Proposition Research Articles

Dummettʼs Anti-Realism about Mathematical Statements

Dummettʼs Anti-Realism about Mathematical Statements

Markov chain modelling of ordered Rayleigh fading channels in non‐orthogonal multiple access wireless networks

AbstractA first‐order finite‐state Markov chain (FSMC) typically models the Rayleigh fading channel in the open literature because the first‐order FSMC is analytically tractable and can derive closed‐form results. Non‐orthogonal multiple access (NOMA) has been recognised as a novel wireless technology that addresses challenges in the next generation of mobile communications. According to the power‐domain NOMA protocol, channels in the NOMA wireless network are sorted by the channel gain. Then considering NOMA, there is insufficient information on how to further form a suitable model for ordered Rayleigh fading channels based on the first‐order FSMC. Given the mathematical statement on how to model the order statistics of multidimensional Markov chains for ordered Rayleigh fading channels, the authors consider these order statistics as a Markov chain, and propose specific processes of representing the state space and constructing the transition probability matrix accordingly. Numerical and simulation results validate the mathematical correctness and accuracy of these novel processes. In addition, for ordered Rayleigh fading channels, the performances of various methods of partitioning the entire signal‐to‐noise ratio range are compared. The performance comparison results are the same as those obtained for the individual unordered Rayleigh fading channel.

Assessing the content typicality and construct of Persian language proficiency test (PLPT) for non-Persian speakers: a corpus-informed study

Drawing on a growing body of research on the interface between corpus linguistics and second/foreign language testing and assessment, we adopted Peykare, a large-scale, annotated, Persian written language resource to evaluate the content (i.e., coverage and typicality) and construct validity of a Persian language proficiency test developed for certification of proficiency in Persian as a foreign language (PFL) of non-native speakers. Designed at the Research Center for Intelligent Signal Processing (RCISP), Peykare contains 35,058 text files over five linguistic varieties and 24 different registers of contemporary Persian. This study addresses how corpora, as rich database resources, can practically be applied to test validation purposes and insightfully inform the test life cycle. The results of content validity phase revealed evidence supporting content representativeness, relevance, and typicality of the test. The linkage between the corpus-extracted criterial features or parameters and those covered by the test was not, however, strongly evidenced by items measuring ezafeh constructions, homographs/homophones, PRO (proposition), and POST (postposition). The analysis of content typicality indicated chunks that did not closely conform to the corpus typical output. The construct validity phase, assessing the test hypothesized factor structure (i.e., hierarchical, unitary, correlated, and uncorrelated models) in two randomly split samples of PFL learners from Asian and European countries (N=121), showed that the correlated model fit the data best in both samples. The results supported the presence of distinctive factors of receptive skills, providing empirical evidence for score interpretations of the corpus-based test.

Standardized PUEBI EYD V’s Regularity in Formal Writing of Mathematical Existential Statement Consequences on Scientific Ontological Theorization to Indonesian Scientific Community

<p class="Abstract">This paper discusses the consequences of the latest PUEBI EYD
 V regulations for scientific ontological theorization through analyzing the
 semantical metaphysical commitment it reflects when we write formal mathematical
 statements using purely mathematical symbol (e.g., “there are 22 aardvarks”).
 This paper shows that PUEBI EYD V commits to mathematical Platonism
 metaphysically. This commitment brings harm to observable entities ontological
 nature in scientific theorization as shown in nominalism projects of philosophy
 of mathematics. Scientific theories - and even mathematical theories - should
 always reject the existence of independent objects for there exists only
 structures (as truth-value). Authors use nominalism stance as a methodology to
 reject the metaphysical commitment of PUEBI and defend the formal usage of
 writing mathematical statement without number symbols (“there are twenty-two
 aardvarks”, etc.).<o:p></o:p></p>

