Open Mathematical Problems Research Articles

Collatz Conjecture Solution

The Collatz conjecture, also known as the 3n + 1 conjecture, stands as one of the most enduring unsolved problems in mathematics. Proposed by German mathematician Lothar Collatz in 1937, the conjecture poses a deceptively simple question: can iteratively applying two basic arithmetic operations—dividing even numbers by 2 and multiplying odd numbers by 3 and adding 1—lead any positive integer to converge to the value 1? Despite extensive computational verification for vast ranges of integers, a proof or disproof remains elusive. This abstract explores the historical background, various attempts at proving the conjecture, and its implications across mathematical domains.

Open modelling problems: cognitive barriers and instructional prompts

Open mathematical modelling problems that can be solved with multiple methods and have multiple possible results are an important part of school curricula in mathematics and science. Solving open modelling problems in school should prepare students to apply their mathematical knowledge in their current and future lives. One characteristic of these problems is that information that is essential for solving the problems is missing. In the present study, we aimed to analyze students’ cognitive barriers while they solved open modelling problems, and we evaluated the effects of instructional prompts on their success in solving such problems. A quantitative experimental study (N = 263) and a qualitative study (N = 4) with secondary school students indicated that identifying unknown quantities and making numerical assumptions about these quantities are important cognitive barriers to solving open modelling problems. Task-specific instructional prompts helped students overcome these barriers and improved their solution rates. Students who were given instructional prompts included numerical assumptions in their solutions more often than students who were not given such prompts. These findings contribute to theories about solving open modelling problems by uncovering cognitive barriers and describing students’ cognitive processes as they solve these problems. In addition, the findings contribute to improving teaching practice by indicating the potential and limitations of task-specific instructional prompts that can be used to support students’ solution processes in the classroom.

Epidemic Waves in a Stochastic SIRVI Epidemic Model Incorporating the Ornstein–Uhlenbeck Process

The worldwide data for COVID-19 for active, infected individuals in multiple waves show that traditional epidemic models with constant parameters are not able to capture this kind of disease behavior. We solved this major open mathematical problem in this report. We first consider the disease transmission rate for the stochastic SIRVI epidemic model, which satisfies the mean-reverting Ornstein–Uhlenbeck (OU) process, and we propose a new stochastic SIRVI model. We then showed the existence and uniqueness of the global solution and obtained sufficient conditions for the persistent mean and exponential extinction of infectious disease, which have not been given before. In the second part, we derive a nonlinear system of differential equations for the time-dependent transmission rate from the deterministic SIRVI model and present an algorithm to compute the time-dependent transmission rate directly from the given active, infected individuals’ data. We then show that the time-dependent transmission obtained from and perturbed by the Ornstein–Uhlenbeck process could be represented after using a smoothing technique using a finite linear combination of a Gaussian radial basis function, which was obtained from our algorithm. This novel computer-assisted proof provides a theoretical basis for other epidemic models and epidemic waves. Finally, some numerical solutions of the stochastic SIRVI model are presented using COVID-19 data from Saudi Arabia and Austria.

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.

The Role of Open Mathematical Problems in the Formation of the Competence of Plausible Reasoning

Today, modern Internet technologies allow you to access the necessary information at any time from anywhere. In such a situation, the ability to find what is needed from different sources and critically revise it becomes of particular importance, which, in turn, leads to the formation of basic knowledge. Consequently, when preparing young people for real life, the need to develop their critical attitude and thinking is urgent. As a rule, in real life, the initial data is not always clear, the goal is not set clearly, and there may be cases when the way out of this situation is also not the only one. Therefore, a person often faces open-type problems. Such cases can be taken as an interpretation of mathematical open problems. Indeed, open tasks are understood as tasks in which the initial data or the final goal are not clearly formulated. At the same time, students are given the freedom to set tasks to a certain extent, choose a way of thinking and enter additional parameters necessary for its solution. The article discusses plausible reasoning in the process of solving open problems and its role in the formation of students' competencies on plausible reasoning. In addition, various examples of how plausible reasoning arises when solving open problems are given, and an algorithm of actions for composing and solving open problems is given. Here the algorithm is only as an orienting basis, since there are no algorithms for solving open problems. It can be seen from the algorithm that when forming the competence of plausible reasoning, students should be able to solve closed and open problems and combine them in a certain effective sequence. Thus, it is concluded that the analysis of the compilation and solution of open mathematical problems plays an important role in the formation of the competence of plausible reasoning of future mathematics teachers and has a great influence on the preparation of young people for changing real life.

