Mathematical Examples Research Articles

On q-Hermite–Hadamard–Mercer and midpoint-Mercer inequalities for general convex functions with their computational analysis

This paper establishes some new inequalities of Hermite–Hadamard–Mercer type for [Formula: see text]-convex functions in the framework of [Formula: see text]-calculus and classical calculus. Some new [Formula: see text]-midpoint-Mercer type inequalities for the [Formula: see text]-differentiable [Formula: see text]-convex functions are also proved. Moreover, the computational analysis of newly established inequalities for convex and [Formula: see text]-convex functions are given to prove that the bounds of this paper are better than those of the existing ones. It is also shown with mathematical examples that the newly established inequalities are valid for [Formula: see text]-convex functions in [Formula: see text]-calculus.

SORM‐Enhanced Inverse Reliability Analysis for Geotechnical Multiobjective Reliability‐Based Design Optimization

ABSTRACTThe first‐order reliability method (FORM) is mostly employed in the existing geotechnical reliability‐based design (RBD) methods due to its computational simplicity and efficiency. However, the first‐order Taylor approximation of the limit state surface (LSS) may result in significant errors, especially in cases of highly nonlinear LSS characterized by substantial curvatures. Therefore, FORM‐based RBD methods require a modification of the curvatures to enhance the accuracy of the probabilistic constraints, specifically by converting the target reliability index into a more precise target failure probability. Correspondingly, reliability index‐based design is converted into failure probability‐based design. In this study, the parabolic second‐order reliability method (SORM), which avoids the Hessian calculations, is adopted to improve the accuracy of probabilistic constraints beyond what is achievable with FORM. The proposed SORM‐enhanced RBD method accounts for the curvature information of the nonlinear LSS, modifying the target reliability index to align with the exact target failure probability through the application of SORM. Moreover, by incorporating an implicit coupling function, multiobjective RBD can be effectively implemented without any additional surrogate model. Furthermore, the proposed RBD method is readily extended to reliability‐based design optimization (RBDO) through integration with an optimization strategy. The proposed RBDO method demonstrates a more precise convergence of the probabilistic constraints, surpassing the accuracy of FORM‐based RBDO methods. Notably, the proposed SORM‐enhanced RBDO method not only significantly improves accuracy but also bypasses the necessity for Hessian computation, which remains both the second‐order accuracy and first‐order efficiency. The feasibility of the proposed method is demonstrated through a mathematical example and three practical geotechnical design examples.

Mixed finite element analysis for a modified Poisson–Fermi interface problem accounting for electrostatic correlations

The Bazant–Storey–Kornyshev (BSK) theory [1–3] recently developed an important continuum framework to expound the nonlocal dielectric permittivity of ionic liquids due to electrostatic correlations, leading to a fourth-order modified Poisson–Fermi equation to model the electrostatic potential field in the solvent (e.g., the electrolyte), while the standard second-order Poisson–Fermi equation is still valid in modeling the electrostatic potential field in the solute. Thus an interface problem is formed between the fourth-order and second-order Poisson–Fermi equations through the interface of solvent and solute with jump coefficients, which has exerted tremendous impacts on applications of electrokinetics, electrochemistry, biophysics, and etc. In this paper, a type of mixed finite element method is developed to solve the proposed interface problem once for all variables: the electrostatic potential, electric field, electric displacement field, electrostatic stress as well as interactional force in the electrolyte in an accurate fashion, and its optimal convergence properties are analyzed for all variables in their respective norms. Numerical experiments are carried out through self-defined mathematical examples to validate all attained theoretical results. Furthermore, as a part of modeling verification for the presented interface problem that models the BSK theory, a practically physical example is investigated to validate the necessity of introducing the fourth-order modified Poisson–Fermi equation to describe the electrostatic correlation effects due to charge reversal phenomenon.

Some Mathematical Examples of Emergent Intuitive Local Time Flow

After reviewing important historical and present day ideas about the concept of time, we develop some instances of mathematical examples where, from the interaction of concepts that model interactions of things in the observable world, time flow emerges in an intuitive and local interpretation. We present several instances of emergence of time flow in mathematical contexts, to wit, by specific parametrisation of deterministic and stochastic curves or of geodesics in Riemann manifolds.

