Mathematical Explanation Research Articles

A noetic account of explanation in mathematics

Abstract We defend a noetic account of intramathematical explanation. On this view, a piece of mathematics is explanatory just in case it produces understanding of an appropriate type. We motivate the view by presenting some appealing features of noeticism. We then discuss and criticize the most prominent extant version of noeticism, due to Inglis and Mejía Ramos, which identifies explanatory understanding with the possession of well-organized cognitive schemas. Finally, we present a novel noetic account. On our view, explanatory understanding arises from meeting specific explanatory objectives. We defend a cluster-concept account of explanatory objectives and identify four important subfamilies within the relevant network of resemblance relations. The resulting view is objectivist (in the sense that it takes explanatory success to be a matter of observer-independent fact), broader in scope than why-question-based accounts, compatible with empirical findings on experts’ explanatory judgments, and capable of generalizing (with appropriate provisos) to scientific explanation as a whole. It thus fulfills Friedman’s half-century-old demand for a general and objectivist theory, which accounts for the link between explanation and understanding.

Students' thinking dispositions and teacher's approaches in conflict situations: a case study of a secondary school mathematics class in Japan

PurposeThis study aims to depict the process of creating a culture of thinking in relation to developing thinking dispositions, focusing on conflicts between students and teachers at the beginning of the school year.Design/methodology/approachThis study conducted a year-long fieldwork in a secondary school in Tokyo, Japan, including participatory observation of a mathematics class and a semi-structured interview with a teacher. The classroom discourse is analyzed to interpret the students’ thinking dispositions and the teacher's approaches in context using eight lessons. A situation in which the teacher attempts to change the method or direction of students’ thinking is defined as a conflict.FindingsFive conflict situations are found, namely (1) listening, (2) mathematical explanation, (3) disclosure, (4) giving up and (5) examining ideas. They reveal three characteristics about students’ thinking dispositions: closed-minded, correct answer-ism and mathematical immaturity. Against such students’ thinking, the teacher tried to make them listen carefully to, understand and examine peer ideas, aiming to discuss mathematically. Through these conflicts, the classroom culture moves toward a community of thinking and discussing mathematically.Originality/valueThis study emphasizes the cultural and contextual aspects of thinking dispositions as shaped by a classroom culture. By depicting “conflict” as a new analysis unit, this study reveals students’ thinking dispositions and how they are enacted in what situations and how teacher’s approaches to them create a culture of thinking.

Mathematical and Numerical Explanation of the Nonlinear Acoustic Wave Interaction in Acousto-Optical Cells

Mathematical and Numerical Explanation of the Nonlinear Acoustic Wave Interaction in Acousto-Optical Cells

Exploring Nonlinear Fractional (2+1)-Dimensional KP–Burgers Model: Derivation and Ion Acoustic Solitary Wave Solution

The purpose of this study is to conduct a complete examination into the fractional (2+1)-dimensional nonlinear KP-Burgers (KP-B) model in the setting of quantum plasma. The main goal is to present a mathematical explanation and derivation for the fractional (2+1)-dimensional nonlinear KP-B model. We used the reductive perturbation technique to obtain the fractional (2+1)-dimensional ion acoustic solitary wave, which leads to the nonlinear KP-B model. To solve the fractional space-time KP-B model, we used the modified sub-equation technique and the extended hyperbolic function methodology. This methodology covers rational, periodic wave, and hyperbolic function solutions. Ion acoustic waves were explored in relation to the effects of ion pressure and an external electric field. The effects of density and fractional order on the properties of a single solution are examined. The plasma system is defined as an unmagnetized epi plasma system composed of relativistic ions, positrons, and nonextensive electrons that can be found in a range of astrophysical and cosmological environments. The solution variables include electron-positron and ion-electron temperature ratios, electron and positron nonextensivity strength, ion kinematic viscosity, positron concentration, and the weakly relativistic streaming factor. The fractional order and associated plasma properties have a considerable influence on the phase velocity of ion acoustic waves. When the fractional order equals one, the obtained results are consistent with known outcomes. This work makes a substantial contribution to our understanding of nonlinear events in quantum plasmas, particularly ion acoustic waves. It provides a solid approach for solution development and analysis, with implications for a variety of astrophysical and cosmological scenarios.

