Mathematical Facts Research Articles

Meta-Analysis of the Effect of Technology-Based Mathematical Fact Practice on Mathematics Outcomes

Meta-Analysis of the Effect of Technology-Based Mathematical Fact Practice on Mathematics Outcomes

Proof Complexity and Beyond

Proof complexity is a multi-disciplinary research area that addresses questions of the general form “how difficult is it to prove certain mathematical facts?” The current workshop focussed on recent advances in our understanding that the analysis of an appropriately tailored concept of “proof” underlies many of the arguments in algorithms, geometry or combinatorics research that make the core of modern theoretical computer science. These include the analysis of practical Boolean satisfiability (SAT) solving algorithms, the size of linear or semidefinite programming formulations of combinatorial optimization problems, the complexity of solving total NP search problems by local methods, and the complexity of describing winning strategies in two-player round-based games, to name just a few important examples.

Wittgenstein on mathematical facts

AbstractThe status of mathematical facts has long been taken to be unclear in Wittgenstein's philosophy of mathematics, and often, it seems that he wants to eliminate mathematical facts in favour of facts about our beliefs or behaviour. In this paper, I argue that by reading Wittgenstein as a radical conventionalist, we can give a reading of the relevant passages according to which Wittgenstein doesn't deny that there are mathematical facts, but rather denies that one needs a metaphysical account of what mathematical facts are and how they relate to the world that goes beyond the minimal claims of radical conventionalism—that empirical facts about how we would find natural to project our training into new cases and the constitution of concepts by our agreement are enough. At the end of the paper, I discuss the implications of this reading on how to understand a rule's determination of its own application.

Geometrical interpretation of isospin subalgebras in SU(3)

Ever since new tetrads in four-dimensional Lorentzian curved spacetimes were introduced, many outstanding properties and results have been proven regarding Riemannian geometry, gauge theory or group theory. These tetrads might carry spacetime-kinematic information about particle multiplets. Among other properties, these new tetrads, locally and covariantly diagonalize stress–energy tensors. In the case where particles have an electromagnetic field, the local planes of gauge symmetry served as a tool in order to propose a new gravitational-kinematic interpretation of particle multiplets. The core reason stems from the mathematical fact that all local gauge symmetries associated to the Standard Model have been proven to be isomorphic to local groups of tetrad transformations in four-dimensional Lorentzian spacetimes. Therefore, the different stress–energy tensors have been proven to determine the local gauge geometry as a natural part of Riemannian geometry. The stress–energy tensors determine locally the orthogonal planes such that the rotation on either of them of the local tetrad vectors that span these planes is isomorphic to local tetrad gauge transformations. All these properties put together allow for a reinterpretation presented previously of particle states as particle gravitational-kinematic-spacetime tetrad states in the asymptotically flat limit. We proceed in this paper to study the case where there are [Formula: see text], [Formula: see text] and [Formula: see text] isospin subalgebras or submultiplets within the context of this new Riemannian interpretation of gauge symmetries.

