Mathematical Discovery Research Articles

Algorithm-assisted discovery of an intrinsic order among mathematical constants

In recent decades, a growing number of discoveries in mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces. As computers become more powerful, an intriguing possibility arises-the interplay between human intuition and computer algorithms can lead to discoveries of mathematical structures that would otherwise remain elusive. Here, we demonstrate computer-assisted discovery of a previously unknown mathematical structure, the conservative matrix field. In the spirit of the Ramanujan Machine project, we developed a massively parallel computer algorithm that found a large number of formulas, in the form of continued fractions, for numerous mathematical constants. The patterns arising from those formulas enabled the construction of the first conservative matrix fields and revealed their overarching properties. Conservative matrix fields unveil unexpected relations between different mathematical constants, such as π and ln(2), or e and the Gompertz constant. The importance of these matrix fields is further realized by their ability to connect formulas that do not have any apparent relation, thus unifying hundreds of existing formulas and generating infinitely many new formulas. We exemplify these implications on values of the Riemann zeta function ζ (n), studied for centuries across mathematics and physics. Matrix fields also enable new mathematical proofs of irrationality. For example, we use them to generalize the celebrated proof by Apéry of the irrationality of ζ (3). Utilizing thousands of personal computers worldwide, our research strategy demonstrates the power of large-scale computational approaches to tackle longstanding open problems and discover unexpected connections across diverse fields of science.

영재 사사과정에서 공학도구를 활용한 미적분 및 미분방정식 탐구 지도 사례

The purpose of this study is to present a year-long instructional case study of middle school mathematically gifted students in Science Gifted Education Center using technological devices to explore mathematical models based oncalculus and differential equation, and to draw implications for gifted education. In the classes conducted in this study, technological devices were used to compute expressions and visualize graphs. The use of technological devices reduced the amount of time spent on calculations and graphing, allowing more time for interpretation and discussion of models. As a result, students understood the calculus and differential equation models and interpreted the models in various ways using graphs based on abundant discussion time. The results of this study show that it is possible to explore calculus and differential equation models at the middle school level by using technological devices. Furthermore, visualization with technological devices provides opportunities for observation and discovery of mathematics and allows for plentiful discussion and interpretation by securing time for interaction among students, so there is a need for more active use of technological devices in gifted classes.

The imaginary unit described as an algebraic ambiguity

A complex number is formed by a real part and an imaginary part, the latter being composed of a real number multiplied by the imaginary unit “i,” defined as the square root of . For this reason, imaginary numbers cannot be placed on the real number line. The geometric representation of complex numbers is done in the so-called complex plane, composed of a real axis and an imaginary axis, orthogonal to the real line. However, from a philosophical point of view, the fact that the imaginary axis is not identified with any axis in three-dimensional Euclidean space raises ontological questions that transcend the well-established mathematical formalism of the set of complex numbers: are imaginary numbers just a sophism, an arbitrary invention, or a true mathematical discovery? Are imaginary numbers a glimpse of a parallel reality that is beyond our perception? In this article, I seek to demonstrate that the imaginary unit can be interpreted as an algebraic ambiguity that arises when representing certain vectors in the real plane as one-dimensional variables. Consequently, we will see how it is possible to express complex numbers in terms of real numbers.

Attitudes in mathematical discovery processes:

This paper’s purpose is to investigate the attitude of students in mathematical discovery processes in terms of the handling of counterexamples. By understanding this attitude as a kind of scientific attitude, it consists of different aspects that become visible in the behaviour during a mathematical discovery process. Since such a process is particularly complex, the author’s interest is to use the concept of attitude as an explanation for students’ behaviour that occurs when dealing with conflicts such as counterexamples. Semi-structured interviews with sixth graders of a German Gymnasium were conducted and analysed in a qualitative and interpretative way. As a result, the case study of Alex and Milo is presented. Based on the framework that observable behaviour is influenced by an underlying attitude, there are drawn conclusions about Alex’s and Milo’s attitudes adopted in the mathematical discovery process and their impact on the process is elaborated.