Singularity formation to the 2D Cauchy problem of nonbarotropic magnetohydrodynamic equations without heat conductivity

AbstractThe question of whether the two‐dimensional (2D) nonbarotropic compressible magnetohydrodynamic (MHD) equations with zero heat conduction can develop a finite‐time singularity from smooth initial data is a challenging open problem in fluid dynamics and mathematics. Such a problem is interesting in studying global well‐posedness of solutions. In this paper, we proved that, for the initial density allowing vacuum states, the strong solution exists globally if the density and the pressure are bounded from above. Our method relies on weighted energy estimates and a Hardy‐type inequality.

Analysis of Polynomial Time and Non-Polynomial Time of Algorithms

Abstract: The P vs NP problem is one of the most significant open problems in computer science and mathematics. This problem asks whether every problem that can be solved in polynomial time can also be verified in polynomial time. The purpose of this research paper is to explore the P vs NP problem and its relevance in the analysis of algorithms. We will discuss the techniques used to design and analyze algorithms, such as divide-and-conquer, dynamic programming, and greedy algorithms, and their relation to the P vs NP problem. We will also examine some examples of polynomial-time algorithms and NP-hard problems and analyze their time and space complexity, addition to which we will analyze the NP-Complete Problem and finally looking the current scenario of the P & NP and stating its significance.

Plausible Reasoning by Solving Open Mathematical Problems

The article deals with the problem of forming plausible reasoning of students through open mathematical problems, justifies its necessity in preparing young people for real life.As a rule, in real life, the initial data are not always clearly formulated, the goal is not clearly set, and there are situations when there is not the only way out of this situation. Therefore, a person often faces open-type tasks. Open tasks are understood as tasks in which the initial data or the final goal are not clearly formulated. At the same time, students are to some extent given the freedom to set a task, choose a way of thinking and introduce additional parameters necessary for its solution. Within the framework of this study, a review of existing scientific literature, publications published in scientific journals was conducted, which necessitated the analysis of plausible reasoning among students, as well as the compilation and solution of open mathematical problems. The algorithm of actions for solving open problems and an example of identifying the possibili

An Asymptotic Approach for the Collatz Conjecture

lt is well known that the Collatz Conjecture is one of the unsolved problems in mathematics. Supercomputer simulation has confirmed that the Collatz conjecture is correct for all positive integers unti1 2.95*10^20. Recently, Terence Tao proved that almost all orbits of the Collatz map attain almost bounded values. It is to say that the Collatz conjecture is almost correct for almost all positive integers. This paper proposes an asymptotic approach for the Collatz Conjecture by the mathematical induction and probability theory. The result of this paper means that the Collatz conjecture is correct for almost all positive integers. Keywords: Collatz Conjecture, asymptotic approach, probability, mathematical induction.

Вершинно-зважена гіпотеза Сеймура

Seymour's conjecture is one of the most famous unsolved mathematical problems in graph theory, which was formulated by Paul Seymour in 1990. This problem is also known as the «second neighborhood problem». A directed graph models a social network in which no two people knoweach other at the same time. This conjecture states that there will be at least one person for whom acquaintances of acquaintances will be no less than ac-quaintances. Definitions and basic theorems of graph theory are described in [1-3]. For an arbitrary graph, Seymour's hypothesis remains unsolved, but there are already proofs for partial cases and for some types of graphs, which are given in [4-6].Seacrest Tyler [5] investigated Seymour's conjecture for graphs with weighted arcs.In [6] showed that every simple digraph without loops or digons con-tains a vertexvfor which the second neighborhood is greater than or equal to the first multiplied by a certain constant.In [7] provided sufficient conditions under which there must exist some vV(D), as wellas examine properties of a minimal graph which does not have such a vertex. We show that if one such graph exists, then there exist in-finitely many strongly-connected graphs having no such vertex.The relevance of the chosen research topic is determined by the rapid pace of development of modern graph theory, which is associated with the expansion of its scope of use: business, logistics, tourism and, most im-portantly, modeling of various networks. One extension of the conjecture is to consider vertex-weighted di-graphs. In this paper we introduce a version of the conjecture for vertex-weighted digraphs and proved that the Seymour conjecture is equivalent to conjecture for vertex-weighted digraphs.

Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications

This small collaborative workshop brought together experts from the Sino-German project working in the field of advanced numerical methods for hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the convergence of numerical methods and proper solution concepts were addressed as well.

Исследование асимптотической последовательности высот правильных симплексов при увеличении размерности евклидова пространства

The problem of spherical packing and the problem of the distribution of points on a sphere are well-known unsolved mathematical problems. In particular, the problem of the distribution of points on a sphere is included in the well-known list of unsolved mathematical problems of Steve Smale. Accordingly, a significant interest for the study is both the direct search for solutions to problems and research in areas related to known problems. In particular, the problem of spherical dense packing and coating packing is directly related to the study of regular, irregular and layered lattices, problems of noise-resistant coding, synthesis and analysis of broadband signals, mathematical research of multidimensional spaces, crystallography, methods for solving differential equations and calculating n-dimensional integrals, and many other particular problems and entire fields of knowledge. The paper presents the results of studies of some properties of n-dimensional regular simplex in Euclidean space. The vectors forming the n-dimensional regular simplex define the fundamental basis of the hexagonal lattice. The subject of the study is some properties and regularities of a vector object forming a regular simplex in n-dimensional Euclidean space, the novelty of the results lies in a formulated and proven theorem, several formulas that allow recursive and non-recursive calculations of some characteristics of n-dimensional regular simplex and spherical hexagonal packages. The object of the study is an n-dimensional regular simplex. Main results a theorem on the asymptotic character of the sequence of heights of regular simplex with increasing dimension of space is formulated and proved, the consequences of the theorem and remarks supplementing the theorem are formulated. The conclusion of the article contains additions and numerical examples with comments for some research results. The practical significance lies in the proposed mathematical tools that contribute to the development of the theory of spherical packing and the study of the properties of geometric objects in n-dimensional Euclidean space.

Пилотный эксперимент по развитию исследовательских умений школьников при обучении математике в условиях инклюзии

Introduction. The implementation of educational programs in a modern school involves the formation of universal educational actions, including those actions that ensure the successful implementation of research activities, the so-called research skills in all categories of students. Unfortunately, there is currently an increasing trend in the number of children and adolescents with disabilities, which actualizes the problem of introducing inclusive education. The potential of mathematics as a school discipline can be used to form and evaluate the research skills of schoolchildren under conditions of inclusion. The purpose of the article is to present the experience of organizing a pilot experiment on the formation of research skills of ninth grade students in teaching mathematics under inclusion conditions and to analyze its results. Research methods. The analysis and generalization of scientific literature on the theory of problem-based learning, on the problems of mathematical training of children with disabilities and the use of interactive teaching methods are applied. Empirical methods were used: questioning of teachers and teachers of mathematics of educational organizations; observation; design of didactic tools to ensure the process of formation of research skills of all categories of students; pilot experiment and analysis of its results. The study involved the experimental ninth grade of school No. 55 in Novokuznetsk, in which there was a student with disorders of the musculoskeletal system (DMS). In the experiment, a binomial criterion was used to process the results. Results. The students of the experimental group were involved in the process of forming research skills, while the key feature of the methodology used is its reliance on a specially developed system of multi-level research open mathematical problems and differentiated educational research with the possibility of their implementation by means of a digital educational environment in online and offline format. A statistically significant positive dynamics of the development of research skills of ninth graders, including a student with disabilities, was revealed: Zem>Zcrit (2.46>1.64). Conclusion. The methodological techniques for the formation of research skills of schoolchildren in teaching mathematics under conditions of inclusion are summarized; the conclusion is made about improving the quality of mathematical education and promising areas for further research.