Exploring Sustainability Instruction Methods in Engineering Thermodynamics Courses: Insights from Scholarship of Teaching and Learning

It is important that engineers are educated to consider sustainability in their work. Thermodynamics is a fundamental course required in several engineering majors that has a natural connection to sustainability topics (e.g., energy and limits on efficiency). This work examined how sustainability was included in university-level engineering thermodynamics courses, based on 18 peer-reviewed papers that described Scholarship of Teaching and Learning studies. This review found that environmental issues were included in 15 courses, social issues in 9 courses, and economic issues in 5. There were 11 papers that included topics related to one or more of the United Nations’ Sustainable Development Goals (SDGs), with 8 of the 17 SDGs represented by one or more papers. The learning outcomes from the courses provided many examples of cognitive outcomes at all six levels of Bloom’s taxonomy. In contrast, affective domain outcomes were generally not explicit. Methods of integrating sustainability topics included mathematical examples, labs, projects, service-learning, application-based learning, simulation tools, and book reviews. These examples should inspire instructors to foster sociotechnical mindsets toward engineering, which are a key to educating engineers who value sustainability and who will advocate for its importance in engineering.

Fractal-fractional estimations of Bullen-type inequalities with applications

The study of inequalities inside fractal domains has been stimulated by the growing interest in fractional calculus for the applied and mathematical sciences. This work uses extended fractional integrals in fractal domains to prove new Bullen-type inequalities for differentiable convex functions. By showing how these operators and inequalities can convert classical inequalities into fractal sets, it fills a vacuum in the literature on fractal-fractional integral inequalities. Using extended fractional integral operators, we derive new fractal-fractional estimates and Bullen-type inequalities, accompanied by thorough mathematical derivations and graphical validations. We show optimal results derived from new Bullen-type inequalities for fractal domains. We confirm our results with mathematical examples and graphs, improving models for complicated problems in fractal contexts. In particular, our findings have important ramifications for special means on fractal sets, probability density functions, and quadrature formulas, as well as for future study and applications.

Relativity and the Magnitude of Velocities

In modern physics, relativity is primarily understood through the framework of Einstein’s theory. According to the special theory of relativity (STR), Galilean relativity serves as a very accurate approximation at low velocities, where ‘relativistic effects’ can be neglected. However, at high velocities, these effects become significant and are accurately described by STR. This paper challenges this understanding, arguing that the opposite is true. In Galilean relativity, the relativity of time is substantial and cannot be ignored. Thus, relative time is more negligible in STR than in Galilean relativity. By comparing Galilean relativity and STR through physical, mathematical, and specific examples, two conclusions are drawn: first, the current categorization of relativity theories based on velocity magnitude is artificial and inaccurate; second, the relativity of time can be ignored in special relativity, while it remains significant and cannot be overlooked in Galilean relativity.

Student approaches to generating mathematical examples: comparing e-assessment and paper-based tasks

Student approaches to generating mathematical examples: comparing e-assessment and paper-based tasks

Quantitative method for the probability of structural damage based on moment theory

In the context of uncertainty in the inverse problem of damage identification, this study proposes a moment theory based probabilistic damage identification (MbPDI) method. Noise errors in the identification process are treated as uncertain parameters, while structural stiffness parameters are treated as functions of noise errors. A limit state function for damage identification is defined, and its probability is computed. Utilizing the concept of moment theory, the expansion series is performed for the limit state function around the most probable point of failure, with gradient information derived using sensitivity matrices. Based on the gradient information and uncertainty inputs, damage probabilities for various elements can be obtained. Additionally, this paper addresses the calculation of structural damage probabilities under the non-normal distribution of random variables by transforming non-normal variables into equivalent normal variables. The proposed probability index and expected index of structural damage in the probabilistic damage identification method has a clear physical meaning, and the calculation has high accuracy. The proposed method is validated through a mathematical example and a numerical simulation. Meanwhile, the influence of damage extent, noise level, and modal order on identification results are analyzed. Experimental validation of the method is also presented.