From knowledge acquired at academic mathematics courses to significant changes in instructional practices

This non-interventional study investigates the contribution of university mathematics to teaching high-school mathematics. Data sources included interviews with five teachers who taught high-school mathematics before, during, and after their academic mathematics studies. All teachers provided tangible examples of fundamental changes in instructional practices that they explicitly linked to new knowledge acquired in the academic mathematics courses. The domain of analysis in general, and the topics of integrals and derivatives in particular, were central in the teachers’ illustrations of changes they made in their teaching, although other mathematical topics and domains were also mentioned. The reported changes were mostly associated with emphasis on mathematical explanations, exposition of two key elements of the deductive structure of mathematics: definition and proof, an increased focus on formal mathematics, and portrayal of mathematics as a wide and varied discipline. The study results are discussed in light of the relevant literature.

String art with repeating decimals

Rational numbers are typically represented as decimals or fractions. This paper explores the artistic potential of these representations, focusing on converting decimal expansions of rational numbers into visual forms via string art. By dividing a circle into 10 equal parts numbered from zero to nine, repeating decimals can be transformed into visual paths defined by the sequence of digits. This approach was used to create a pair of artworks and several interactive GeoGebra animations. The visual appeal of the designs is complemented by a mathematical explanation using number theory results.

Enhancing understanding of the rotating magnetic field in electric machines through active learning and visualization

AbstractThis paper presents a method aimed at improving comprehension of AC electric machine principles by facilitating the learning of the Rotating Magnetic Field (RMF) through visualization tools provided by Finite Element Method (FEM) software. First, traditional methods used in textbooks to explain RMF in electric machines are reviewed, with an analysis of various instructional strategies. Acknowledging the limitations of these conventional approaches and the inherent complexity of RMF comprehension, a novel visualization method is proposed. Understanding RMFs in electric machines is fundamental for electrical engineers due to their crucial role in electric machine design and optimization. While textbooks typically rely on mathematical explanations and simple sketches, advancements in computer and software technology offer opportunities to utilize finite element tools for enhanced comprehension. Through dynamic animations and interactive simulations, emphasis is placed on prioritizing conceptual understanding over mathematical descriptions. Furthermore, this paper explores the development of pre‐simulation models in FEM tools to facilitate RMF learning in AC electric machines, and the process of creating a model to teach rotating magnetic fields in electric machines is carefully outlined. This approach holds promise for engineers seeking a deeper understanding of electric machines and can also be utilized by educators to enhance their teaching methodologies.

Elementary students' argumentative and explanatory mathematical writing about fractions

AbstractMathematical writing (MW) is an important yet underutilized form of mathematics discourse. Given the lack of attention to MW, there are few resources to support the implementation and assessment of MW. This study responds to the lack of resources by describing a MW unit focused on fractions that was implemented during a summer school program to support rising fourth grade students with explanatory and argumentative writing. Findings from the study revealed that students used different forms of reasoning about fractions to compose their mathematical argument and explanation. We also identified five categories that captured how students communicated a mathematical argument and explanation: writing organization, precise mathematical vocabulary, mathematics content, mathematical representation, and characteristics specific to a mathematical argument/explanation. The five categories led to the development of an analytic rubric for argumentative and explanatory writing.

