Mathematical Diagrams Research Articles

The manuscript diagrams of Theodosios’ Spherics

This paper is a study of the visual characteristics of the medieval manuscript diagrams found in Theodosios’ Spherics. It provides a new taxonomy of eight types of diagrams found in the manuscript sources, and examines the way in which geometric information is encoded in the manuscript diagrams using a number of visual encodings. It then argues that these diagrams can be used to address ongoing research debates about whether the striking visual characteristics of medieval diagrams in mathematical texts are the result of deliberate choices by the premodern authors, the result of accidents of transmission, or both. Along the way, the paper identifies a further type of transmission error that is prevalent in the medieval transmission of mathematical diagrams.

Who's Afraid of Mathematical Diagrams?

Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions.   Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs.  In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that diagrams form genuine notational systems, and I argue that this explains why they can play a role in the inferential structure of proofs without undermining their reliability.  I then consider whether diagrams can be essential to the proofs in which they appear.@font-face{font-family:"Cambria Math";panose-1:2 4 5 3 5 4 6 3 2 4;mso-font-charset:0;mso-generic-font-family:roman;mso-font-pitch:variable;mso-font-signature:-536870145 1107305727 0 0 415 0;}@font-face{font-family:Calibri;panose-1:2 15 5 2 2 2 4 3 2 4;mso-font-charset:0;mso-generic-font-family:swiss;mso-font-pitch:variable;mso-font-signature:-536859905 -1073697537 9 0 511 0;}p.MsoNormal, li.MsoNormal, div.MsoNormal{mso-style-unhide:no;mso-style-qformat:yes;mso-style-parent:"";margin:0in;line-height:200%;mso-pagination:widow-orphan;font-size:12.0pt;font-family:"Calibri",sans-serif;mso-fareast-font-family:Calibri;}.MsoChpDefault{mso-style-type:export-only;mso-default-props:yes;font-family:"Calibri",sans-serif;mso-ascii-font-family:Calibri;mso-fareast-font-family:Calibri;mso-hansi-font-family:Calibri;mso-bidi-font-family:Calibri;}.MsoPapDefault{mso-style-type:export-only;line-height:200%;}div.WordSection1{page:WordSection1;}

Application of random forest-based decision tree approach for modeling fully developed turbulent flow in rough pipes

A random forest (RF) -based decision tree programming methodology was aimed for modeling fully developed turbulent flow conditions in rough pipes. In the present computational study, a flexible RF-based soft-computing strategy was applied for the estimation of the required pipe diameter (D) and Darcy–Weisbach friction factor (λ or f) obtained from the iterative solution of the implicit Colebrook–White equation for five basic pipeline design variables considered in sizing problems (Type 3) of pipe distribution systems. The prediction performance of the implemented RF-based model was assessed more than 15 different statistical goodness-of-fit parameters and useful mathematical diagrams such as box-and-whisker-plots and spread plots. The statistical metrics corroborated the superiority of the RF-based approach in predicting both the required pipe diameter (R2 = 0.9793, MAE = 0.0287 m, RMSE = 0.03833 m, SEE = 0.0326 m, IA or WI = 0.9933, CV(RMSE) or SI = 0.0595, NSE = 0.9753, LMI = 0.8482, and AIC = -1954.6438 for the testing dataset) and friction factor (R2 = 0.9576, MAE = 0.0011, RMSE = 0.0023, SEE = 0.0018, IA or WI = 0.9851, CV(RMSE) or SI = 0.0660, NSE = 0.9478, LMI = 0.8500, and AIC = -3646.7124 for the testing dataset). The descriptive statics suggested that the 25% percentile values (Q1), median values (Q2), and 75% percentile values (Q3) of RF-predicted values of D and λ and the corresponding actual values of these responses were found to be very close. The proposed RF-based model was also tested against additional some dataset obtained from the relevant literature. The validation results indicated that the applied decision tree-based method produced realistic estimations and acceptable statistics (i.e., R2 = 0.9624, MAE = 0.0598 m, and RMSE = 0.0708 m for D values, and R2 = 0.9130, MAE = 0.0043, RMSE = 0.0052 for λ values) even at extreme L values greater than 2000 m. This study demonstrated the importance and ability of the applied soft-computing strategy to accurately predict D and λ values and eliminated error-prone steps of the traditional iterative approach.