Ninety-Nine Variations on a Proof

Reviews in Common Knowledge generally seek to be more cool and edgy than their subjects, an impossibility in this case. Ording takes a mathematical statement and reaches it in ninety-nine different ways. This book is quite literally a page-turner: most of the arguments take the recto page, with comments on their verso. One keeps cycling back and forth between the mathematical inventiveness of the recto and the philosophical elegance of the verso. The ambition is huge—to construct a mathematical counterpart to Queneau's masterpiece Exercises de Style—and, remarkably, this ambition is fulfilled.It is interesting that Queneau's variations are much closer to each other than Ording's are. Queneau often feels free to provide, as a variation, the very same text with a fundamentally grammatical transformation (switching the order of the descriptions, changing their tenses, and so forth). Most of the time, Ording feels obliged to come up with a significantly different argument. Indeed, his fixed core is less constraining: instead of following the same narrative arc in different forms, he merely has to reach the same endpoint. Ording's book is rather like ninety-nine different jokes leading to the same punch line. To be clear: it is perfectly possible to produce the precise same narrative arc, in mathematics, via different forms. Switching the order is a normal way in which such exercises in mathematical style are actually produced (mathematicians do have to decide whether to write “P, therefore Q” or to write “Q, since P”). Evidently, Ording felt that producing many such purely grammatical and formal transformations would feel tedious, which is fair enough but is not precisely a contrast between mathematics and literature. Queneau's variations are, indeed, tedious; delightfully tedious; delightful in their tediousness. Joyce as a tasting menu. Some readers of modernist literature can take delight in a certain tediousness, but Ording has no illusions concerning the audience for mathematics, and, to make his own variations appealing, he had to make them much more interesting than Queneau's. The result is brilliant and should be read by anyone interested in mathematics, literature, and their many parallels and variations.

Standardized PUEBI EYD V’s Regularity in Formal Writing of Mathematical Existential Statement Consequences on Scientific Ontological Theorization to Indonesian Scientific Community

<p class="Abstract">This paper discusses the consequences of the latest PUEBI EYD
 V regulations for scientific ontological theorization through analyzing the
 semantical metaphysical commitment it reflects when we write formal mathematical
 statements using purely mathematical symbol (e.g., “there are 22 aardvarks”).
 This paper shows that PUEBI EYD V commits to mathematical Platonism
 metaphysically. This commitment brings harm to observable entities ontological
 nature in scientific theorization as shown in nominalism projects of philosophy
 of mathematics. Scientific theories - and even mathematical theories - should
 always reject the existence of independent objects for there exists only
 structures (as truth-value). Authors use nominalism stance as a methodology to
 reject the metaphysical commitment of PUEBI and defend the formal usage of
 writing mathematical statement without number symbols (“there are twenty-two
 aardvarks”, etc.).<o:p></o:p></p>

The architecture of a reasoning system for Defeasible Deontic Logic

We present the architecture of Houdini-2.0, a reasoning system that computes the extension of a defeasible deontic theory given as input, the process of computing the consequences of the rules expressed in the theory itself. The decision process is a sceptical, non-monotonic, and it allows us to determine which prescriptive behaviours are in force (obligations, permissions, prohibitions) along with propositional ones. The system is based on pre-existing algorithmic solutions, and it is implemented as an online platform to deploy the results of a computation in several use cases, including those that pertain legal domain.

ПОСТАНОВКА ЗАДАЧИ ОПТИМИЗАЦИИ МАРШРУТОВ ГРУЗОВЫХ МУЛЬТИМОДАЛЬНЫХ ПЕРЕВОЗОК

The article deals with the urgent task of increasing the efficiency of multimodal transportation of goods based on the optimization of delivery routes. The task is formulated in terms of the subject area of interest to the transportation customer. The mathematical statement of the problem is presented in the form of a graph model that takes into account dynamic changes in the requirements of both the consignor and the consignee. According to the authors, this approach to solving the logistics problem allows the multimodal operator to respond to changes in a timely manner and gives some advantages when choosing an effective route in the face of sanctions pres-sure and border closures.

