Conjecture Research Articles

Machine Learning Discovers Invariants of Braids and Flat Braids

We use machine learning to classify examples of braids (or flat braids) as trivial or non-trivial. Our machine learning takes the form of supervised learning, specifically multilayer perceptron neural networks. When they achieve good results in classification, we are able to interpret their structure as mathematical conjectures and then prove these conjectures as theorems. As a result, we find new invariants of braids and prove several theorems related to them. This work evolves from our experiments exploring how different types of AI cope with untangling braids with 3 strands, this is why we concentrate mostly on braids with 3 strands.

Towards a polarized semantics for assertion and denial

Abstract Recent logic and linguistic literature suggest that certain forms of denial cannot be reduced to a simple assertion of negation. In particular, the existence of mathematical and empirical conjectures offers a basis for refuting Frege’s thesis of equivalence between denial and assertion of negation. Following this line of thought we develop a formal framework with two primitive illocutionary operators for assertion and denial, where denial is not a simple negation of assertion. We introduce a semantics for the operators mentioned above. Finally, we discuss the relations between asserted and denied contents.

Two conjectures on 3D Voronoi structures: a toolkit with biomedical case studies.

3D Voronoi scaffolds are widely applied in the field of additive manufacturing as they are known for their light weight structural resilience and share many topological similarities to various natural (bone, tumours, lymph node) and synthetic environments (foam, functionally gradient porous materials). Unfortunately, the structural design features that promote these topological similarities (such as the number of vertices) are often unpredictable and require the trial and error of varying design features to achieve the desired 3D Voronoi structure. This article provides a toolkit, consisting of equations, based on over 12 000 3D Voronoi structures. These equations allow design features, such as the number of generating points (G), to be efficiently and accurately predicted based on the desired structural parameters (within ±3G). Based on these equations we are proposing, to the best of our knowledge, two new mathematical conjectures that relate the number of vertices or edges, and the average edge length to G in Voronoi structures. These equations have been validated for a wide range of parameter values and Voronoi network sizes. A design code is provided allowing any of over 12 000 structures to be selected, easily adjusted based on user requirements, and 3D printed. Biomedical case studies relevant to T-cell culturing, bone scaffolds and kidney tumours are presented to illustrate the design code.

Editorial for Journal of AppliedMath (Volume 1, Issue 4)

Numerous fields employ applied mathematics to find solutions to a wide range of problems. Mathematical ideas and expertise are fascinating. Researchers never give up on completing such theories and knowledge to apply them to solving the most recent and challenging problems. Readers will find this issue to be a captivating investigation into applied mathematics from multiple aspects. For instance, some researchers finished the earlier mathematical theories and confirmed the classical mathematical conjecture, and other researchers created mathematical models to address frequently occurring situations. This issue may offer readers helpful information.

Analisis Kemampuan Berpikir Matematis Siswa pada Pembelajaran Aritmatika Sosial Ditinjau dari Model Pembelajaran dan Self Efficacy Siswa

The main problem in this study is students' ability to think mathematically in solving mathematical problems in learning social arithmetic in their class through minimum guidance-based learning (PBL and DL). This research uses a mixed method to look at the description of students' mathematical thinking abilities and the tendency of these abilities based on students' self-efficacy levels. The results showed that the increase in students' mathematical thinking skills at each level of low self-efficacy based on n-gain calculations. In addition, there is no significant difference in the scores for improving students' mathematical thinking skills based on the learning model and the level of self-efficacy. There is no significant interaction between the Learning Model and the Level of Self-efficacy in determining the increase in the average score of students' mathematical thinking abilities. Other findings show that students with low self-efficacy tend to have limitations in the ability to think mathematically in the process of conjecture and convincing. On the other hand, students with moderate and high self-efficacy have more complete mathematical thinking abilities, including specialization, generalization, conjecture, and convincing. However, in the process of conjecture and convincing, two sub-processes are found, namely knowledge modification using factual knowledge, contextual tools, or substantive thinking, as well as the process of selecting relevant information in solving problems. Therefore, the two groups are divided into four groups based on the way students use conjecture and convincing mathematical thinking abilities, namely: (1) students use specialization mathematical thinking skills based on the knowledge learned from the teacher, (2) students use specialization mathematical thinking abilities based on knowledge acquired in everyday life, (3) students use convincing mathematical thinking abilities by examining the formula used or based on knowledge learned from the teacher, and (4) students use convincing mathematical thinking abilities based on the knowledge obtained in everyday life (factual knowledge).