Exploring the linguistic factors influencing concept identification

ABSTRACT This study investigates how the semantic differences, if any, in the Arabic and Hebrew terms for “tangent line” and “rectangle” affect students’ understanding and application of these concepts, as evidenced by their definitions, identification of examples, and explanations given during performance test tasks. We conduct our research with Arabic and Hebrew-speaking high school students enrolled in mathematics classes in Israel. Our goal is to explore how language influences how students define and recognize concepts and examples of geometric shapes. Specifically, we examine how the specific meaning associated with concept names in Arabic and Hebrew impacts the process of defining and identifying these mathematical concepts examples, including the reasoning students use for examples identification.

An adaptive multi-output Gaussian process surrogate model for large-scale parameter estimation problems

PurposeParameter estimation in complex engineering structures typically necessitates repeated calculations using simulation models, leading to significant computational costs. This paper aims to introduce an adaptive multi-output Gaussian process (MOGP) surrogate model for parameter estimation in time-consuming models.Design/methodology/approachThe MOGP surrogate model is established to replace the computationally expensive finite element method (FEM) analysis during the estimation process. We propose a novel adaptive sampling method for MOGP inspired by the traditional expected improvement (EI) method, aiming to reduce the number of required sample points for building the surrogate model. Two mathematical examples and an application in the back analysis of a concrete arch dam are tested to demonstrate the effectiveness of the proposed method.FindingsThe numerical results show that the proposed method requires a relatively small number of sample points to achieve accurate estimates. The proposed adaptive sampling method combined with the MOGP surrogate model shows an obvious advantage in parameter estimation problems involving expensive-to-evaluate models, particularly those with high-dimensional output.Originality/valueA novel adaptive sampling method for establishing the MOGP surrogate model is proposed to accelerate the procedure of solving large-scale parameter estimation problems. This modified adaptive sampling method, based on the traditional EI method, is better suited for multi-output problems, making it highly valuable for numerous practical engineering applications.

Extension of K-nearest neighbors and introduction of an applicable prediction criterion for a novel Monte Carlo simulation-based method in structural reliability

Efficiency is an important issue in structural reliability analysis. In this paper, a novel method is proposed to increase the efficiency of Monte Carlo Simulation by means of a machine learning algorithm. Using K-nearest neighbors method, the safe or failure region of many generated samples can be predicted without evaluating the limit state function. The proposed method represents three novel parts as its main core. In the first part of the proposed method, the classical K-nearest neighbors algorithm has been extended to a new fuzzy level such that the proposed method could predict even if all K neighbors are not in the same region. In previous similar methods, there is a prerequisite that K neighbors must be all in the failure region or all in the safe region. This extension to the fuzzy level is carried out by assigning a specific failure weight to each neighbor. In the second part of the proposed method, an innovative diagnostic criterion has been defined to check whether the region of a sample could be predicted or the sample has to be evaluated in the limit state function. The criterion describes the relative position of the newly generated sample with respect to the neighbors. This newly defined criterion is an effective alternative for the laborious action of drawing the convex hull around K neighbors, which is done as a time-consuming part in previous similar methods. Promotion of the accuracy or efficiency in the proposed method could be set using the adjusting parameter of the represented criterion. Since the implementation of the algorithm for each individual sample needs much time, in the third part, the proposed method introduces a technique to perform function evaluations or predictions in clusters. This part effectively reduces the computational burden, i.e. the number of function evaluations and run time. The performance of the proposed method, applying three novel and newly represented parts, has been verified using a number of mathematical and structural examples.

Inventory model having preservation technology with fix lifetime under two level trade credit policy

Supply-chain management involves moving storage supplies from origin to consumption, with manufacturers running production based on quadratic demand, distributors and retailers monitoring inventory. When a new product is released, demand often rises linearly and then declines dramatically when an alternative becomes available. Shortages are not allowed. Players' inventory will decrease at a rate of (1/(1+m-t)), where m is fixed lifetime, greater than the replenishment time. Deteriorating goods experience constant mass loss or usefulness, but preservation technology can help the damaged item to be consumed. Retailers with direct customer relationships can reduce stock spoilage through good warehouses. Manufacturers' storage systems have a higher deterioration rate. Two-tier trade credit financing is examined in this model. Distributors offer specific credit terms to stores, while manufacturers provide a grace period for invoicing. Distributors and retailers must pay interest on unsold inventories if invoices aren't settled on time. An integrated storage system reduces costs by minimizing costs through multiple shipments from manufacturers to distributors and retailers, and by adjusting replenishment times for each player. The resolution process is designed so that the supply chain operator gets the best possible decision. Therefore, results are authorized using mathematical examples for different scenarios. Management decisions are suggested.