Principles of Creating Multi-objective Quality Models for Software Systems

This article proposes an original approach at a mathematical level to the creation of multi- objective quality models for software systems. The research is based on the study and generalization of quality modeling trends of software systems and user needs in order to determine optimal principles for constructing such models. The article provides mathematical explanations that play a key role in identifying and formalizing the principles of creating multi-objective quality models of software. An important aspect is the consideration of quality model construction principles at the mathematical level, allowing for a more precise assessment and analysis of various aspects of software quality. The research results indicate that incorporating mathematical principles into the creation of multi-objective quality models for software systems can have a significant practical impact. It is established that on a practical level of developing multi-objective quality models for software systems, consideration of the principles of creating multi-objective quality models can have numerous practical implications. Specifically, the application of metrics within established principles allows for a comprehensive view of software quality and identifies areas requiring attention and improvement. This helps developers and software quality engineers make informed decisions regarding system improvement and optimization. The research has shown that creating high-quality models of quality requires attention to various aspects, from user needs to testing and continuous improvement, as well as the use of mathematical methods for their formalization and analysis. The developed principles of creating multi-objective quality models at the level of mathematical models allow the use of these models to assess and analyze various aspects of software quality, representing each model using a corresponding function that determines the relationship between quality metrics and the quality of the software itself. It is expected that further development and implementation of these principles will contribute to improving software development processes and ensure high quality of the resulting software.

Epistemological justification

AbstractEpistemological justification is a way of thinking that manifests itself through perturbation-resolution cycles revolving around the question why and how was a piece of mathematical knowledge conceived? The paper offers a conceptual framework for constituent elements of epistemological justification. The framework provides: (a) a theoretical basis for epistemological justification, (b) criteria for its occurrence, and (c) analysis of its relation to mathematical explanation. The criteria are illustrated by a series of learning-teaching events taken from teaching experiments aimed at investigating the learning and teaching in linear algebra. The contribution of the proposed framework is three-fold: (a) it addresses a critical aspect of proof understanding not explicitly addressed in the literature; (b) it goes beyond the traditional treatment of mathematical understanding and production into questions about learners’ conceptualization of the origins of mathematical knowledge; and (c) it theorizes instructional approaches that can advance this conceptualization among students.

Enhancement of heat transfer through alloys and nanoparticle over a melted sheet: A Koo and Kleinstreuer (K-K) model

Alloys are used in a wide variety of contexts, including aircraft building, the military, industry, medicine, and manufacturing. Alloys including tools, equipment, cars, and structures. Furniture, shipping containers, and medical supply storage containers are among products that benefit from the usage of aluminium alloys. To further enhance simultaneous heat transmission, scientists have developed nano-fluids, which are essentially nano-particle suspensions in a regular fluid. Ferric Sulfate may be added to soil to improve its quality and is also used to clean water. Iron (3+) sulphate is an iron (III) and sulfate (III) combination with a 3:2 ratio of iron (III) to sulfate (III) ions. As a catalyst, mordant, and astringent, it serves several purposes. It is a sulphate of a metal and a molecule of iron. Due of these purposes, we employ the K-K model to study how alloys and nanoparticles affect surface flow and heat transfer. By making use of the boundary layer approach and the flow assumptions, the partial differential equations (PDEs) were generated by approximating the Navier-Stokes equation. This system is transformed into ordinary differential equations (ODE's) using similarity transformations. Using the BVP4C method, a system with non-dimensions is defined. Mathematical and pictorial explanations are provided for the consequences of the key physical variables. It is founded that the magnitude of the temperature field is directly proportional to the R value; a higher value indicates more heat conduction to the liquid. Additionally, the velocity of the fluid lessened by the impact of melting effect.

Lipschitz energy functional for anisotropic diffusion applications

Image denoising remains a key research problem because of its potential role as a pre-processing component in image processing, computer vision, and machine learning tasks. Of the available approaches for image denoising, those inspired by anisotropic diffusion processes have been a center of discussion for decades. Despite the efforts and promising results achieved by diffusion-inspired denoising methods, we noted insufficient attention on the design of energy functionals for anisotropic diffusion equations. Most researchers consider heuristic approaches to design diffusivity functionals, a practice that cannot provide mathematical explanations on why their approaches work. The current research presents a strictly convex and Lipschitz energy functional that guarantees a unique solution for an evolutionary process. Based on this functional, we derive an anisotropic diffusion equation for image denoising applications. Experimental results show that an algorithm corresponding to the proposed equation is computationally efficient, and generates informative and visually appealing images with competitive values of peak signal-to-noise ratio and structural similarity. Guided by the compelling properties of our energy functional, we provide an additional insight to describe quality of the results. Implementation codes and test datasets of the proposed approach are publicly accessible at the MATLAB File Exchange (https://www.mathworks.com/matlabcentral/fileexchange/160108-lipschitz-diffusion-inspired-energy-functional).