Algorytm regulacji geometrycznych sieci niwelacyjnych

The natural way to reduce the duration of measurement of a levelling network is to cut down on the number of levelling lines without damaging the quality of the final results. The main objective of the study is to demonstrate that this is possible without any lack of accuracy, if some mathematical facts regarding the average of both measurements of the line elevations are taken into account. Based on 60 paired random samples of size 1000, derived from different continuous distributions, e.g., N (0, 1), U (-1.732, 1.732) and Gamma (1, 1), each of them with theoretical standard deviation σ=1, it was found that the averages of each pair form new distribution with standard deviation σ≈0.707. However, the samples, which were formed by selecting the nearest to the known theoretical expectation from both measurements and their average have distributions, which standard deviations tend to σ≈0.53, σ≈0.46 and σ≈0.43 for the U (-1.732, 1.732), N (0, 1) and Gamma (1, 1) distributions, respectively. Therefore, if we choose the more appropriate value from the “first”, the “second” measurement and their average, we will increase the accuracy of the network almost 2 times in comparison to the accuracy, yielded by the only use of the averages. If our network contains n lines, the process of finding of these elevation values, which leads to the best fit of the network, is based on 3n single adjustments of the network. In addition, we can minimize the impact of the shape of the network on the final standard errors of the adjusted heights or geopotential numbers of the nodal benchmarks in the network, if we apply some iterative procedures, e.g., Inverse Distance Weighting (IDW), Inverse Absolute Height Weighting (IAHW), etc. In order to check the above explained algorithm, the Second Levelling of Finland network was adjusted in three variants. In the first variant, the whole network was adjusted as a free one. The classical weights w=L-1 were used. In the second variant, the network was separated into two parts. Applying 312 and 314 independent adjustments, the selection of the best fitted values of line elevations was done and the network was adjusted by using them. The IDW and IAHW with power parameter p=5 were finally applied. In the third variant, the network was separated in four parts. Applying 313, 312, 316 and 312 independent adjustments, the new selection of the line elevations was done and the network was adjusted by them. The IDW (p=6.5) and IAHW (p=6) were executed. Comparison of the standard errors of the adjusted geopotential numbers in the separate variants revealed that there was no statistically significant difference between the results, yielded in the second and the third variant. However, these variants produced 3-5 times increase of the accuracy in comparison to the classical first variant. The best results were obtained in the second variant with IAHW, where the mean value of the standard errors of the adjusted geopotential numbers is below 1.4 mgpu.

On the stability of a Caputo fractional order predator-prey framework including Holling type-II functional response along with nonlinear harvesting in predator

In this study, we look at a fractional-order two-species predator-prey framework that uses Caputo Sense's fractional derivative to try to understand its dynamics, which include a simplified Holling type II functional response and nonlinear harvesting in the predator. Initially, we establish certain mathematical facts, such as the solutions of fractional order dynamical systems being unique, non-negative, bounded, and existing. The dissipativeness of the FDE system solution is addressed. Our investigation also includes a look at the local stability requirements for every possible equilibrium point. As a bifurcation constraint, the order of derivatives has been considered in our discussion of the existence condition of Hopf bifurcation. Our analysis has been strengthened by the incorporation of fractional Hopf bifurcation, and the dynamics of the proposed model are influenced by selective non-linear harvesting. Furthermore, the investigation focuses on the impact of the predator's pace of harvesting about the complexities of the prey-predator relationship. Through the use of certain setting values, empirical evidence indicates as the non-integer order framework displays stable to unstable, when the non-integer order is increased. The analytical findings are shown by various instances presented in the numerical part.

Standalone gradient measurement of matrix norm for programmable unitary converters

Programmable unitary converters are powerful tools for realizing unitary transformations, advancing the fields of computing and communication. The accuracy of these unitary transformations is crucial for maintaining high fidelity in such applications. However, various physical artifacts can impair the accuracy of the synthesized transformations. A commonly employed approach uses the system’s gradient to restore accuracy. Matrix norm is used to define error between matrices, and minimization of this norm using the gradient restores the accuracy. Although this gradient can indeed be physically measured using external equipment, it leads to a rather complex optical system. In this study, we propose a standalone method for measuring matrix norm gradients, where “standalone” means that no additional optical equipment is needed. This method is based on the mathematical fact that the central difference, which is generally used for the approximation of differentiation, can yield exact differentiation for any unitary converters. Furthermore, we introduce a new matrix distance that is suitable for optimizing unitary converters that use intensity detectors at the output. This distance also yields the exact differentiation with the central difference. Numerical analysis demonstrates that our method exhibits orders of magnitude higher tolerance to measurement noise than prior similar approaches.