How to Pose Problems on Periodicity and Teaching with Problem Posing

Research on how to pose good problems in mathematical science is rarely touched. Inspired by Kilpatrick’s “Where do good problems come from?”, the current research investigates the problem of the specific problem posed by mathematicians in mathematical sciences. We select a recent mathematical conjecture of Yang related to periodic functions in the field of functions of one complex variable. These problems are extended to complex differential equations, difference equations, differential-difference equations, etc. Through mathematical analysis, we try to reproduce the effective strategies or techniques used by mathematicians in posing these new problems. The results show that mathematicians often use generalization, constraint manipulation, and specialization when they pose new mathematical problems. Conversely, goal manipulation and targeting a particular solution are rarely used. The results of the study may have a potential impact and promotion on implementing problem-posing teaching in primary and secondary schools. Accordingly, teachers and students can be encouraged to think like mathematicians, posing better problems and learning mathematics better. Then, we give some examples of mathematical teaching at the high school level using problem-posing strategies, which are frequently employed by mathematicians or mathematical researchers, and demonstrate how these strategies work. Therefore, this is a pioneering research that integrates the mathematical problem posing by mathematicians and the mathematical problem posing by elementary and secondary school math teachers and students. In addition, applying constraint manipulation and analogical reasoning, we present four unsolved mathematical problems, including three problems of complex difference-related periodic functions and one problem with complex difference equations.

New explorations and remarkable inequalities related to Fortune’s conjecture and fortunate numbers

Fortune's conjecture (named after the social anthropologist Reo Franklin Fortune) is an extremely elegant mathematical conjecture that always remains an open problem in number theory. It is a conjecture about prime numbers, which leads to the so-called "fortunate numbers" (not to be confused with "lucky numbers"): Reo F. Fortune predicted that no fortunate number is composite. This conjecture impresses us all as mathematicians, that's why we decided that it will be the subject of this paper, which has many objectives and very interesting findings, among them: - Highlighting numerous properties of the fortunate numbers. - Giving a proof of Fortune's conjecture in a particular case, by two different methods one of which is original. - Presenting many counterexamples that reinforce the previous point, when the satisfied hypothesis in that particular case, is not met. - Proving a new remarkable inequality that all the 3000 first (known until now) fortunate numbers perfectly fulfill. Despite our continuous research (quite recently) on the subject of Fortune’s conjecture, we have never found a mathematical reference with a variety of ideas dealing with this conjecture, so we hope that this paper will be the first scientific work containing multiple ideas, comments, results and goals, and contributing significantly to find a definitive solution of Fortune's conjecture, therefore this paper may advance the field.

Making mathematics together by modeling shared experiences

IntroductionThis article illustrates a pedagogical approach to integrating models and modeling in Geometry with mathematics teacher-learners (MTLs). It analyzes the work of MTLs in a course titled “Computers, Teaching, and Mathematical Visualization” (or “MathViz”), which is designed to engage MTLs in making mathematics together. They use a range of both physical and virtual models of 2-manifolds to formulate and investigate geometric conjectures of their own.ObjectivesThe article articulates the theoretical basis and design rationale of MathViz; it analyzes illustrative examples of the discourse produced in collaborative investigations; and it describes the impact of this approach in the students’ own voices.MethodsMathViz has been iteratively refined and researched over the past 6 years. This study focuses on one iteration, aiming to capture the phenomenological experience of the MTLs as they structured and pursued their own mathematical investigations. Video data from two class sessions of the Fall 2021 iteration of the course are analyzed to illustrate the discourse of collaborating students and the nature of their shared inquiry. Excerpts from this class’s Learning Journals are then analyzed to capture themes across students’ experience of the course and their perspectives on its impact.ResultsAnalysis of students’ discourse (while investigating cones) shows how they used models and gesture to make sense of geometric phenomena; forged connections with investigations they had conducted throughout the course on different surfaces; and articulated and proved mathematical conjectures of their own. Analysis of students’ Learning Journals illustrates how experiences in MathViz contributed to their conceptualization of making mathematics together, using a variety of models and technologies, and developing a set of practices that that they could introduce with their future students.DiscussionAn argument is made that this approach to collective mathematical investigation is not only viable and valuable for MTLs, but is also relevant to philosophical reflections about the nature of mathematical knowledge-creation.

Literature and Artificial Intelligence

The essay proposes a ‘literary’ perspective on the history of artificial intelligence. On the one hand, this means a literary-historical retrospective of the decades since the Second World War, but it also emphasizes the basic probabilistic trait of artificial intelligence, which becomes recognizable especially against the background of current technological developments. The space of literature is the horizon of possibility that accompanies what is real for man in early modern times and to which his activity is directed. When ‘intelligent machines’ today turn to their environment in the mode of mathematical conjecture, this marks not only an epistemic, but also an anthropological caesura.