Out of Sight, but Not Out of Mind: A Diagrammatic Conversation on Relational Drawing

Abstract The authors reflect on their ongoing dialogue about connections between drawing in the practice of art and in the practice of mathematics. They show how visualizing relational connections in mathematical diagrams can be used to contextualize aspects of Heath’s drawings. The account provides a sequence of loops, reflecting on drawings where relational marks coexist with those in which they do not. Diagrams are the vehicle for this dialogue across the disciplines of art and science, acting as boundary objects supported by a common visual vocabulary of loops, connections, change over time, negative space, and a shared interest in exploring ideas through the physicality of practice.

Implementation of the Learning Management Model Based on Cognitive Development Theory to Enhance Mathematical Problem-Solving Ability for Prathomsuksa 6 Students

This study investigates the implementation of the learning management model based on cognitive development theory to enhance mathematical problem-solving ability for Prathomsuksa 6 students. The sample group assigned by cluster random sampling was thirty Prathomsuksa 6 students of Chumchon Yot Kaeng Songkhro School. The research instruments consist of learning achievement tests and mathematical problem-solving ability tests (pre-test and post-test). The statistical analyses were percentage, mean, standard deviation, and t-test (dependent samples) was employed for testing hypothesis. The results showed that students’ learning achievement after learning with the learning management model was significantly higher than before learning at a statistical level of .05. Also, the students had the mathematical problem-solving ability after learning significantly higher than before learning at a .05 level. When considering in each aspect, it was found that all of the mathematical problem-solving ability after learning was higher than before learning at a statistical level of .05, namely, 1) finding of the problem relationship; 2) writing of mathematical diagrams; 3) problem solving; and 4) traceability.

What are mathematical diagrams?

Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term “mathematical diagram” is used in diverse ways. I propose a working definition that carves out the phenomena that are of most importance for a taxonomy of diagrams in the context of a practice-based philosophy of mathematics, privileging examples from contemporary mathematics. In doing so, I move away from vague, ordinary notions. I define mathematical diagrams as forming notational systems and as being geometric/topological representations or two-dimensional representations (or both). I also examine the relationship between mathematical diagrams and spatiotemporal intuition. By proposing an explication of diagrams, I explain (away) certain controversies in the existing literature. Moreover, I shed light on why mathematical diagrams are so effective in certain instances, and, at other times, dangerously misleading.

SIMULATION OF TROLLEBUS STEERING SYSTEM WITH ELECTRIC POWER STEERING BASED ON ROLLING ROTOR SWITCHED RELUCTANCE MOTOR

The analysis of power steering which used on modern rolling stock is carried out. Their main shortcomings are identified. Given the requirements for the steering of trolleybuses, a solution to increase its efficiency is proposed. Based on the developed mathematical model and functional diagrams of the trolleybus steering system with a electric power steering based on rolling rotor switched reluctance motor, a simulation model of trolleybus steering was created using the Matlab Simulink package. The peculiarity of the simulation model is taking into account the mass and size characteristics of the rolling stock, the parameters of the suspension of the steered axle, the impact of the road surface and the speed of the trolleybus, changes in the parameters of the electric motor magnetic system during operation. The rolling rotor switched reluctance motor which is offered for use as the electric power steering of the LAZ E183D1 trolleybus is calculated. The calculation of the magnetic system of the engine by the finite element method with the subsequent approximation of the obtained results is carried out. With the help of the developed simulation model the simulation of the trolleybus steering system with electric power steering based on rolling rotor switched reluctance motor was performed. Time diagrams of transients in the trolleybus steering system are obtained and their processing and analysis are carried out.

Postgraduate Mathematics Education Students' Perceptions of Technology-based Tools and Resources: Exploring the Influences of Connectivism and the Three Worlds of Mathematics