A Mixed Reality Environment for Mathematics

Throughout history, mathematicians and geometricians have relied on visualizations to convey abstract concepts effectively, including describing, discussing, studying, and teaching Mathematics. In Mathematics education, visualizations are still used whenever possible to support teaching, inspire students, and satisfy their need to see abstract Mathematical facts. In recent years, the integration of virtual reality (VR) and augmented reality (AR) in contemporary Mathematics education offers a fascinating and extremely motivating new tool for teachers, allowing students to visualize Mathematics in three dimensions. This article provides an overview of immersive environments developed in recent years to support geometry education, focusing on the combined use of VR and AR. By examining various methodologies and outcomes, we explore the potential of mixed reality in transforming how students engage with mathematical concepts. The procedures and outcomes of the methods of applications are discussed and described below. Keywords—Augmented Reality, Virtual Reality, computer generated environment, geometry education, sentenced-based math problems, interactive playgrounds

Canonical‐Cell Tilings and their Atomic Decorations

AbstractThe canonical cell tiling is a geometrical framework that uses four kinds of basic polyhedra, called the canonical cells, to model the packing of atoms and clusters in icosahedral quasicrystals and related periodic approximants. Over the past three decades, it has become increasingly clear that this framework is the most sensible approach to describe related structures, albeit technically much less tractable than the Ammann‐Kramer‐Neri tiling, which is the simplest icosahedral tiling geometry based on the two Ammann rhombohedra. Geometrical arrangements of cells pose a number of combinatorial problems that cannot be handled using simple linear algebra, making it infeasible to determine structures using the standard six‐dimensional scheme. This up‐to‐date review begins with the motivation, definition, and mathematical facts about the canonical cell tiling. Then the reader is taken through the zoo of concrete structures, from smaller periodic approximants to larger ones, along with an overview of the techniques and heuristics used to study them. The recent discovery of a quasiperiodic canonical cell tiling is also briefly illustrated. The latter half of this review surveys the atomistic modeling of real atomic structures in all the three existing structural families based on the decoration concept of the canonical cell tiling.

A conventionalist account of distinctively mathematical explanation

Distinctively mathematical explanations (DMEs) explain natural phenomena primarily by appeal to mathematical facts. One important question is whether there can be an ontic account of DME. An ontic account of DME would treat the explananda and explanantia of DMEs as ontic items (ontic objects, properties, structures, etc.) and the explanatory relation between them as an ontic relation (e.g., Pincock, 2015; Povich, 2021). Here I present a conventionalist account of DME, defend it against objections, and argue that it should be considered ontic. Notably, if indeed it is ontic, the conventionalist account seems to avoid a convincing objection to other ontic accounts (Kuorikoski, 2021).

EPR-Net: constructing a non-equilibrium potential landscape via a variational force projection formulation.

We present EPR-Net, a novel and effective deep learning approach that tackles a crucial challenge in biophysics: constructing potential landscapes for high-dimensional non-equilibrium steady-state systems. EPR-Net leverages a nice mathematical fact that the desired negative potential gradient is simply the orthogonal projection of the driving force of the underlying dynamics in a weighted inner-product space. Remarkably, our loss function has an intimate connection with the steady entropy production rate (EPR), enabling simultaneous landscape construction and EPR estimation. We introduce an enhanced learning strategy for systems with small noise, and extend our framework to include dimensionality reduction and the state-dependent diffusion coefficient case in a unified fashion. Comparative evaluations on benchmark problems demonstrate the superior accuracy, effectiveness and robustness of EPR-Net compared to existing methods. We apply our approach to challenging biophysical problems, such as an eight-dimensional (8D) limit cycle and a 52D multi-stability problem, which provide accurate solutions and interesting insights on constructed landscapes. With its versatility and power, EPR-Net offers a promising solution for diverse landscape construction problems in biophysics.

Developing patterns of students' mathematical literacy processes: Insights from cognitive load theory and design-based research

This research aims to develop patterns of students' mathematical literacy processes based on the Cognitive Load Theory (MLCLT). Research using Design-Based Research (DBR) aims to design and develop learning components in the form of resulting learning patterns arranged in 4 stages. The research results show that a CLT-based mathematical literacy process domain pattern has been found. For the aspect of formulating situations mathematically, the pattern of mathematization was produced, and it resulted in some students having good communication skills. For the element of employing mathematical concepts, facts, procedures, and reasoning, the pattern of application was produced, and it showed an increase in students' literacy processes for indicators such as mathematizing, representation, reasoning, argument, devising strategies for solving problems, using symbolic, formal, and technical language and operations, as well as using mathematical tools. The pattern for the aspect of interpreting, applying, and evaluating mathematical outcomes is the reflection pattern. In this pattern, students are already accustomed to having indicators of mathematical literacy processes related to devising strategies for solving problems, using symbolic, formal, and technical language and operations, as well as using mathematical tools so that they can produce real solutions to real-world problems.

