Mathematical Form Research Articles

Identification of Students' Misconceptions Using a Four-Tier Multiple Choice Diagnostic Test on Colligative Properties of Solutions

Misconception is an event of a wrong concept received by students after following a learning process, the concept received is not in accordance with existing scientific studies. This study aims to identify students' misconceptions on the colligative properties of the solution and the factors that cause misconceptions. This type of research is descriptive quantitative. The instrument used is a four-tier multiple choice diagnostic test. This study involved 34 students of class XII MIA at MA Khulafaur Rasyidin Sungai Raya. The results showed that the level of misconceptions of students had an average percentage of 34% with a moderate misconception category, where a total of 65% students experienced misconceptions in the sub-topic of vapor pressure depression, 26,30% students experienced misconceptions in the sub-topic of freezing point depression, 33,30% students experienced misconceptions in the sub-topic of boiling point elevation, and 26% students experienced misconceptions in the sub-topic of osmotic pressure. Based on the results of the interview, the factors causing misconceptions in this study consist of internal and external factors. Internal factors come from students, that are students provide concept ideas spontaneously based on their understanding without being investigated first, students provide incorrect reasoning, students are confused in connecting one concept with another, lack of interest and ability of students in learning chemistry. External factors come from the context and teaching methods, where one of the learners' discussion partners has misconceptions, the teacher is more dominant in providing chemical explanations in mathematical form, and limited experiments. Those misconceptions can be reduced by presenting the topic in an interesting way, using an experiment, or using multiple representations approach.

Cross-spectral purity of nonstationary vector optical fields: A similarity with stationary fields

This study establishes a reduction formula for nonstationary cross-spectrally pure vector (or electromagnetic) light fields with any spectral bandwidth. In general, for nonstationary cross-spectrally pure electromagnetic fields, the reduction formula of the two-time Stokes parameters normalized by the corresponding usual Stokes parameters is not similar to the one for stationary fields since the correlation properties for nonstationary fields depend on two-time instants t1 and t2, in addition to their time difference. Here, we derive a reduction formula by including time-integrated correlations that share a similar mathematical structure to that of reduction formulas used for stationary vector fields if the normalized spectra of the interfering fields are identical. Additionally, we examine the requirement of strict cross-spectral purity for nonstationary electromagnetic fields using a time-integrated coherence function, which exhibits a mathematical form similar to that of strict cross-spectral purity in stationary vector fields. This investigation sheds light on the cross-spectral purity of pulse-type fields, which holds potential applications in statistical optics.

Modeling information flow from multispectral remote sensing images to land use and land cover maps for understanding classification mechanism

ABSTRACTInformation on Land Use and Land Cover Map (LULCM) is essential for environment and socioeconomic applications. Such maps are generally derived from Multispectral Remote Sensing Images (MRSI) via classification. The classification process can be described as information flow from images to maps through a trained classifier. Characterizing the information flow is essential for understanding the classification mechanism, providing solutions that address such theoretical issues as “what is the maximum number of classes that can be classified from a given MRSI?” and “how much information gain can be obtained?” Consequently, two interesting questions naturally arise, i.e. (i) How can we characterize the information flow? and (ii) What is the mathematical form of the information flow? To answer these two questions, this study first hypothesizes that thermodynamic entropy is the appropriate measure of information for both MRSI and LULCM. This hypothesis is then supported by kinetic-theory-based experiments. Thereafter, upon such an entropy, a generalized Jarzynski equation is formulated to mathematically model the information flow, which contains such parameters as thermodynamic entropy of MRSI, thermodynamic entropy of LULCM, weighted F1-score (classification accuracy), and total number of classes. This generalized Jarzynski equation has been successfully validated by hypothesis-driven experiments where 694 Sentinel-2 images are classified into 10 classes by four classical classifiers. This study provides a way for linking thermodynamic laws and concepts to the characterization and understanding of information flow in land cover classification, opening a new door for constructing domain knowledge.