As we embrace the Fourth Industrial Revolution, transformations in education necessitate the use of technology-based pedagogy. This study explored postgraduate mathematics education students' perceptions of using technology-based tools and resources. This qualitative and interpretive study was conducted at a university in KwaZulu-Natal, South Africa. The study was framed utilizing the theory of connectivism and the three worlds of mathematics theory. Connectivism is a learning theory that is considered to be strengthened by existing technology-based tools and is founded on the notion that ubiquitous Information Communication Technology impacts teaching and learning. The three worlds of mathematics theory support the view that the development of mathematical thinking is founded on perception, operation and reasoning. Postgraduate mathematics education students engaged in interactive workshops using different online platforms. Subsequently, a sample of participants was interviewed using a semi-structured interview schedule. The interviews were conducted via various online platforms. Data that were generated were analyzed thematically. The results revealed two main themes. These themes focused on the strengths and challenges of using technology-based tools and resources to learn mathematics as perceived by the participants. The results of this study provide evidence that technology-based tools and resources may also be considered by lecturers when teaching mathematics topics that involve drawing complex mathematical diagrams and solving problems based on these complex mathematics diagrams. Also, the results support the view that to alleviate challenges associated with using technology-based tools and resources, lecturers need to ensure that the technology-based tools and resources being used are user-friendly, inexpensive, readily available and data-efficient. Moreover, this study support the view that lecturers need to consider attending professional development workshops and seminars that focus on integrating technology effectively in educational environments. This study has relevance globally and nationally when considering the implications of using technology-based tools and resources for learning mathematics.

A Research Synthesis Using Instrumental Learning Trajectories: Knowing How and Knowing Why

In the current study the theoretical notion of instrumental learning path or trajectory is analyzed through examples based on a research synthesis. I point out the role of instrumental decoding in a static or dynamic environment, and how the competence of the participants (students –researcher/ teacher) can influence the holistic result of the learning process by creating interdependencies/intra-dependencies during the construction of instrumental learning trajectories. Instrumental trajectories are not just construction instructions, or a set of information which provides the properties of the figure as the figure is constructed. Instrumental trajectories can show us the interdependencies/intra-dependencies that exist or can be created between different tools. Dynamic Geometry allows for the creation of interdependencies and intra-dependencies between mathematical objects, diagrams and tools. In the sections that follow, I shall present three examples of instrumental learning trajectories, in which the interdependencies among the tools and meanings are analyzed. My aim is to combine and synthesize different primary qualitative research studies and make their results more generalizable. Keywords: instrumental decoding, interdependencies/intra-dependencies, instrumental learning trajectories DOI: 10.7176/IKM/11-3-02 Publication date: April 30 th 2021

Instructional Exposure of Senior High School Students to Approaches that Promote Critical Thinking and Problem-Solving Skills

This descriptive study assessed students’ exposure to activities that promote the acquisition of critical thinking and problem-solving skills. The study involved 150 randomly selected senior high school students. Results reveal that students have a “great extent” of exposure to instruction that promotes the acquisition of the desired mathematics skills, particularly on the content, strategies, assessment, and instructional materials. Students are exposed to a “great extent” to activities that allow them to predict, gather and organize information, derive conclusions, make judgment or decisions, discuss and justify solutions; however, have a “low extent” of exposure to problem posing. Students were exposed to a “very great extent” to the step by step process in dealing with problems. The students have a “great extent” of exposure to problem-solving strategies that promote verbal-logical, visual-spatial, and organizing skills, however, to a low extent in restructuring and rethinking skills. Students’ exposure to the use of manipulatives or mathematical models, calculators, creative pictures and diagrams, worksheets, online materials, creative PowerPoint presentations, varied textbooks, and scholarly materials in mathematics problem solving was to a “great extent” but “low” on the use of mathematics software and videos. Along assessment, students have a “great extent” of exposure to assessment on knowledge, comprehension, application, synthesis, and evaluation skills but to a “low extent” on assessment that requires metacognitive and reflective thinking.

Establishment the program and mathematical diagrams of embankment stability analysis on soft soil reinforced by soil cement columns

In the calculation of improvement of embankments on soft soils, the geometrical parameters of soil cement columns such as the length L, diameter d, the distance between the columns D, greatly affect to the stability of embankments (settlement S, coefficient of stability Fs) and treatment efficiency. On the basis of the multivariate correlation equation predicting total settlement (S), residual settlement after treatment (DS) based on the unit weight of soil , the height of embankments H, the length L, diameter d, the distance between the columns D after statistical analysis on p_value index and R2 with the following models: Linear, Quadratic combined with the Bishop method on slope stability analysis, the authors have built the code of stability analysis for the embankment on soft ground reinforced with soil cement columns system. At the same time, we have developed mathematical diagrams for the analysis of the influence of each pair of column geometry factors on the settlement of the roadbed.

Why Were Greek Mathematical Diagrams Schematic?