Generalizing the eight levels theorem: a journey to Mersenne prime discoveries and new polynomial classes

Mersenne primes, renowned for their captivating form as 2 p − 1 , have intrigued mathematicians for centuries. In this paper, we embark on a captivating quest to unveil the intricate nature of Mersenne primes, seamlessly integrating methods with the Eight Levels Theorem. Initially, we extend the Eight Levels Theorem and introduce an innovative approach that harmoniously combines arithmetic and differential techniques to compute the coefficients of the polynomial expansions of x n + y n in terms of binary quadratic forms. This endeavor leads us to the genesis of novel polynomial sequences as we scrutinize the coefficients within this expansion. Our research unearths previously uncharted connections between Mersenne numbers and the derivatives of specific polynomial sequences. By forging this linkage, we not only enhance our understanding of Mersenne primes but also bridge the divide between well-established sequences in number theory and differential equations. This broadens the applicability of our findings across diverse scientific domains, revealing fresh avenues for exploration in number theory and beyond. These polynomial bridges serve as conduits between these sequences, unveiling exciting prospects for future research. This interdisciplinary exploration opens up exciting possibilities for the broader implications of Mersenne primes, extending their significance beyond the realm of pure mathematics. In our paper, we also delve into the intriguing influence of the Golden Ratio constant, which unveils segments reminiscent of the beauty found in nature, adding an unexpected dimension to the world of arithmetic. In this harmonious journey, the allure of Mersenne primes resonates through the symphony of mathematical discovery.

Mathematical discoveries from program search with large language models

Large language models (LLMs) have demonstrated tremendous capabilities in solving complex tasks, from quantitative reasoning to understanding natural language. However, LLMs sometimes suffer from confabulations (or hallucinations), which can result in them making plausible but incorrect statements1,2. This hinders the use of current large models in scientific discovery. Here we introduce FunSearch (short for searching in the function space), an evolutionary procedure based on pairing a pretrained LLM with a systematic evaluator. We demonstrate the effectiveness of this approach to surpass the best-known results in important problems, pushing the boundary of existing LLM-based approaches3. Applying FunSearch to a central problem in extremal combinatorics—the cap set problem—we discover new constructions of large cap sets going beyond the best-known ones, both in finite dimensional and asymptotic cases. This shows that it is possible to make discoveries for established open problems using LLMs. We showcase the generality of FunSearch by applying it to an algorithmic problem, online bin packing, finding new heuristics that improve on widely used baselines. In contrast to most computer search approaches, FunSearch searches for programs that describe how to solve a problem, rather than what the solution is. Beyond being an effective and scalable strategy, discovered programs tend to be more interpretable than raw solutions, enabling feedback loops between domain experts and FunSearch, and the deployment of such programs in real-world applications.

Relative frequencies of perovskite solar cells strengthened by composite materials: Application of deep neural networks for solving the vibration problem

Perovskite solar cells have shown competitive power conversion efficiencies (PCE) with the potential for better performance when compared to the most widely used photovoltaic technology. Perovskites may deteriorate when exposed for an extended amount of time to light, heat, or electrical current, as well as when they interact with moisture and oxygen. To increase stability, researchers are examining deterioration in both the perovskite material itself and the surrounding device layers. Cell durability has to be increased in order to produce perovskite solar goods for sale. Improved cell durability is critical for the development of commercial perovskite solar products. So, improving the dynamic properties of this kind of green energy device is a key issue in engineering. In the current report, the stability zone and relative frequency change of the perovskite solar cells reinforced by three-dimensional functionally graded material (3D-FGM) as the last layer is presented. For exact modeling of the perovskite solar cell structure due to its size (micro size), it is crucial to take the size effect into account. Modified couple stress theory (MCST) as a size-dependent theory for micro-scale systems with curvature derivate parameters is offered. Once the study’s findings have mastered the mathematical discoveries, they are instructed to produce the machine learning technique’s outputs. In actuality, the fast-converging numerical approach is used to derive the characteristics of the design points. The momentum-based optimizer is used to find the optimal values of the parameters influencing how the fully connected neural network functions. The method used in this research is efficient since it quickly and properly trains the multi-layered network. The funding now available may be used to promote future research on the importance of the dynamic stability of perovskite solar cells employing 3D-FG material as the constructed layer.

Introduction to Mathematical Language Processing: Informal Proofs, Word Problems, and Supporting Tasks

Abstract Automating discovery in mathematics and science will require sophisticated methods of information extraction and abstract reasoning, including models that can convincingly process relationships between mathematical elements and natural language, to produce problem solutions of real-world value. We analyze mathematical language processing methods across five strategic sub-areas (identifier-definition extraction, formula retrieval, natural language premise selection, math word problem solving, and informal theorem proving) from recent years, highlighting prevailing methodologies, existing limitations, overarching trends, and promising avenues for future research.

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.