A novel multiple linearization method for reliability analysis based on evidence theory

Evidence theory has been applied in structural reliability assessment to address epistemic uncertainties, given its capability to deal with imprecise, incomplete and even contradictory information. An important challenge in structural reliability assessment based on the framework of evidence theory is its expensive computational cost in multidimensional problems. In this work, a modified multiple linearization method is proposed to identify a set of hyperplane approximations to the limit state function. It provides a clear guidance for the hyperplanes by evaluating curvature magnitude of the original limit state at the most probable point along each axis in the rotated standard normal space, and shows good stability and satisfactory accuracy in nonlinear reliability analysis based on evidence theory. The belief measure and plausibility measure of the structure are assessed by the hyperplanes through the intersection or union of the failure domains. In the meantime, the classification and division of focal elements in failure set, safety set and uncertain set will be made through the segment edge of the failure and safety domain to reduce the computation burden. Finally, four mathematical examples and engineering cases are explored to illustrate the methodology and show its benefits.

The Role of Core Sample Geometry on Countercurrent Spontaneous Imbibition: Mathematical Foundation, Examples, and Experiments Accounting for Realistic Geometries

Summary Countercurrent spontaneous imbibition (SI) experiments are among the most common multiphase experiments performed on porous rock samples. Although the samples rarely are designed to give linear flow, they are often modeled and interpreted using mathematical descriptions assuming linear flow. In this work, the goal is to derive general understanding of how imbibition into different sample geometries behaves compared with linear (1D) imbibition. Using the mathematical theory of N-volume spheres (N being the dimension), we consider core samples as N-spherical and quantify their geometry by a dimension N and length scale L. For the special cases, N=1,2,3, we obtain linear, radial, and spherical flow; however, we treat the dimension as an arbitrary real number for cases not adhering to either of these regimes. Particularly, for rectangular or cylindrical core plugs, a continuous range of dimensions is produced. Theoretical calculations of N and L of arbitrary sample shapes are derived based on relations with area per volume and derivative of area with respect to volume. They correctly produce limit cases and physically meaningful values for symmetrical, cylindrical, and rectangular geometries. The differential equation for countercurrent imbibition in N-dimensions is derived and solved with numerical examples. Also, a simplified analytical solution assuming piston-like displacement is derived to get illustrative relations between system parameters (including dimension) and recovery and front position. Predicted recovery profiles of the N-dimensional (N-D) solution overlap consistently with numerical simulations (by an alternative simulator) into cylindrical geometries with a wide range of height/diameter ratios and viscosity ratios. At early time, the saturation profiles are self-similar (look the same plotted against position divided by square root of time) and identical regardless of dimension. As long as the profiles are self-similar near the open boundary, recovery is proportional to the square root of time. For N=1, this lasts long after the front has reached the closed boundary, while for N >1, it can happen long before the boundary has been reached. The same time scale was applicable for all geometries, stating especially that doubling the length scale increased the time of the entire recovery profile by a factor 4. As long as recovery is proportional to square root of time, at a given time, it is also proportional to dimension over length, NL, and the time needed to reach same level of recovery is proportional to LN2. Permeability anisotropy (lower vertical than horizontal permeability) could effectively be modeled using an effective increased height, which further was captured by the dimension and length scale. Literature and in-house experimental data were matched by the model and used to validate model predictions such as the variation in time, shape of recovery curves with changes in dimension, and the importance of accounting for the dimension (geometry) during data interpretation. The model is valid for all wetting states but assumes negligible gravity and compressibility effects.

THE MATHEMATICAL EXAMPLE OF GNOMONS IN ARISTOTLE, PHYSICS 3.4, 203a10–16

Abstract This article examines a complex passage of Aristotle's Physics in which a Pythagorean doctrine is explained by means of a mathematical example involving gnomons. The traditional interpretation of this passage (proposed by Milhaud and Burnet) has recently been challenged by Ugaglia and Acerbi, who have proposed a new one. The aim of this article is to analyse difficulties in their account and to advance a new interpretation. All attempts at interpreting the passage so far have assumed that ‘gnomons’ should indicate ‘odd numbers’. In this article it is argued that the usage of ‘gnomon’ related to polygonal numbers, which is normally considered late, could be backdated to at least the fifth/fourth centuries b.c.; in particular, it explains the link between the philosophical explanandum and the mathematical explanans in Aristotle's passage.