Mathematics symbol instruction and senior secondary students’ achievement in word problems: A quasi-experimental study

An assessment of the possible effect of mathematics symbol instruction on achievement in word problems was the main thrust of the study. Two null hypotheses were tested in the study.<b> </b>The non-equivalent group pre-test-post-test quasi-experimental research design was adopted. The multistage sampling technique was used to select a sample of 387 participants from 6,740 senior secondary class one students in the 81 government-owned coeducational secondary schools in Calabar education zone of Cross River State. The treatment class was exposed to mathematics symbol instructional strategy for solving word problems while the control class was exposed to regular word problem teaching, otherwise called the conventional method. “Students mathematics achievement test” was the instrument used for data collection after validation by experts. The reliability coefficient of 0.90 was determined using the Kuder-Richarson 20 formula. Primary data were obtained after informed consent from the participants. The null hypotheses were all tested at the .05 alpha level using the analysis of covariance. Findings revealed a significant effect of the mathematics symbol instructional strategy on students’ word problem achievement in secondary schools. Male students perform significantly better in word problems than their female counterparts when taught using the mathematics symbol instruction. These findings discussed fundamental theoretical, practical, and research implications. It was recommended, among other things, that mathematics teachers concentrate on demystifying the abstraction in mathematics by deliberately adopting mathematics symbol instructional strategy in interpreting mathematical statements and sentences into their symbolic notations before engaging in their computation.

Nonlinear bending of polygonal functionally graded plates resting on elastic foundation

To investigate nonlinear bending of the functionally graded (FGM) plates with complex shape and resting on elastic foundation a variational-structural method (RFM) is proposed. Mathematical statement of nonlinear boundary value problems of plate bending is carried out in the framework of the classical geometrically nonlinear plate theory. To solve a sequence of linear boundary value problems obtained as a result of the linearization of the original nonlinear system of equations by the method of successive loadings and the Newton method, the method of R-functions was applied. Testing, as well as comparison of the obtained results with the results of other authors, made it possible to establish the reliability and effectiveness of the developed approach and apply it to study the stress-strain state (SSS) of thin plates with complex shape. A computational experiment was carried out for thin hexagonal plates with mixed boundary conditions for various types of external load and elastic foundation characteristics. The relationship between the greatest deflection of the plate and the load is obtained. The results are presented in the form of graphs.

Problem solution analysis on finding the velocity distribution for laminar flow of a non-linear viscous flushing fluid in the annular space of a well

Modern drilling fluids are non-linear viscous media with an initial shear stress. In classical scientific works on hydromechanical modeling of drilling fluids motion in pipes and annular channels the Shvedov – Bingham approximation and Ostwald – de Waale power-law model were used, which did not fully account for behavior of technological fluids in a wide range of shear rates. This article presents a numerical solution for a mathematical model of drilling fluid motion of the three-parameter Herschel – Bulkley rheological model in the annular space of the well. The Herschel – Bulkley model in the rheological equation takes into account the presence of initial shear stress and a tendency for viscosity to change with shear rate, which distinguishes it from the Ostwald – de Waale and Shvedov – Bingham models. The target function in solving the equation of motion is the velocity distribution in the radial direction of the upward flow of the flushing fluid. The analysis of obtained solution is based on the theory of velocity profile influence on quality of cuttings removal during wellbore cleaning. Due to peculiarities of mathematical statement of the task, which supposes necessity of differential equation of motion solution, Wolfram Mathematica computational software has been used as a calculation tool. The analysis of numerical solution allowed to draw conclusions about the possibility of its application in evaluation of velocity profile when drilling fluid moves in annular space of the well. The possibility for application of modified excess coefficient as a relative quantitative parameter for evaluation of velocity profile uniformity was substantiated.

Problem of motion cueing along linear degrees of freedom on flight simulators

On the basis of Gibson's perception theory, a set of characteristic attributes of human motion perception is determined. On the basis of the system approach, the mathematical statement of the motion cueing problem along linear degrees of freedom on flight simulators of non-maneuvering aircraft is substantiated. This statement guarantees a motion cueing as close as possible to the real set of characteristic attributes of the motion cueing: nature, beginning time, direction, duration, and intensity of perception.