Literasi Matematika sebagai Kompetensi Utama di Abad 21 dan Implikasinya pada Pendidikan di Sekolah Dasar

Students will not master mathematics material if they have difficulty in understanding concepts. To understand concepts, literacy skills are needed. Someone who can understand concepts well and is also skilled in mathematics should be sensitive to mathematical concepts related to problems. From this awareness then comes the strategy to formulate the problem in mathematical form and then solve it. Formulating, applying and interpreting are all part of this process. The purpose of this research is to explore the concept of mathematical literacy and analyse the role of mathematical literacy in the 21st century, examining some of the factors that are thought to influence mathematical literacy achievement and mathematical connection skills. The results showed that mathematical literacy can be realised through contextualised learning in achieving the goal of mathematical literacy, teachers must engage students in important activities that help them relate academic lessons to the real-world contexts they face. Mathematical literacy can be realised in learning that is concrete and teachers should teach by linking learning with students’ daily lives by doing contextual learning or doing the Realistic Mathematics Learning movement, both of which are considered good in improving the mathematical literacy skills of elementary school student.

Numerical modelling for design of microchannel reactors: Application to hydrogen production from methanol by steam reforming

Microchannel reactors involve chemical-reaction engineering problems in their design and operation, with heat and mass transfer considerations dominating their practicability and efficiency. Study of the fundamental transport phenomena necessitates their description in sophisticated mathematical form. The present study seeks to address this particular problem by investigating the effects of various factors on the transport processes in a heterogeneously catalysed microchannel steam reforming reactor. Numerical modelling for reactor design was performed using a model that accounted for complex catalyst dynamics. Dimensionless number analyses were conducted to understand the nature of the transport processes. The reactor performance was evaluated by means of various parameters, and design recommendations were made. Different operating limit lines were mapped out, and various definitions of energy efficiency were distinguished. The results indicated that the use of microchannel technology enables the achievement of unusually favourable conversion. While mass-transfer limitations are dominant, the reaction is practically feasible at millisecond contact times. A maximum output power in excess of 70 watts per channel can be reached, and energy efficiencies up to about 70 percent are available. The operating limit line can be changed as desirable to maintain the desired product yield throughout the range of conditions. While an optimum velocity for maximum efficiency exists, the flow rate and catalyst loading must be carefully adjusted to simultaneously control the conversion and maximum power within certain needed limits.

Simulation examining the factors influencing capillary wick transport in a refrigerant direct cooling system for power battery packs

The effectiveness of power battery refrigerant direct cooling systems of electric vehicles incorporating capillary wicks is directly determined by these wicks’ transport performance. The Fries–Dreyer equation describes wicking behavior, but there is a significant gap between its predictions and the experimental results as reported in the literature. This work examines the factors influencing transport performance in an unconsolidated capillary wick with spherical particles. A mathematical and physical model is developed, the latter using the COMSOL software platform. Both the developed mathematical form and the numerically simulated results of this model are closer to the experimental results than those obtained using the Fries–Dreyer equation. The simulation results enable optimizing the equilibrium height and capillary time numbers providing a fitted Fries–Dreyer equation that is then used to analyze the influence of saturation, inclination angle, wick particle diameter, and tortuosity on the liquid rise mass and velocity and the equilibrium height, and the effects are in close but not perfect accord with experimental data. To narrow the gap, the Fries–Dreyer equation is further optimized using the numerically simulated results, substantially improving the accord with the experimental results.

Colloquium 5 Commentary on Katz

Abstract In this response to Emily Katz’s “Aristotle’s Rejection of Mathematized Metaphysics,” I raise questions about her central interpretive claim that mathematical forms cannot, for Aristotle, appear among first principles of nature. Topics addressed include the notion of priority, especially in the sciences; the relationship between natural change and material realization; and the general nature and scope of mathematical explanations for physical phenomena.