Classification of open mathematical problems and their role in academic achievement and motivation of students

Problems of developing subject education and science have become more acute during different social changes. The most problematic is the teaching of exact disciplines, including mathematical disciplines (mathematics, algebra, geometry), which form knowledge for working in high-tech industries. It is worth noting that the methodology of teaching mathematics has accumulated a large unsystematized portfolio of new techniques and methods. Thus, open problems aimed at developing students’ creativity are used more actively in the lessons. Tasks that require students to use non-standard approaches to the solution affect academic success. But a teacher of mathematics needs to have a classification of problems in the educational process to choose tasks of an open type correctly. In this regard, the article deals with substantiating the classification of open problems for teaching mathematical disciplines and developing methodological guidelines for lessons that help increase learning motivation and academic achievement of students. The leading research methods are the following: observation of teachers’ methodological work, talks with teachers, analysis of methodological developments, educational documentation and teachers’ surveys. Statistical processing of the results using Wilcoxon W test. In 2020-2021 the authors carried out an experiment with 358 students from 16 classes. Based on its results, the authors of the article have identified a classification of open problems for teaching mathematical disciplines; developed and implemented methodological guidelines for lessons of mathematics that contribute to developing students’ creativity in the academic progress. The authors evaluated the effectiveness of using the selected classification of open problems for teaching mathematical disciplines. They concluded about the increase of group motivation among students and academic performance in mathematical disciplines. The practical use of the classification of open problems allows the teacher to differentiate educational problems in mathematics, taking into account the relationship between their logical structure and the level of ability achieved by the student to solve problems in mathematics. Methodological guidelines developed by the authors can be used to help increase the learning motivation and academic achievement of students while preparing for mathematics lessons.

About Inverse Laplace Transform of a Dynamic Viscosity Function.

A dynamic viscosity function plays an important role in water hammer modeling. It is responsible for dispersion and decay of pressure and velocity histories. In this paper, a novel method for inverse Laplace transform of this complicated function being the square root of the ratio of Bessel functions of zero and second order is presented. The obtained time domain solutions are dependent on infinite exponential series and Calogero–Ahmed summation formulas. Both of these functions are based on zeros of Bessel functions. An analytical inverse will help in the near future to derive a complete analytical solution of this unsolved mathematical problem concerning the water hammer phenomenon. One can next present a simplified approximate form of this solution. It will allow us to correctly simulate water hammer events in large ranges of water hammer number, e.g., in oil–hydraulic systems. A complete analytical solution is essential to prevent pipeline failures while still designing the pipe network, as well as to monitor sensitive sections of hydraulic systems on a continuous basis (e.g., against possible overpressures, cavitation, and leaks that may occur). The presented solution has a high mathematical value because the inverse Laplace transforms of square roots from the ratios of other Bessel functions can be found in a similar way.

Deep Learning Architectures for Approximating Goldbach’s Function in New Regions

Goldbach conjecture is one of the most famous open mathematical problems. He asserts that: Every even number greater than two is the sum of two prime numbers. The Goldbach function receives an even number and returns the number of different ways to write it as an unordered sum of two prime numbers. We developed a simple multilayer perceptron that attempts to predict Goldbach’s function. This simple model performs well when trained and tested on numbers up to 4 million. However, as expected, the model’s performance significantly deteriorates when trained on smaller numbers (up to 4 million) but tested on larger numbers (4–10 million). To overcome this problem, we present two novel deep learning architectures. In these architectures, we introduce two types of multiplication layers, which we believe are more appropriate for solving mathematical relations. We show that both architectures significantly outperform the simple multilayer perceptron when trained on smaller numbers and tested on larger numbers. We further improve the performance of the deep learning architectures by using a known analytically derived estimation that is used in order to normalize the model’s output.