An Examination of Learning Using Fourier Analysis of Mathematical Models of Consciousness

The concept of consciousness remains quite possibly the ultimate mystery for humanity. It is somehow linked to the ability to learn and experience the external environment. The process of learning is of utmost importance to any species and yet remains largely not understood. While many experts from varying fields throughout history have attempted to quantify and measure learning, all failed to capture the process with a mathematical explanation. The purpose of this paper is to present three different models of consciousness and their related Fourier transform and evaluate the ability of each to capture some of the behavior that is understood about the conscious experience. There is some evidence that the experience of consciousness as a function of time can be measured in an expression of phase. With this hypothesis, the technique of Fourier transform becomes a useful tool to examine a mathematical model of consciousness that can be transformed from the phase plane to the linearization of time. This paper summarizes some current research on consciousness and learning and perception of time. This paper then presents three different models of consciousness represented as a pulse function, Dirac Delta function, and exponential decay function and examines each model utilizing Fourier transform. Finally, this paper concludes that the intersection between psychology, neuroscience and cognitive science can be made by utilizing analysis tools in the field of applied mathematics.

Method for measuring radiation patterns of digital antenna arrays

Formulation of the problem. The Digital Beamforming Antenna Arrays (DBAA) are widely used as the directional antennas in radio systems today. However, such arrays typically do not have a radiofrequency output. Consequently, the standard methods for measuring the Radiation Patterns (RP), based on the transmitting of a monochromatic probing signals, reception of the signals in the far field, as well as measuring the output power of the antenna array, cannot be used to measure the RP of the DBAA. The method considered in this work involves the transmitting a probing signal modulated by a pseudonoise sequence and processing the DBAA output signal at the zero intermediate frequency (base-band) using a matched filter or a correlator. Target. A method for measuring the radiation pattern of the DBAA is considered, based on the correlation processing of its output signal. Results. The paper presents a description of the setup for measuring the DBAA RP, as well as a mathematical explanation of the measurement procedure. It is shown that the accuracy of the RP measurement depends on the Signal-to-Noise Ratio (SNR) at the outputs of the receivers of the DBAA channels, and the number of array channels and the number of symbols of the pseudonoise sequence used for the modulation of the radiated signal. The error in the RP measurement decreases by 3 dB each time as the number of the DBAA channels or the number of symbols of the pseudonoise sequence used in the measurements is doubled. This result is confirmed by simulation the measurement of RP of the eight- and sixteen-channel linear DBAA using pseudo-noise sequences of maximum length (M-sequences) with a number of symbols equal to 1023 and 2047. Practical significance. The proposed method does not require the usage of the up-converter to convert the base-band DBAA output signal from the base-band to a carrier frequency, and it also does not require the usage of a power meter for measurement the analog signal at the output of this converter.

Correlations of Cross-Entropy Loss in Machine Learning.

Cross-entropy loss is crucial in training many deep neural networks. In this context, we show a number of novel and strong correlations among various related divergence functions. In particular, we demonstrate that, in some circumstances, (a) cross-entropy is almost perfectly correlated with the little-known triangular divergence, and (b) cross-entropy is strongly correlated with the Euclidean distance over the logits from which the softmax is derived. The consequences of these observations are as follows. First, triangular divergence may be used as a cheaper alternative to cross-entropy. Second, logits can be used as features in a Euclidean space which is strongly synergistic with the classification process. This justifies the use of Euclidean distance over logits as a measure of similarity, in cases where the network is trained using softmax and cross-entropy. We establish these correlations via empirical observation, supported by a mathematical explanation encompassing a number of strongly related divergence functions.