Some explorations of linear algebra

It is a fundamental problem in quantum information whether a particular quantum state of a composite system is entangled. It has enormous potential in quantum error correction, quantum cryptography, and quantum teleportation applications. This problem can be transferred in the form of a mathematical conjecture in language of linear algebra. In this paper, the authors explain the important applications, convenience, efficiency of using linear algebra in math physics, and computer science. The authors give some examples of linear algebra used in various areas, including datum coordinate system finding a location, encryption and decryption algorithm, storage of images, classical mechanics, and quantum physics. The authors list the definition of the matrix, real matrix, complex matrix, diagonal matrix, identity matrix, scalar matrix, trace, rank, and determinants of matrices. The authors explored Laplace expansion, transpose, inverse, conjugate of a matrix and addition, multiplication between matrices and between a scalar and a matrix, Kronecker product and their properties of matrices, the definitions and solving method of eigenvalue and eigenvector, and the diagonalisation and its conditions of matrices. The authors introduce the applications of matrix transformations and operations in programs. The authors explain two encryption and decryption algorithms based on linear algebra and their strengths and weaknesses.

Confirming Mathematical Conjectures by Analogy

Analogy has received attention as a form of inductive reasoning in the empirical sciences. Its role in mathematics has, instead, received less consideration. This paper provides a novel account of how an analogy with a more familiar mathematical domain can contribute to the confirmation of a mathematical conjecture. By reference to case-studies, we propose a distinction between an incremental and a non-incremental form of confirmation by mathematical analogy. We offer an account of the former within the popular framework of Bayesian confirmation theory. As for the non-incremental notion, we defend its role in rationally informing the prior credences of mathematicians in those circumstances in which no new mathematical evidence is introduced. The resulting framework captures many important aspects of the use of analogical inference in the domain of pure mathematics.

The Decision Problem for Effective Procedures

The “somewhat vague, intuitive” notion from computability theory of an effective procedure (method) or algorithm can be fairly precisely defined even if it is not sufficiently formal and precise to belong to mathematics proper (in a narrow sense)—and even if (as many have asserted) for that reason the Church–Turing thesis is unprovable. It is proved logically that the class of effective procedures is not decidable, i.e., that there is no effective procedure for ascertaining whether a given procedure is effective. This result is proved directly from the notion itself of an effective procedure, without reliance on any (partly) mathematical lemma, conjecture, or thesis invoking recursiveness or Turing-computability. In fact, there is no reliance on anything very mathematical. The proof does not even appeal to a precise definition of ‘effective procedure’. Instead, it relies solely and entirely on a basic grasp of the intuitive notion of an effective procedure. Though the result that effectiveness is undecidable is not surprising, it is also not without significance. It has the consequence, for example, that the solution to a decision problem, if it is to be complete, must be accompanied by a separate argument that the proposed ascertainment procedure invariably terminates with the correct verdict.

Effective Procedures

The “somewhat vague, intuitive” notion from computability theory of an effective procedure (method) or algorithm can be fairly precisely defined, even if it does not have a purely mathematical definition—and even if (as many have asserted) for that reason, the Church–Turing thesis (that the effectively calculable functions on natural numbers are exactly the general recursive functions), cannot be proved. However, it is logically provable from the notion of an effective procedure, without reliance on any (partially) mathematical thesis or conjecture concerning effective procedures, such as the Church–Turing thesis, that the class of effective procedures is undecidable, i.e., that there is no effective procedure for ascertaining whether a given procedure is effective. The proof does not even appeal to a precise definition of ‘effective procedure’. Instead, it relies solely and entirely on a basic grasp of the intuitive notion of such a procedure. Though the result itself is not surprising, it is also not without significance. It has the consequence, for example, that the solution to a decision problem, if it is to be complete, must be accompanied by a separate argument that the proposed ascertainment procedure is, in fact, a decision procedure, i.e., effective—for example, that it invariably terminates with the correct verdict.

Transition from complex numbers to complex function and series.

Complex analysis, traditionally known as the theory of functions of a complex variable, is a branch of mathematical analysis that investigates functions of complex numbers. If n is odd, there is only one order n root, and if n is even, there are only two order n roots. Things change in complex numbers, though. Keep in mind that complex numbers have closed algebra. As a result, there exist n roots of order n always.Dating back to the 16th century when Italian mathematicians Girolamo Cardano and Raphael Bombelli first observed complex numbers when they were trying to solve a certain algebra and developed by Cauchy and Riemann in 19th century, complex analysis nowadays has played a more and more importance role on mathematics, physics and engineering, especially in the fields of algebraic geometry, fluid dynamics, quantum mechanics and so on. In this paper, the birth and discovery of complex number will be discussed. Starting from discovery of complex number, the properties of complex function and power series are elaborated which are foundation for mordern mathematical conjectures.

Reasoning by Analogy in Mathematical Practice

Abstract In this paper, we offer a descriptive theory of analogical reasoning in mathematics, stating general conditions under which an analogy may provide genuine inductive support to a mathematical conjecture (over and above fulfilling the merely heuristic role of ‘suggesting’ a conjecture in the psychological sense). The proposed conditions generalize the criteria of Hesse in her influential work on analogical reasoning in the empirical sciences. By reference to several case studies, we argue that the account proposed in this paper does a better job in vindicating the use of analogical inference in mathematics than the prominent alternative defended by Bartha.