Numerical analysis of the thermal-elastic-plastic state of electrical contact gasket made from dispersion-reinforced composite materials using the fusioned deposition modeling method

The article presents the results of a theoretical study on determining the suitability of using electrical contact gaskets (ECG) in industrial conditions (Castner direct graphitization furnaces), made of coke-pitch composite using additive technologies based on the Fused Deposition Modeling (FDM) method.
 Numerical analysis of the physical state of ECG under the conditions of industrial application was performed on the basis of the mathematical statement of the thermo-elastic-plasticity problem and the algorithm of implicit inverse mapping of its solution based on the finite element method in the Mathcad programming environment. To construct the geometry and tetraid mesh of the ECG model, a freely open program code was used - the Gmsh CAD system for grid generation, and for the visualization of the results of physical field calculations, the free open program code ParaView was used.
 The study of the thermo-elastic-plastic state of ECG under the conditions of the graphitization process in Castner furnaces was carried out in the temperature range up to 900 °С, at which the thermoplastic properties of the material are manifested. At the same time, such physical fields of ECG were analyzed as temperature distribution, resulting displacements, equivalent elastic stresses according to Mises, equivalent total, elastic and plastic deformations according to Mises, and the volume fraction of ECG material in a plastic state, depending on the temperature level and radial gradient temperatures (radial temperature difference) of electrical contact gaskets.
 Numerical simulation of the thermo-elastic-plastic state of the ECG was carried out under the conditions of force loading by external pressure on the lateral surface of the gasket of 2.5 MPa and different values ​​of the radial temperature difference in the range of 15–90 °C in the temperature range up to 900 °C. It was established that: at the stages of formation of semi-coke and coke in the material of the coke-pitch composite ECG under the thermomechanical conditions of operation of the Kastner furnace, a margin of strength of not less than unity was obtained; with the subsequent increase in the temperature level in the Castner furnace to 3000 °C and above, the raw material of ECG, as a result of thermal destruction, has already turned into coke and, therefore, its mechanical properties have become close to the mechanical properties of electrode blanks in columns that are subjected to graphitization. This gives reason to assert that the ECGs will not be mechanically destroyed during the entire graphitization campaign of the Castner furnace.
 On the basis of the analysis of the results of numerical simulation, the possibility of using ECGs made from dispersion-reinforced composite materials (coke-pitch mixtures) by the FDM method in the technology of graphitizing electrode products according to the Сastner method is substantiated.

Application of Graphic Expression in the Proof of Set Theoretical Statements

Mathematical statements are proved using verbal reasoning. An addition to the traditional way of presenting evidence is proposed. The progress of the proof is presented graphically. There are almost no verbal explanations. The student can visually trace the development of the proof in time (consecutive steps to obtain the desired result) and in space (branching that arise in the course of the proof, and which require a separate study). This way of presenting evidence has been developed and used for many years in teaching the Discrete Mathematics course to students of computer specialties of the Siberian State Automobile and Highway University.

A conventional and digital mathematical board game design and development for use by students in learning arithmetic

This paper reported the design and development of a conventional and digital mathematical board game for use by students in learning arithmetic. At the time of research, there is no significant indication that a mathematical board game is available in scientific and published patent documentation. The availability of mathematical board games for students’ drills and practice in arithmetic, especially in mathematical statement construction, would benefit them, as this competency is an essential life skill. This research was conducted through the design and development research method with the procedure of users’ need analysis, researcher as developer capability analysis, product design, product development, field testing in its natural setting environment, and the prototype. The board game prototype was developed in both conventional printed and digital versions. The field testing for the conventional printed version was conducted at secondary school classes with 34 and 36 students, respectively, while for the digital by selected participants. The field testing shows that the developed mathematical board game can work as expected in its natural setting environment.

ANALISIS SOAL-SOAL MATEMATIKA TIPE HIGHER ORDER THINKING SKILLS (HOTS) UNTUK MENDUKUNG KEMAMPUAN PENALARAN MATEMATIS SISWA SMP

The ability to think logically and draw conclusions using logic is known as mathematical reasoning. Students understand how to use mathematical reasoning to manipulate formulas and properties, construct proofs, and explain ideas in mathematical statements. Answering HOTS (Higher Order Thinking Skills) questions improves mathematical reasoning. HOTS questions are higher-order thinking skills questions that require a reasoning process to be demonstrated when knowledge and skills are applied in new or more complex contexts. The purpose of this research was to determine the results of an analysis of HOTS type questions that could aid students' mathematical reasoning abilities. This study takes a qualitative approach, with content analysis serving as the descriptive research method. The HOTS type of question was found in 9 of the 40 Madrasah Exam questions examined. HOTS questions are those that assist students in developing their mathematical reasoning skills.Keywords: Mathematical Reasoning Ability, HOTS Type Questions, Madrasah Exam Questions.