THE CREATION OF A REGRESSION MODEL BASED ON APPROXIMATION AND DISPERSION ANALYSIS OF STATISTICAL DATA

Problem statement. The most difficult issue when creating statistical models is the choice of a mathematical form of connection, that is, an analytical function that connects the elements of the system being studied. The form of the connection equation is established on the basis of theoretical, technological considerations or intuition. When it is difficult to imagine the dependence in advance, then a correlation field of points is built for two signs, the location of which on the plane determines the direction of action and the form of communication. For a deep and comprehensive study of statistical relationships, the concepts of correlation and regression are used. The task of correlation analysis is to establish the direction of action and the form of communication type. The task of regression analysis is to build a mathematical model of regression in the form of the resulting characteristic average value dependence on factor characteristics. The parameters of the regression model should be selected in such a way that the line constructed by the model passes between the points and is located as close as possible to all points of the correlation field, that is, it passes almost through its center. The purpose of the article is to create a regression model based on approximation, correlation and dispersion analysis of observational data. Results. Data approximation for multidimensional samples of active and passive experiments was performed, approximating functions and regression models in general were obtained based on them. On specific examples, the relationship between the factor features was established, the factor feature was selected. The most significant in terms of the connection closeness with the resulting feature, and a multivariate regression model suitable for forecasting was obtained. A multivariate dispersion analysis of the active experiment data was carried out on the research example of the influence on the homogeneity coefficient of concrete cement brand, type of aggregate, test period and “production period: of concrete. Analysis of dispersion shows that the most significant factors are the cement grade, the test period, the “making period” of the concrete and their minor interaction and the type of aggregate. The proposed technique significantly simplifies the process of creating a regression model. Conclusions. The performed calculations prove that: on the basis of the approximation and correlation of the passive experiment observations, it is possible to establish a relationship between the factor characteristics, to choose the factor characteristic that is the most significant in terms of the connection closeness with the resulting characteristic, to obtain a multivariate regression model suitable for forecasting, and this allows simplify the procedure for creating a regression model by successively moving from a simple model to a complex one; dispersion analysis of active experiment data makes it possible to assess the impact of individual factors and the impact of interaction between them. The approximation proves that the regression models of the active experiment data are linear. This is also confirmed by dispersion analysis, because the interaction of factors is insignificant.

The Accuracy of Computational Results from Wolfram Mathematica in the Context of Summation in Trigonometry

This article explores the accessibility of symbolic computations, such as using the Wolfram Mathematica environment, in promoting the shift from informal experimentation to formal mathematical justifications. We investigate the accuracy of computational results from mathematical software in the context of a certain summation in trigonometry. In particular, the key issue addressed here is the calculated sum ∑n=044tan⁡1+4n°. This paper utilizes Wolfram Mathematica to handle the irrational numbers in the sum more accurately, which it achieves by representing them symbolically rather than using numerical approximations. Can we rely on the calculated result from Wolfram, especially if almost all the addends are irrational, or must the students eventually prove it mathematically? It is clear that the problem can be solved using software; however, the nature of the result raises questions about its correctness, and this inherent informality can encourage a few students to seek viable mathematical proofs. In this way, a balance is reached between formal and informal mathematics.

The Relationship between Multiple Representational Skills and Understanding of Physics Concepts in the Pre-Service Science Teacher

The concept of physics has various representations that must be mastered by the Pre-Service Science Teacher (PSST). It is not uncommon for PSST to have poor multiple representation skills, impacting the delivery of concepts in school. Therefore, this study aims to analyze the profile of multiple representation skills and their relation to understanding the concept of PSST Physics. The representations studied include verbal, visual, symbolic, and mathematical forms. Eleven PSST became respondents in the study. Multiple representation and conception measurements use instruments with a three-tier item format containing content, argumentation, and confidence levels. Student responses are analyzed descriptively, qualitatively, and quantitatively. The findings of this study are: 1) 18% of PSST belong to the concept understanding, 27% experienced misconceptions, and the rest (55%) were classified as not knowing the concept or responding by guessing; 2) the highest representation format mastered by students is the visual representation, and the lowest is the mathematical representation; and 3) the ability to multiple representations and understand physics concepts has a significant and perfect correlation with a Pearson Correlation of 0.847. This research implies that learning for PSST can emphasize multiple representation abilities as it affects their understanding of concepts.