A Revıew of Mathematical Conjectures: Exploring Engaging Topics for University Mathematics Students

A Revıew of Mathematical Conjectures: Exploring Engaging Topics for University Mathematics Students

Distillability Problem of 4×4 Monomial Matrices

It is a fundamental problem in quantum information whether a particular quantum state of a composite system is entangled. It has enormous potential in quantum error correction, quantum cryptography, and quantum teleportation applications. This problem can be transferred in the form of a mathematical conjecture called the distillation conjecture. In the first section of this paper, relevant physical and mathematical information is presented, including basic linear algebra knowledge, the statement, and concrete applications of multiple mathematical knowledge like conjugate, eigenvalue, and singular value. Then, we introduce the distillation conjecture in a mathematical version for a more precise mathematical analysis. In an effort to make more significant headway in proving the conjecture, we selected some theories and findings relating to the Kronecker product, Kronecker sum, eigenvalue, and singular value, then evaluated and grouped them. In addition, we provided multiple proofs of the conjecture under varying conditions and made numerous attempts and hypotheses regarding how to establish the conjecture.

Design of quantum optical experiments with logic artificial intelligence

Logic Artificial Intelligence (AI) is a subfield of AI where variables can take two defined arguments, True or False, and are arranged in clauses that follow the rules of formal logic. Several problems that span from physical systems to mathematical conjectures can be encoded into these clauses and solved by checking their satisfiability (SAT). In contrast to machine learning approaches where the results can be approximations or local minima, Logic AI delivers formal and mathematically exact solutions to those problems. In this work, we propose the use of logic AI for the design of optical quantum experiments. We show how to map into a SAT problem the experimental preparation of an arbitrary quantum state and propose a logic-based algorithm, called Klaus, to find an interpretable representation of the photonic setup that generates it. We compare the performance of Klaus with the state-of-the-art algorithm for this purpose based on continuous optimization. We also combine both logic and numeric strategies to find that the use of logic AI significantly improves the resolution of this problem, paving the path to developing more formal-based approaches in the context of quantum physics experiments.

Analysis of student mathematical investigations ability on transformation geometry in terms of cognitive style

Mathematical investigation is an activity that can encourage an experimental activity, collect data, make observations, identify patterns, make and test conjectures and make generalizations used to improve skills and develop students' mathematical thinking processes optimally. Different cognitive styles can affect students' ability to think and reason, especially in solving mathematical investigative problems. Therefore, this paper will examine mathematical investigations' ability in reflective and impulsive cognitive styles in qualitative descriptive analysis. The subjects in eleventh-grade senior high school at SMA Negeri 2 Mataram, Indonesia. Students were selected using a purposive sampling technique, and six students were selected as subjects in the interview consisting of three reflective students and three impulsive students. The instruments used are mathematical investigation tests, Matching Familiar Figure MFFT tests, and interview guidelines. The results showed that students with a reflective cognitive style were more thorough and systematic in writing down the answers to each point and always thought first in solving problems. Most students went through 4 stages of mathematical investigations: specialization, conjecture, generalization, and justification. While students with impulsive cognitive styles mostly managed to go through 3 stages of mathematical investigations, specialization, conjecture, and generalization, due to a lack of accuracy in solving questions and providing as simple answers as possible according to the question request.

On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation* *Supported by RSF Grant 18-41-05003.

The tunnelling effect predicted by Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors separated by a narrow dielectric. It states existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modelled by a family of differential equations on two-torus depending on three parameters: B (abscissa), A (ordinate), ω (frequency). We study its rotation number ρ(B, A; ω) as a function of (B, A) with fixed ω. The phase-lock areas are the level sets L r ≔ {ρ = r} with non-empty interiors; they exist for (Buchstaber, Karpov, Tertychnyi). Each L r is an infinite chain of domains going vertically to infinity and separated by points. Those separating points for which A ≠ 0 are called constrictions. We show that: (1) all the constrictions in L r lie on the axis {B = ωr}; (2) each constriction is positive: this means that some its punctured neighbourhood on the axis {B = ωr} lies in Int(L r ). These results confirm experiments by physicists (1970ths) and two mathematical conjectures. We first prove deformability of each constriction to another one, with arbitrarily small ω and the same , using equivalent description of model by linear systems of differential equations on (Buchstaber, Karpov, Tertychnyi) and studying their isomonodromic deformations described by Painlevé 3 equations. Then non-existence of ghost constrictions (i.e., constrictions either with , or of non-positive type) with a given ℓ for small ω is proved by slow-fast methods.