Mathematical explanations for knee osteotomies: "Dear engineer, how would you explain it in a simple way?"

Knee osteotomies are effective procedures to treat different deformities and to redistribute the load at the joint level, reducing the risk of wear and, consequently, the need for invasive procedures. Particularly, knee osteotomies are effective in treating early arthritis related to knee deformities in young and active patients with high functional demands, with excellent long-term results. Precise mathematical calculations are imperative during the preoperative phase to achieve tailored and accurate corrections for each patient and avoid complications, but sometimes those formulas are challenging to comprehend and apply. Four specific questions regarding controversial topics (planning methods, patellar height, tibial slope, and limb length variation) were formulated. An electronic search was performed on PubMed and Cochrane Library to find articles containing detailed mathematical or trigonometrical explanations. A team of orthopedic surgeons and an engineer summarized the available Literature and mathematical rules, with a final clear mathematical explanation given by the engineer. Wherever the explanation was not available in Literature, it was postulated by the same engineer. After the exclusion process, five studies were analyzed. For three questions, no studies were found that provided mathematical analyses or explanations. Through independent calculations, it was demonstrated why Dugdale's method underestimates the correction angle compared to Miniaci's method, and it was shown that the variation in patellar height after osteotomy can be predicted using simple formulas. The five included studies examine postoperative variations in limb length and tibial slope, providing formulas applicable in preoperative planning. New formulas were independently computed, using the planned correction angle and preoperatively obtained measurements to predict the studied variations. There is a strict connection among surgery, planning, and mathematics formulas in knee osteotomies. The aim of this study was to analyze the current literature and provide mathematical and trigonometric explanations to important controversial topics in knee osteotomies. Simple and easy applicable formulas are provided to enhance the accuracy and outcomes of this surgical procedure.

Investigation on work hardening rate of Al-Zn-Mg-Cu alloy under solid solution process and thermomechanical coupled multi-step fast aging process

In this paper, the work hardening rate of Al-Zn-Mg-Cu alloy in the conditions of solid solution and thermo-mechanical coupled fast aging were investigated separately, and the main research content and results are as follows. Under the solid solution processes at different temperatures, the uniaxial tensile test curves of work hardening rate are relatively dispersed at a lower strain rate, and the work hardening rate increases with the increase of solution temperature. Then, as the strain rate increasing, the work hardening rate curves at each solution temperature gradually converge, until all curves intersect at a common intersection point. The above mechanism is attributed to the comprehensive effect of precipitates, solid solutions, and dynamic strain aging in the matrix. For the thermo-mechanical coupled aging, the work hardening rate has a great different with that of solid solution process. Consequently, the influence of pre-deformation on the work hardening rate were investigated, which concludes the curve shape type and the softening coefficient. The introduction of dislocation have some significant influences on the work hardening rate. Firstly, it makes the shape of work hardening rate curves change from shape 1 to shape 2. On the other hand, as the pre-deformation increasing, the variation law of softening coefficient shows an abnormal trend that is different from the ordinary law, which is due to the coordinated effect between dislocation density and dislocation induced precipitation. Furthermore, in order to reveal the internal mechanism in depth, the microstructure are observed by TEM (Transmission electron microscope). Finally, a targeted set of analytical models for work hardening rate were established, especially the mathematical explanation of the multicollinearity phenomenon of work hardening rate curves in the solid solution condition.

Explaining empirical dynamic modelling using verbal, graphical and mathematical approaches.