Basic Symbolic Number Skills, but Not Formal Mathematics Performance, Longitudinally Predict Mathematics Anxiety in the First Years of Primary School.

Mathematical anxiety (MA) and mathematics performance typically correlate negatively in studies of adolescents and adults, but not always amongst young children, with some theorists questioning the relevance of MA to mathematics performance in this age group. Evidence is also limited in relation to the developmental origins of MA and whether MA in young children can be linked to their earlier mathematics performance. To address these questions, the current study investigated whether basic and formal mathematics skills around 4 and 5 years of age were predictive of MA around the age of 7-8. Additionally, we also examined the cross-sectional relationships between MA and mathematics performance in 7-8-year-old children. Specifically, children in our study were assessed in their first (T1; aged 4-5), second (T2; aged 5-6), and fourth years of school (T3; aged 7-8). At T1 and T2, children completed measures of basic numerical skills, IQ, and working memory, as well as curriculum-based mathematics tests. At T3, children completed two self-reported MA questionnaires, together with a curriculum-based mathematics test. The results showed that MA could be reliably measured in a sample of 7-8-year-olds and demonstrated the typical negative correlation between MA and mathematical performance (although the strength of this relationship was dependent on the specific content domain). Importantly, although early formal mathematical skills were unrelated to later MA, there was evidence of a longitudinal relationship between basic early symbolic number skills and later MA, supporting the idea that poorer basic numerical skills relate to the development of MA.

Stiffness-based stress–strain model of FRP-confined high-strength and ultra-high strength concrete column with various corner radii

This study introduces a unified design-oriented stress–strain model tailored for fiber-reinforced polymer (FRP)-confined high-strength concrete (HSC) with diverse cross-sectional shapes. The proposed stress–strain model incorporates more suitable mathematical forms, incorporating post-peak axial stiffnesses as crucial parameters, thereby enabling the model to accurately capture the three types of post-peak stress–strain behavior observed in FRP-confined HSC: strain-hardening, strain softening, and strain softening recovering to strain hardening. This variation in post-peak axial stiffness arises due to insufficient FRP confinement at the initial peak strength point, as well as changes in the confinement stiffness ratio stemming from the altered dilation behavior of FRP-confined HSC during the post-peak stage. To address the nonuniform effects of FRP confinement, the concept of effective confinement rigidity was defined, unifying the behavior of FRP-confined circular and square HSC. Additionally, the model incorporates the ultimate condition and the characteristic point on the post-peak branch, representing the change in FRP confinement stiffness, to establish the stress–strain model for FRP-confined HSC. The model parameters were evaluated using available experimental results, and the stress–strain curves derived from the proposed model were compared with existing models in the literature, revealing the proposed model's accuracy and rationality.

Frequency-domain expression of sound pressure field radiated by monopole source with general moving trajectory

Frequency-domain expressions of sound pressure field radiated by moving source were derived successfully only for some special moving trajectories in the literature. In this paper, a frequency-domain expression of sound pressure field radiated by monopole source with general moving trajectory is firstly derived. This expression has a simple mathematical form, where the sound pressure spectrum is written as a Fourier transformation of free-field frequency-domain Green functions along the trajectory. This expression is subsequently verified through theoretical and numerical verification. In theoretical verification, this expression is degenerated into the known analytical solution in each case with special moving trajectories. In numerical verification, the sound pressure spectra calculated via this expression are consistent with those calculated via a time-domain numerical method in cases without known analytical solutions. This expression is finally applied to enable the mode composition beamforming method to localize a rotating source with time-variant rotation speed.