Mathematical Literacy of Junior High School Students in Solving Problems PISA Content Quantity

<p style="text-align: justify;">Mathematical literacy is defined as students’ capacity to formulate, employ and interpret mathematics in a variety of contexts. It includes reasoning mathematically and using mathematical concepts, procedures, facts and tools to describe, explain and predict phenomena. It assists individuals in recognising the role that mathematics plays in the world and to make the well-founded judgements and decisions needed by constructive, engaged and reflective citizens. Quantity content includes the ability in reasoning quantitatively, representing something into numbers, understanding mathematical steps, counting by heart and making assessments. The questions on quantity content are mostly implemented in everyday life, such as in exchanging currency exchange rates, determining bank interest, shopping, calculating taxes, measuring time, measuring distance and others. The purpose of this article is to describe the level of mathematical literacy ability of junior high school students in solving Quantity content PISA questions. The data from this study were taken from the results of the PISA test questions on quantity content and interviews. Subject taking is based on the test results of students who have the highest scores, as it is expected to get the results of the analysis of students’ high literacy skills. In this research, the researcher is using indicators of mathematical literacy level according to PISA. Based on the results of research and discussion, it is concluded that there are students who have high mathematical literacy skills equivalent to level 5.</p>

Is the secrecy of the parametric configuration of slot machines rationally justified? The exposure of the mathematical facts of games of chance as an ethical obligation

Is the secrecy of the parametric configuration of slot machines rationally justified? The exposure of the mathematical facts of games of chance as an ethical obligation

Totality, Regularity, and Cardinality in Probability Theory

Abstract Recent developments in generalized probability theory have renewed a debate about whether regularity (i.e., the constraint that only logical contradictions get assigned probability $0$ ) should be a necessary feature of both chances and credences. Crucial to this debate, however, are some mathematical facts regarding the interplay between the existence of regular generalized probability measures and various cardinality assumptions. We improve on several known results in the literature regarding the existence of regular generalized probability measures. In particular, we give necessary and sufficient conditions for the existence of regular generalized probability measures defined on the whole powerset of any sample space.

Static, dynamic and interactive elements in digital teaching materials in mathematics

Contemporary comprehensive mathematics teaching material covering whole courses has developed substantially from the early versions that roughly were ‘books as pdf’ with some complementary online material. In teaching materials that are offered in online web portals (digital teaching platforms) a variety of dynamic and interactive elements can be utilised, offering new ways to engage with mathematics. Despite this recent development, the variety of affordances of the digital environment are utilised to a surprisingly small extent. The pros and cons with digital teaching materials in mathematics are debated, and publishers advertise with arguments about algorithms that lay out an ideal learning path and about joyful content. Critical for students’ learning while working with teaching materials is however that they find it meaningful to use the materials, a persistence in the interaction with the materials, and furthermore that the willingness to explore mathematics remains. In this study students’ interaction with digital teaching material with various kinds of dynamic and interactive elements supplementing the static parts in the presentation of new content is explored. Differences in students’ attention to mathematical facts, essential in the problem solving, is captured using an eye-tracker. Analyses of differences in attentive behaviour depending on the kind of digital element that are used for presentation reveal that the type of digital element that students attend the least to is static elements. Differences in what is offered to and what is demanded from a reader when mathematical facts are presented using various digital elements is discussed and potential implications from the results are suggested.