A FEW POINTS ABOUT AXIOMATICS OF EDUCATION

Postulates and axioms have been used as a means of knowledge since ancient times. The well-known ancient Greek thinkers Aristotle, Plato, Euclid, and others used this cognitive tool. As an example, we all know – Euclid’s postulates. Although mathematicians consider him the pioneer of mathematics, one of the most famous mathematicians, nevertheless, it is more appropriate to consider Euclid as a Greek philosopher, who presented the most important postulates and axioms in his book "Elements". The prevailing opinion is that in essence it is exactly enough to prove all mathematical statements. Even though that the most perfect axiomatics is classical Euclidian geometry, there is an alternative here, i.e., modern non-Euclidean geometry.

MATHEMATICAL MODELLING OF THE MAGNETIC SYSTEM BY A.N. TIKHONOV REGULARIZATION METHOD

A nonlinear magnetostatic inverse problem is investigated for a case when it is needed to create a required magnetic field using conductors which coordinates vary on condition that current value is equal in all the conductors. It is known that the same problems fall under the category of noncorrect problems. Mathematical statement of this type of nonlinear magnetostatic inverse problems is given. The proposed numerical algorithm using the regularization method by A.N.Tikhonov permits to overcome comparatively easily the difficulties connected with badly conditioned equation systems to which usually the magnetostatic inverse problems reduce. This algorithm allows one to calculate the existing winding geometry of superconducting dipole ironless megnet which ensures a magnetic field homogeneity up to 10-5 within a rectangular aperture.

“How to meme it”: reverse engineering the creative process of mathematical Internet memes

Mathematical Internet memes are examples of how the creative thrust characterising the Web 2.0 environment reaches the field of mathematics, translating mathematical statements into a new digital form endowed with an epistemic potential that is capable of initiating a process of mathematical argumentation. The research presented in this paper aims to shed light on the creative process of mathematical memes, contributing to building a body of knowledge on mathematical memes that, prospectively, could enable educators to profit from these objects in their teaching. Theoretically, this is based on a widened concept of creativity that focuses on the connection linking digital culture with mathematics, and on distinguishing and merging three perspectives to disclose the meanings of mathematical memes. Methodologically, the process of mathematical memes’ creation is investigated through a reverse engineering approach on a dataset of about 2100 items collected in a 3-year-long ethnographic observation within online communities. The result is a heuristic action model of the creation process, that is validated by creating two new mathematical Internet memes that are shared online within the observed communities to explore if they retain the mathematical and epistemic characteristics of Web-found ones.

The role of epistemic emotions in undergraduate students’ proof construction

Proofs as epistemic tools are central to mathematical practice, as they establish and provide explanations for the validity of mathematical statements. Considering the challenge that proof construction poses to learners of all ages, prior research has investigated its cognitive determinants, but the impact of affective-motivational experiences on proof construction has been insufficiently investigated. Emotions related to knowledge acquisition (i.e., epistemic emotions) are assumed to play a key role in epistemic processes. In this study we investigated how the performance of 80 mathematics undergraduate students in a geometric proof construction task relates to the epistemic emotions experienced during proof construction. Controlling for geometry knowledge, we included control and value appraisals as antecedents in our investigation of epistemic emotions, and attention and motivation as mediators of their effects on proof construction performance. The results indicate that positive as well as negative emotions are influenced by students’ appraisals, also indicating an interaction of both appraisal dimensions. Primarily enjoyment and curiosity mediate the effects of these appraisals on attention and motivation. These two markers of the proof construction process, in turn, mediate the effects of enjoyment and boredom on proof construction performance. In this study we investigated systematically the role of epistemic emotions in geometric proof construction and we offer insights that complement the existing research on the cognitive determinants of proof performance. Moreover, this study extended research on epistemic emotions into the area of proof construction, an epistemic process central to mathematics.