Dynamical analysis of nonlinear combined drug therapy model for HIV infection: Bayesian regularization technique intelligent networks

The study aims to provide an overview of the design and analysis of the differential system of the Nonlinear Combined Drug Therapy Mathematical (NCDTM) model for HIV Infection. This is achieved by utilizing the artificial Neural Network with Bayesian Regularisation technique intelligent network (ANNs-BRTIN), which is a stochastic method that enhances accuracy, reliability, and efficiency in the dynamics calculation process. The NCDTM model is defined along with experiments employing integer and nonlinear mathematical forms via four classes Where the number of uninfected cells in the blood and represented by A(t) the countable number of free virus particles in the blood is represented by S(t) when RTI therapy is administered denotes the number of infected cells in the Pre-RT phase with η dosage represented by I(t), D(t) represents when RTI therapy is administered using, reflects the number of infected cells in the post-RT phase η dosage. The numerical computing of the NCDTM system is performed with Adams methods, and these results are fed to the proposed ANNs-BRTINs for finding the approximated solution of four distinct instances by incorporating 15% of the data for testing and validation and 85% for training. The presented ANNs-BRTINs accuracy is illustrated by comparing the results from the acquired dataset of the Adam method for sundry scenarios. The use of regression for testing and residual, state transitions, mean square error, error histograms, and correlation in numerical replications of the ANNs-BRTINs is also explored to confirm their capacity, validity, consistency, correctness, and competence.

Scaffolding practices for modelling instruction in STEM-related contexts: insights from expert and novice teachers

AbstractMathematical modelling (MM) plays a pivotal role in the integration of Science, Technology, Engineering and Mathematics (STEM) into school studies. This current empirical study suggests using a four-step solution plan as a scaffolding tool during the instruction of MM tasks in a STEM context in formal school mathematics. The study goals are twofold; first to recognise MM-oriented scaffolding practices of teachers during observations of their instruction of MM tasks. Second, to explore the differences in type and quantity of the recognised scaffolding practices, while comparing MM task instruction in a STEM context of expert and novice teachers. Using a multiple case study design, we conducted an in-depth study of MM scaffolding practices implemented during instruction of three MM tasks in a STEM context (the ‘Mobileye’, ‘Gamma correction’, and ‘GPS’ MM tasks) taught by three expert teachers and five novice teachers, in a total of five lessons for each group. Findings revealed three types of practices to support a particular solution plan step, enact the transition between solution plan steps, and motivate students' MM learning. A similar distribution of practices between expert and novice teachers was revealed, while support in each step of the solution plan was evident to all, especially during the mathematical search step. The study provides a perspective on the necessary practices that may take place in modelling instruction in a STEM context.

Extending the cystatin C based EKFC-equation to children - validation results from Europe.

A new cystatin C based European Kidney Function Consortium (EKFCCysC) equation was recently developed for adults, using the same mathematical form as the previously published full age spectrum creatinine based EKFC-equation (EKFCCrea). In the present study the cystatin C based EKFC-equation is extended to children, by defining the appropriate cystatin C rescaling factor QCysC. Rescaling factor QCysC for cystatin C was defined as: a) 0.83mg/L, exactly as it was defined for young adults in the adult equation, and b) a more complex QCysC-age relationship based on 4th degree cystatin C-age polynomials after evaluation of data from Uppsala, Stockholm and Canada and aggregated data from Germany. The EKFCCysC equation was then validated in an independent dataset in European children (n = 2,293) with measured GFR, creatinine, cystatin C, age, height and sex available. The EKFCCysC with the simple QCysC-value of 0.83 had a bias of -7.6 [95%CI -8.4;-6.5] mL/min/1.73 m2 and a P30-value of 85.8% [95%CI 84.4;87.3] equal to the EKFCCysC with the more complex 4th degree QCysC-value. The arithmetic mean of the EKFCCrea and EKFCCysC with the simple QCysC of 0.83 had a bias of -4.0 [95%CI -4.5;-3.1] mL/min/1.73 m2 and P30 of 90.4% [95%CI 89.2;91.6] similar to using the more complex 4th degree QCysC-polynomial. Using exactly the same QCysC of 0.83mg/L, the adult EKFCCysC can easily be extended to children, with some bias but acceptable P30-values. The arithmetic mean of EKFCCrea and EKFCCysC results in bias closer to zero and P30 slightly over 90%.