Empirical dynamic modelling (EDM) is becoming an increasingly popular method for understanding the dynamics of ecosystems. It has been applied to laboratory, terrestrial, freshwater and marine systems, used to forecast natural populations and has addressed fundamental ecological questions. Despite its increasing use, we have not found full explanations of EDM in the ecological literature, limiting understanding and reproducibility. Here we expand upon existing work by providing a detailed introduction to EDM. We use three progressively more complex approaches. A short verbal explanation of EDM is then explicitly demonstrated by graphically working through a simple example. We then introduce a full mathematical description of the steps involved. Conceptually, EDM translates a time series of data into a path through a multi-dimensional space, whose axes are lagged values of the time series. A time step is chosen from which to make a prediction. The state of the system at that time step corresponds to a 'focal point' in the multi-dimensional space. The set (called the library) of candidate nearest neighbours to the focal point is constructed, to determine the nearest neighbours that are then used to make the prediction. Our mathematical explanation explicitly documents which points in the multi-dimensional space should not be considered as focal points. We suggest a new option for excluding points from the library that may be useful for short-term time series that are often found in ecology. We focus on the core simplex and S-map algorithms of EDM. Our new R package, pbsEDM, enhances understanding (by outputting intermediate calculations), reproduces our results and can be applied to new data. Our work improves the clarity of the inner workings of EDM, a prerequisite for EDM to reach its full potential in ecology and have wide uptake in the provision of advice to managers of natural resources.

TTCF4LAMMPS: A toolkit for simulation of the non-equilibrium behaviour of molecular fluids at experimentally accessible shear rates

We present TTCF4LAMMPS, a toolkit for performing non-equilibrium molecular dynamics (NEMD) simulations to study the fluid behaviour at low shear rates using the LAMMPS software. By combining direct NEMD simulations and the transient-time correlation function (TTCF) technique, we study the fluid response to shear rates spanning 15 orders of magnitude. We present two examples for simple monatomic systems: one consisting of a bulk liquid and another with a liquid layer confined between two solid walls. The small bulk system is suitable for testing on personal computers, while the larger confined system requires high-performance computing (HPC) resources. We demonstrate that the TTCF formalism can successfully detect the system response for arbitrarily weak external fields. We provide a brief mathematical explanation for this feature. Although we showcase the method for simple monatomic systems, TTCF can be readily extended to study more complex molecular fluids. Moreover, in addition to shear flows, the method can be extended to investigate elongational or mixed flows as well as thermal or electric fields. The high computational cost needed for the method is offset by the two following benefits: i) the cost is independent of the magnitude of the external field, and ii) the simulations can be made highly efficient on HPC architectures by exploiting the parallel design of the algorithm. We expect the toolkit to be useful for computational researchers striving to study the nonequilibrium behaviour of fluids under experimentally-accessible conditions. Program summaryProgram title: TTCF4LAMMPSCPC Library link to program files:https://doi.org/10.17632/hh2rkcxbrf.1Developer's repository link:https://github.com/edwardsmith999/TTCF4LAMMPSLicensing provisions: GNU General Public License 3Programming language: Python 3Nature of problem: Measuring the nonequilibrium behaviour of bulk and confined fluids under experimentally accessible strain rates in non-equilibrium molecular dynamics (NEMD) simulations.Solution method: Creating a Python-based code that utilises the transient-time correlation function method and the LAMMPS software to enable the bulk fluid properties (e.g. viscosity) and confined fluid interfacial properties (e.g. shear stress and slip velocity) to be computed at low shear rates with NEMD.

Likelihood-based generalization of Markov parameter estimation and multiple shooting objectives in system identification

This paper considers system identification (ID) of linear and nonlinear non-autonomous systems from noisy and sparse data. We analyze an optimization objective derived from Bayesian inference for the dynamics of hidden Markov models. We then relate this objective to that used in several state-of-the-art approaches for both linear and nonlinear system ID. In the former, we analyze least squares approaches for Markov parameter estimation, and in the latter, we analyze the multiple shooting approach. We demonstrate the limitations of the optimization problems posed by these existing methods by showing that they can be seen as special cases of the proposed optimization objective under certain simplifying assumptions: conditional independence of data and zero model error. Furthermore, we observe that the proposed approach has improved smoothness and inherent regularization that make it well-suited for system ID and provide mathematical explanations for these characteristics’ origins. Finally, numerical simulations demonstrate a mean squared error over 8.7 times lower compared to multiple shooting when data are noisy and/or sparse. Moreover, the proposed approach identifies accurate and generalizable models even when there are more parameters than data or when the system exhibits chaotic behavior.