Was There a Period of Latent Development of COVID-19 in St. Petersburg? Mathematical Simulation Results and Facts

Was There a Period of Latent Development of COVID-19 in St. Petersburg? Mathematical Simulation Results and Facts

Assessing ChatGPT’s Proficiency in Quantitative Risk Management

The purpose and novelty of this article is to investigate the extent to which artificial intelligence chatbot ChatGPT can grasp concepts from quantitative risk management. To this end, we enter a scholarly discussion with ChatGPT in the form of questions and answers, and analyze the responses. The questions are classics from undergraduate and graduate courses on quantitative risk management, and address risk in general, risk measures, time series, extremes and dependence. As a result, the non-technical aspects of risk (such as explanations of various types of financial risk, the driving factors underlying the financial crisis of 2007 to 2009, or a basic introduction to the Basel Framework) are well understood by ChatGPT. More technical aspects (such as mathematical facts), however, are often inaccurate or wrong, partly in rather subtle ways not obvious without expert knowledge, which we point out. The article concludes by providing guidance on the types of applications for which consulting ChatGPT can be useful in order to enhance one’s own knowledge of quantitative risk management (e.g., using ChatGPT as an educational tool to test one’s own understanding of an already grasped concept, or using ChatGPT as a practical tool for identifying risks just not on one’s own radar), and points out those applications for which the current version of ChatGPT should not be invoked (e.g., for learning mathematical concepts, or for learning entirely new concepts for which one has no basis of comparison to assess ChatGPT’s capabilities).

스마트폰 활동 기반 중학교 이차함수 교수학습 자료 개발 및 적용

Objectives In this study, teaching and learning materials were developed for middle school students to explore
 quadratic functions through smartphone-based activities. After the middle school students performed the activities,
 the results of the activities were analyzed to find out whether they were motivated to learn mathematics and
 what mathematical facts they found.
 Methods Teaching and learning materials for quadratic function for exploration activities were developed using
 GeoGebra for 5 2nd grade students at “A” middle school located in a small city. Afterwards, the developed teaching
 and learning materials were applied to middle school students, and the students' results were analyzed in
 terms of motivation, mathematical discovery, and content understanding.
 Results After analyzing the middle school quadratic function curriculum and various textbooks, smartphone activity-
 based quadratic function teaching and learning materials were developed. In addition, as a result of analyzing
 the results of applying the developed teaching and learning materials to students, students were motivated to explore
 quadratic functions using GeoGebra on their smartphones. In addition, while moving graphs of quadratic
 function and observing the changes in function formula, the experience of reasoning about the translation of the
 function was experienced. In addition, students gained more understanding and reasoning experience through
 geometry-centered teaching and learning activities such as drawing points, drawing graphs, and moving them in
 GeoGebra.
 Conclusions Using the teaching-learning materials developed in this study, middle school students were able to
 gain motivation for learning and infer and explore the contents of quadratic functions.

Evaluation of Students' Difficulties in Learning Mathematics in Complex Variable Material

Prospective mathematics teachers are required to be able to master facts and be able to give examples that exist in everyday life. A prospective mathematics teacher is expected to be skilled and mature in using media when operating, explaining and demonstrating direct objects that exist in any complex variable material. The aim of this study is to analyze the obstacles and difficulties of students in Mathematics. Two indicators for assessing mathematics learning outcomes: 1) Direct objects (Facts, Concepts, Skills, and Principles), and 2) Indirect objects (Internet and friends). Mixed methods research method. The subject of the complex variable mathematics course and the student object. Qualitative data collection techniques by observation, interviews, documentation and quantitative survey method collection techniques by distributing instruments. Analysis with, presentation, reduction, presenting bar charts, data synchronization and drawing conclusions. The findings, students' difficulties lie in the direct object, namely mastery of mathematical facts 95.55% and concept difficulties 91.11%, principle (formula) difficulties 91.11% and algorithms 75.55%. Students who get good grades online are assisted by indirect objects, namely the internet and other student discussion partners. In conclusion, students' obstacles and difficulties lie in the direct objects of fact indicators (definitions) and concepts. Material design, facts and concepts with group discussion models are learning solutions.