A proposed composite Boussinesq equation for estimating baseflow recessions and storage-outflow relationship

Baseflow recession analysis characterizes the interactions between ground and surface water, and many studies have focused on estimating the mathematical form of the relationship and its parameters. The Boussinesq equations can estimate baseflow recession and storage-outflow relationship. However, these equations assume either linear or nonlinear behavior, depending on the streamflow conditions. The linear equation cannot represent the nonlinear behavior of the baseflow, and the nonlinear equation cannot fit the lowest discharge values. In addition, the nonlinear equation is oversimplified and does not include baseflow from groundwater storage below the stream’s water level. Therefore, we propose a composite Boussinesq equation that includes the linear portion of baseflow below the stream’s water level, includes a nonlinear storage, and fits low discharge values. The performance of the linear, nonlinear, and composite methods are assessed using 1561 USGS stream gaging stations across the contiguous US (CONUS). The regression constants, which represent the aquifer characteristics, are estimated for all gages using the three methods. Results show that the composite equation results in smaller errors for estimating baseflow recession events and the storage-outflow relationship. Finally, four different variables, precipitation, evapotranspiration, average discharge, and drainage area, were analyzed to find the relationship between the watershed conditions and baseflow analysis in terms of the regression constants and the relative errors of the best fit curves. Results show that the nonlinearity increases in drier regions, and the relative error for all three methods increases in wet climate regions with high precipitation and evapotranspiration. The composite equation is a well-performing alternative to linear or nonlinear equations across all basin characteristics, allowing for estimating the full range of baseflow behavior.

Generic examples in undergraduate topology: a case study

Expert mathematicians use examples and visual representations as part of their informal mathematics reasoning when constructing proofs. In contrast, the ways undergraduates reason and work toward creating proofs is an open area of investigation, particularly in advanced mathematics. Furthermore, research on students’ reasoning in topology is largely unexplored. Building on findings from Gallagher [(2020). Identifying Structure in Introductory Topology: Diagrams, Examples, and Gestures. Doctoral Dissertation, West Virginia University, Morgantown, West Virginia. https://researchrepository.wvu.edu/etd/7599/ (MS #8610)], we present a case study of an undergraduate taking a first course in general topology and her use of generic examples in argumentation prior to producing formal proofs and counterexamples. We discuss patterns that emerged in her use of generic examples when proving true statements and disproving false claims. We compare and contrast this student’s generic examples with other generic examples in the literature from other content areas within mathematics, and we argue that generic examples are not one-size-fits-all but rather must be fit-for-purpose. Our results suggest that generic examples may be particularly useful for students writing proofs when they may not have access to specific examples.

MHD hybrid nanofluid flow over a stretched surface with convective boundary conditions: Applications of heat transfer

The main objective of this research is to analytically study an innovative, viscous, time-independent, and incompressible (HFN) hybrid nanofluid flow on a slanderous stretched surface. This study examines thermal conductivity, heat absorption/omission, thermal radiation, and the magnetic effect of in the existence of HFN. Incorporating an external heat source into the heat transfer process increases the effect of the dissipative heat and thermal radiation. The flow characteristics are governed by the mathematical form which is further transferred to ordinary differential equations (ODEs) via appropriate assumptions regarding similarity variables and stream functions. The strong nonlinearity was successfully solved using the analytical technique homotopy analysis method (HAM) for these nonlinear equations. The results and discussion section provides a thorough explanation of the main outcomes of the study. This research focuses on the enhancement of heat transfer and the effects.

An Introduction to the Weak Gravity Conjecture

The Weak Gravity Conjecture holds that gravity must be the weakest force. This is true of the familiar forces in our own universe—electromagnetism, for instance, is many orders of magnitude stronger than gravity. But the bold claim of the Weak Gravity Conjecture is that this statement is true, not only for electromagnetism in our universe, but for any similar force in any consistent universe governed by quantum mechanics. In this brief introduction, aimed at advanced undergraduates or beginning graduate students, we elaborate on the precise definition of the Weak Gravity Conjecture, the evidence for it, and some of the remarkable implications of it. We explain how the Weak Gravity Conjecture may play a role in bridging the gulf between the formal mathematics of string theory and the real-world observations of particle physics and cosmology.