphs to show the validity of the newly established inequalities.

A green realistic inventory model with preservation technology for deteriorating items under carbon emission

Greening efforts are the steps taken to control the hazardous activities by different factories and to take environment-friendly steps which cause the least harm to nature. People buy products more which are green in nature. In this study, an inventory model is articulated in which demand is reliant on selling price and greening efforts. Here, all unit quantity discount is assumed. The purchasing cost is reliant on order size besides holding cost is reliant on time. By using greening efforts, a retailer can earn more profit. The aim of this study is to find a profit function that is taken to be dependent on greening efforts, cycle time, and selling price. The main aim of this paper is to earn more profit by using preservation technology and by allowing a carbon tax. The deterioration rate cannot be ignored so preservation technology is used here to overcome the effect of worsening. To find the optimal answer, an algorithm is developed. Concavity is shown through 3D graphs. Mathematical examples and sensitivity analysis is also done to validate the results.

Hybrid time‐dependent reliability analysis under a mixture of random and interval uncertainties

AbstractA new time‐dependent reliability analysis model with a mixture of random variables and interval variables is proposed in this paper. First, samples of random variables are generated according to their probability density functions. Then, the samples are substituted into the performance function to obtain the interval processes corresponding to these samples. The reliability index of each interval process is computed to finally obtain the upper and lower bounds of failure probability. The adaptive Kriging surrogate model based on the global addition strategy is used in the calculation process to increase the computational efficiency. The accuracy of the computational strategy is demonstrated by a mathematical example and three engineering practical cases. The proposed new hybrid time‐dependent reliability model is compared to the conventional hybrid reliability model. The comparison results show that the proposed model is more convenient to implement than the conventional reliability model. It can accurately evaluate the reliability of the structure and can be applied to real‐world engineering practice thereby providing assistance and guidance for hybrid time‐dependent reliability analysis of structures.

Controller design for discrete-time nonlinear feed-forward cascades with application to bacterial quorum sensing

This paper investigates the semi-global practical asymptotic stability (SPA stability) of a class of nonlinear feed-forward cascade systems. In particular, using general theories presented on the stabilization of sampled-data systems, a SPA stabilizing controller has been designed and the essential conditions for the semi-global asymptotic stability of this class of nonlinear systems have been presented. In doing so, using the approximated discrete-time model of a general form of feed-forward cascade systems in conjugation with the idea of cross-term constructed Lyapunov function, sampled data stabilizing conditions for the discretized system have been investigated and subsequently, the proper SPA stabilizing controller has been derived. To illustrate the effectiveness of the proposed scheme, the designed controller is applied on three examples. First, the framework has been applied to a nonlinear mathematical example and then to the well-known ball and beam system. In the end, the Quorum Sensing mechanism has been investigated as a novel application that extends the use of this set of frameworks to biological systems.

Online processing while monitoring worked-out examples with embedded errors: defining university student profiles

This study explores the process itself of comprehension monitoring of worked-out examples in mathematics. A ‘reversal error’ was embedded in a worked-out example of algebraic nature. Ninety-four engineers in a master’s degree program to become secondary teachers of technology were asked to judge the comprehensibility of the statement and the resolution provided, and to report in writing any incoherence, inconsistency, or error they might detect. The participants’ mental processes throughout the task were operationalized through behavioural variables based on a psychological mechanism proposed for inconsistency detection. The behavioural variables focused on the monitoring of important mathematical processes, the algebraic translation, and the interpretation of the numerical solution of the worked example. The software ‘Read and Answer’ was used to record online data on each participant’s behaviour while monitoring the example, as well as his/her written partial and final reports (the task products). An individual short interview was conducted to increase the reliability of the study. Data from each participant were first analysed. Secondly, data from all the participants were considered together in statistical analyses aimed at relating behavioural variables to task products. Four student monitoring profiles were identified corresponding to different combinations of detection/overlooking the embedded algebraic inconsistency, and detection/overlooking the subsequent inconsistency in the result: ‘competent monitoring’, ‘delayed monitoring’, ‘blocked monitoring’, and ‘poor monitoring’ students. Implications for teaching are discussed.