Visibility of Albert Einstein’s Four-Dimensional Plane of Existence in Sri Aurobindo’s Yogic Vision of ‘Space’ in His Epic “Savitri”

Albert Einstein’s mathematical discovery of the four-dimensional plane of existence, the ‘Spacetime’ or ‘Space-Time Continuum’, as it is called, is an evolutionary breakthrough in the realm of Science. Since the plane of ‘Spacetime’ or ‘Space-Time Continuum’ is a mathematical structure, the most celebrated scientist finds it to be a physical plane of existence. Mysteriously enough, it is not physically visible. Physics is answerless on its invisibility. Transcending Einstein, Sri Aurobindo, a ‘yogi of the unknown eternal heights’ reveals ‘Space’, in his epic Savitri, to be ‘a vast experiment of Soul’, the Divine Consciousness in the being of Man. Furthermore, it is the Soul of the Universe, ‘Vishwatman’, as the Rishis call it. In such a vision of space the mystery of the invisibility of Spacetime disappears. The plane of Spacetime is the plane of integral existence to which mankind is yet to evolve. Man is subject to the divisive principle of Consciousness as a result of which the plane of Spacetime is not visible to him. On account of the subjection to the divisive principle of consciousness Newton could find Space and Time to be two incompatible realities. Man is required to evolve his latent spiritual consciousness to attain the integrality of Consciousness for the vision of the plane of Spacetime. He has to undertake the yoga to attain, first, the silent state of mind to ascend the planes of the spiritual and Supramental Consciousnesses, says Sri Aurobindo. In the silent state of mind, the divisiveness of consciousness ceases to have its hold as a result of which man is able to visualise the plane of Spacetime, the Spacetime which is all the more clearly visible when he continues to ascend the higher planes of the spiritual consciousness. On the plane of the spiritual consciousness there is a movement towards the integrality of consciousness, and consequently ‘Space’ is found to be ‘a vast experiment of Soul’ in this movement, the movement which is, in fact, the ascent of the yogic consciousness.

The Poetry and Music of Science: Comparing Creativity in Science and Art

The Poetry and Music of Science: Comparing Creativity in Science and Art

Embed-Solitons in the Context of Functions of Symmetric Hyperbolic Fibonacci

In this article, we discuss the findings of new developments in a class of new triangular functions that blend the quantity functions of the traditional triangular. Considering the significant role played by the triangular functions in applied mathematics, physics, and engineering, it is conceivable to predict that the theory of new triangular functions will provide us with additional interpretations and discoveries in mathematics and physics. The solutions which consider variable separation based on arbitrary functions are constructed to the (3+1)-dimensional Burgers model by presenting the Fibonacci Riccati technique and the linearly independent variable separation approach. This technique’s fundamental concept is to describe the solution of the Burgers model as a polynomial in the Riccati Equation solution that satisfies the symmetrical hyperbolic and triangular Fibonacci functions. Depending on the choice of suitable functions for variable separation, an abundance of new localized solutions were obtained. Moreover, examples such as embedded solitons, rectangle-solitons, plateau-type ring solitons, taper-like solitons, and their interactions with each other, following the symmetrical hyperbolic and triangular Fibonacci functions, as well as the golden mean, could be explored.

Computer Experiment in Teaching Mathematics

A mathematical experiment has always been a key source for a mathematical discovery. Over the past 50 years, thanks to digital technologies, its role in mathematical research has grown significantly. Digital technologies have opened up fundamentally new opportunities for experimentation in mathematics education, bringing mathematical education closer to mathematical research for the majority of students. This tendency is especially desirable in the modern world, where it becomes possible thanks to digital technologies. The article discusses the results of the authors’ work over the past decades on the application of a computer mathematical experiment at different levels of school and university education. Particular attention is paid to dynamic geometry environments. The possibilities of using computer algebra systems are also discussed. A project concerning schoolchildren’s work on generalizations of Napoleon’s theorem is considered in detail.

COMPUTER EXPERIMENT IN TEACHING MATHEMATICS

Mathematical experiment has always been a key source for mathematical discovery. Over the past 50 years, thanks to digital technologies, its role in mathematical research has grown significantly. Digital technologies have opened up fundamentally new opportunities for experimentation in mathematics education, bringing the majority of students of mathematical education closer to mathematical research. Such an approximation is especially desirable in the modern world, where it becomes possible thanks to digital technologies. The article discusses the results of the work of the authors over the past decades on the application of a computer mathematical experiment at different levels of school and university education. Particular attention is paid to dynamic geometry environments. The possibilities of using computer algebra systems are also considered. The project of schoolchildren’s work on generalizations of Napoleon’s theorem is considered in detail.