Abstract This article presents the case for the claim that Greek mathematical diagrams were schematic. Following a deep dive into the practices and the process of transmission of those diagrams, the article situates Greek diagrammatic practices within the broader context of Greek scribal and readerly practices. The literary papyrus bookroll was produced and read as a tool for the projection of an imagined performance; so was the specialized type of bookroll containing mathematical proofs.

Penrose

We introduce a system called Penrose for creating mathematical diagrams. Its basic functionality is to translate abstract statements written in familiar math-like notation into one or more possible visual representations. Rather than rely on a fixed library of visualization tools, the visual representation is user-defined in a constraint-based specification language; diagrams are then generated automatically via constrained numerical optimization. The system is user-extensible to many domains of mathematics, and is fast enough for iterative design exploration. In contrast to tools that specify diagrams via direct manipulation or low-level graphics programming, Penrose enables rapid creation and exploration of diagrams that faithfully preserve the underlying mathematical meaning. We demonstrate the effectiveness and generality of the system by showing how it can be used to illustrate a diverse set of concepts from mathematics and computer graphics.

The Computer-Aided Rough Patterns of Christopher Alexander

This paper analyzes Christopher Alexander’s combined use of mathematical graphs and hand-made diagrams, and argues that such affinities marked the insertion of roughness into architectural computational thinking. Within the techno-scientific context of American postwar architecture, the techniques of transcription and calculation used by Alexander at the Center for Environmental Structure reveal the progressive erasure of determinacy that took place within an architecture practice with empiricist, mathematic and computational preferences. Rather than establishing an optimized and quantified standard to which architecture had to conform, Alexander’s rough diagrams and mathematical graphs serialized variation and provided room for indeterminacy and contingency within a clearly defined set of rules.

Constructing and Unpacking Diagrams in Geometry

By having students practice constructing diagrams for geometric theorems, teachers can develop students' understanding of mathematical claims, vocabulary, and notation methods. This practice can also strengthen students' ability to interpret mathematical diagrams and recognize their limitations.

Vestiges of the emergence of overspecification and indifference to visual accuracy in the mathematical diagrams of medieval manuscripts

Vestiges of the emergence of overspecification and indifference to visual accuracy in the mathematical diagrams of medieval manuscripts

The Complexity Classes of Angular Diagrams of the Metal Conductivity in Strong Magnetic Fields

We consider angular diagrams describing the dependence of the magnetic conductivity of metals on the direction of the magnetic field in rather strong fields. As it can be shown, all angular conductivity diagrams can be divided into a finite number of classes with different complexity. The greatest interest among such diagrams is represented by diagrams with the maximal complexity, which can occur for metals with rather complicated Fermi surfaces. In describing the structure of complex diagrams, in addition to the description of the conductivity itself, the description of the Hall conductivity for different directions of the magnetic field plays very important role. For the evaluation of the complexity of angular diagrams of the conductivity of metals, it is convenient also to compare such diagrams with the full mathematical diagrams that are defined (formally) for the entire dispersion relation.

Model pathway diagrams for the representation of mathematical models

Mathematical models are the foundation of numerical simulation of optoelectronic devices. We present a concept for a machine-actionable as well as human-understandable representation of the mathematical knowledge they contain and the domain-specific knowledge they are based on. We propose to use theory graphs to formalize mathematical models and model pathway diagrams to visualize them. We illustrate our approach by application to the van Roosbroeck system describing the carrier transport in semiconductors by drift and diffusion. We introduce an approach for the block-based composition of models from simpler components.

Galileo’s Two New Sciences as a Model of Reading Practice

Galileo’s 1638 Two New Sciences, a canonical text of early modern science, is analyzed as a window into period practices of mixed-mathematical reading. Galileo’s depiction of reading reflects common scholarly practices, including those of summarizing, commenting, repeated study, and an interest in mathematical diagrams. With this text, Galileo also attempted to shape his readers’ practices, inciting them to approach topical-based reading strategies with care and to use experiment and experience to validate the written word. It is suggested that the concern with reading practices exhibited in the text derived from circumstances surrounding Galileo’s 1633 condemnation.

Decrease in Power Consumption of Hydraulic Systems of Modern Machine Tools

This paper shows several issues related to decrease in electric power consumption for hydraulic systems of modern machine tools, especially CNCs. A core element in the structure of such systems is the pneumo-hydraulic accumulator. Here, examples are shown for hydraulic systems destined to locking and unlocking of skids and to opening of tool holders. This paper shows mathematical models, diagrams, simulations and certain experimental executions.