Optimal control of Boolean control networks with state-triggered impulses

In the real world, abrupt changes caused by external environment and system interaction can exhibit a pattern resembling an “impulse” response. A novel model, known as the hybrid-index model, has recently been introduced for Boolean control networks (BCNs) to characterize such impulsive responses. The optimal control problem is a fundamental issue in control theory, and many control problems can be classified as special cases of optimal control problems. Nevertheless, the optimal control problem for the hybrid-index model remains unresolved. This study delves into the optimal control problem of impulsive Boolean control networks (IBCNs) under the hybrid-index model framework. Forward completeness serves as a crucial condition to avert Zeno behavior in the hybrid-index model of IBCNs. In contrast to prior studies on the hybrid-index model of IBCNs, the study shows that the optimal control problem of IBCNs can be addressed even in cases where the system is not forward complete. Leveraging the theory of semi-tensor product, both finite-horizon and infinite-horizon optimal control problems of IBCNs are tackled using the quotient mapping method and the recursive method for optimal control of Boolean control networks. The study yields the optimal cost and optimal control law through state feedback. To illustrate the efficacy of the proposed approach, a mathematical example and a biological example known as the λ switch are presented.

What Are and What Are Not Extrema Points? Examining Definitions and Examples

AbstractThis paper reports on five secondary school mathematics prospective teachers’ conceptions of extreme point. The analysis of the data addressed students’ definitions, examples, and evaluation of given examples, with special attention to the related domain. Written assignments and individual interviews uncover salient, erroneous concept images regarding what is and what is not an extreme point. Participants viewed extrema points as points that necessarily satisfy f′ = 0 or as points that are always at a “change in monoticity” of the function. The topic “extreme points” is both an aim and a mean to address broader issues related to mathematical definitions, examples, and nonexamples. We conclude with possible next-step ideas.

Resilience of sustainability for a smart production system to produce biodiesel from waste animal fat

The smart production system is implemented for more effective labor, less waste, and a reduction of energy consumption. Biofuels are significant substitutes for conventional fuels and decreases air pollution and carbon monoxide, can be made through a smart production system with automated machines. Biodiesel can be produced using an assortment of sources that can be both edible and inedible, but animal fat is a sensible alternative among inedible sources because of its affordable price and ease of availability. The model studies the smart biodiesel production system with energy utilization and less carbon emissions to produce pure biodiesel. The multi-objective mathematical model minimizes the total biodiesel supply chain cost followed by minimizing the total biodiesel supply chain environmental impact and enhancing its social impact. The smart production system with a variable production rate produces the biodiesel with less impurities, but still impurities in the biodiesel, that can be detected by a two-stage autonomated inspection system. The impure biodiesel is remanufactured in the biorefineries at the end of every natural production cycle. For the best utilization of energy, various energy costs, such as lighting and air handling costs are calculated. The improved augmented ϵ-constraint approach with lexicographic optimization is utilized to solve the model. The four mathematical examples are studied, and the repercussions of mathematical experiments reveal that the animal fat purchasing cost is 44.32% of the total cost, and the percentage of impurities in biodiesel can be decreased through minimum energy consumption. The managerial insights of the model, graphical representation, and sensitivity analysis of key parameters are provided in this research. The conclusions can be useful to investors in a sustainable smart production system and biofuel policymakers.

SYSTEMATIC LITERATURE REVIEW: CULTURAL INTEGRATION IN LEARNING CONCEPTS WITH AN ETHNOMATHEMATICS APPROACH

In an educational context, the integration of culture in the learning concept is essential to ensure that each student feels recognized and their values ​​are appreciated. The ethnomathematics approach is one way to integrate culture into mathematics learning. By including mathematical examples and problems that are relevant to students' culture, learning will become more meaningful and students will be more motivated to learn. In addition, by understanding mathematical concepts from various cultures, students can develop a deeper understanding of these concepts. It can also help in creating an inclusive learning environment and stimulate students' interest in mathematics. It is important to continue to develop this approach in our educational context to ensure that every student gets a fair and stimulating opportunity to learn mathematics