The joy of mathematical discovery: To the 120th anniversary of Academician A.N. Kolmogorov

This article is dedicated to the memory of the outstanding mathematician of the 20th century, Andrei Nikolaevich Kolmogorov, whose 120th birthday is celebrated this year. The author describes in detail the formation of Kolmogorov as a scientist, the significance of Moscow State University in the scientific life and pedagogical activity of the famous mathematician. Kolmogorov’s contribution to such branches of mathematics as classical analysis, topology, geometry, approximation theory, functional analysis, and probability theory is analyzed, and the contribution of his scientific school to Russian science, including not only mathematics and physics but also the humanities, is shown.

Challenges and suggestions regarding the aesthetic experience of mathematics when teaching and learning fully online

Growth in fully online mathematics education has been steady for decades. This is where essentially all interactions associated with the teaching and learning process are mediated via the online medium. Initially spurred on by technological innovation and a variety of socio-cultural forces, growth has more recently rapidly accelerated due to pandemic restrictions. Many suggest we are on the precipice of profound transformation in mathematics education, the outlines of which remain unknown. In leveraging online education possibilities and advancing online mathematics education, this paper considers aesthetic experiences associated with learning mathematics fully online. First, three assumptions are made to highlight the importance of these experiences. Second, three challenges are identified regarding the current state of fully online mathematics education. Finally, specific instructional design and online teaching strategies are outlined that potentiate experiences of wonder and beauty often associated with mathematical discovery. As its central theme, the paper seeks to answer how we may take advantage of this technology in delivering quality mathematics education without allowing it to diminish our human essence.

Non-traditional assessments to match creative instruction in undergraduate mathematics courses

Assessment is an issue that is central to the lives of educators and students alike. At all levels, from programmatic to course-specific, instructors strive to assess in a way that accurately measures achievement or progress relative to one’s goals. This paper reports on students’ experiences with alternative assessments in our undergraduate mathematics course for non-STEM (Science, Technology, Engineering and Mathematics) majors. The course, Creative Thinking in Mathematics, designed to fulfil the university’s general education requirement for mathematics, was recently revised and renamed to highlight the role of creativity in mathematics. In the course, students engage in mathematics using creative approaches and consider how these approaches connect to other disciplines, their career aspirations, and important mathematical discoveries throughout history. This shift in course emphasis necessitated revisions to assessments in ways that mirrored the course’s instructional approach. This article describes the alternative assessments, outlines the research study, and reports on the results relative to students’ experiences with these assessments, both prior to and during the course, highlighting a need for instructors to reconsider traditional assessments.

Neurocompositional computing: From the Central Paradox of Cognition to a new generation of AI systems

AbstractWhat explains the dramatic progress from 20th‐century to 21st‐century AI, and how can the remaining limitations of current AI be overcome? The widely accepted narrative attributes this progress to massive increases in the quantity of computational and data resources available to support statistical learning in deep artificial neural networks. We show that an additional crucial factor is the development of a new type of computation. Neurocompositional computing adopts two principles that must be simultaneously respected to enable human‐level cognition: the principles of Compositionality and Continuity. These have seemed irreconcilable until the recent mathematical discovery that compositionality can be realized not only through discrete methods of symbolic computing, but also through novel forms of continuous neural computing. The revolutionary recent progress in AI has resulted from the use of limited forms of neurocompositional computing. New, deeper forms of neurocompositional computing create AI systems that are more robust, accurate, and comprehensible.

Workings of science

Concerns about the trustworthiness of science are not confined to fringe groups that reject science entirely. There is substantial unease even among researchers about the reliability of peer review and the reproducibility crisis---where scientific results are not or cannot be tested by replication. Here the authors point out that such worries even apply to mathematics---for many the language of science. This is largely caused by the growing complexity of our knowledge base, where results are more complicated and investigators sometimes have to rely on the results of others that they do not fully understand. This means, as with the sciences, mathematical discoveries increasingly have to be treated as not absolutely reliable, but as part of a process of searching for the truth, and even what it means for something to true.

Comparative Analysis of Science, Problem Solving and Effectiveness of Open Learning Settings for Mathematical Discovery of Circular Materials

Comparative Analysis of Science, Problem Solving and Effectiveness of Open Learning Settings for Mathematical Discovery of Circular Materials