Early Days of Computing at Manchester: Max Newman's Royal Society Project, 1946–1951

Max Newman became Professor of Pure Mathematics at Manchester University in 1945, having led the outstandingly successful Colossus cryptanalytical project at Bletchley Park during the war. He obtained Royal Society funding to set up a Computing Machine Laboratory at Manchester to apply similar digital electronic technologies to the solution of as-yet unsolved problems in pure mathematics. But what kind of (special-purpose) digital computer was likely to be most suitable? Using new evidence, this article charts the progress of Newman's project in some detail, attempting for the first time to set Newman's Colossus-centric vision in the context of John von Neumann's IAS architecture. We contrast mathematical and engineering perspectives. Though Max Newman's original plans were not realized, innovative computer design at Manchester flourished in directions unforeseen by Newman. It was his 1946 initiative that led, somewhat indirectly, to significant computing milestones.

The Non-Trivial Zeros of The Riemann Zeta Function through Taylor Series Expansion and Incomplete Gamma Function

The Riemann zeta <img src=image/13425982_01.gif> function <img src=image/13425982_02.gif> is valid for all complex number <img src=image/13425982_03.gif>, for the line <img src=image/13425982_04.gif> = 1. Euler-Riemann found that the function equals zero for all negative even integers: −2, −4, −6, ... (commonly known as trivial zeros) has an infinite number of zeros in the critical strip of complex numbers between the lines <img src=image/13425982_04.gif> = 0 and <img src=image/13425982_04.gif> = 1. Moreover, it was well known to him that all non-trivial zeros are exhibiting symmetry with respect to the critical line <img src=image/13425982_05.gif>. As a result, Riemann conjectured that all of the non-trivial zeros are on the critical line, this hypothesis is known as the Riemann hypothesis. The Riemann zeta function plays a momentous part while analyzing the number theory and has applications in applied statistics, probability theory and Physics. The Riemann zeta function is closely related to one of the most challenging unsolved problems in mathematics (the Riemann hypothesis) which has been classified as the 8th of Hilbert's 23 problems. This function is useful in number theory for investigating the anomalous behavior of prime numbers. If this theory is proven to be correct, it means we will be able to know the sequential order of the prime numbers. Numerous approaches have been applied towards the solution of this problem, which includes both numerical and geometrical approaches, also the Taylor series of the Riemann zeta function, and the asymptotic properties of its coefficients. Despite the fact that there are around 10¹³, non-trivial zeros on the critical line, we cannot assume that the Riemann Hypothesis (RH) is necessarily true unless a lucid proof is provided. Indeed, there are differing viewpoints not only on the Riemann Hypothesis's reliability, but also on certain basic conclusions see for example [16] in which the author justifies the location of non-trivial zero subject to the simultaneous occurrence of <img src=image/13425982_06.gif>, and omitting the impact of an indeterminate form <img src=image/13425982_07.gif>, that appears in Riemann's approach. In this study, we also consider the simultaneous occurrence <img src=image/13425982_06.gif> but we adopt an element-wise approach of the Taylor series by expanding <img src=image/13425982_08.gif> for all <img src=image/13425982_09.gif> = 1, 2, 3, ... at the real parts of the non-trivial zeta zeros lying in the critical strip for <img src=image/13425982_10.gif> is a non-trivial zero of <img src=image/13425982_11.gif>, we first expand each term <img src=image/13425982_08.gif> at <img src=image/13425982_12.gif> then at <img src=image/13425982_13.gif>. Then in this sequel, we evoke the simultaneous occurrence of the non-trivial zeta function zeros<img src=image/13425982_06.gif>, on the critical strip by the means of different representations of Zeta function. Consequently, proves that Riemann Hypothesis is likely to be true.

Peer-feedback as a part of collaborative problem solving

The article presents a study within the international project named The Social Unit of Learning. This study examines the ways in which students interact with each other when solving an open mathematical problem in small groups (four students a group). Attention is focused on peer feedback, especially on its form and how it affects the process of solving the problem. From our analysis, feedback appears often in the group discussion and takes the form of both positive (confirmation; question asking for clarification, request for clarification, eliciting) and negative reactions (rejection, explained or unexplained objection, doubt, question involving doubt). The analysis also identified a communication pattern that generalizes the course of the group discussion and factors influencing the quality of peer feedback related to the content structure of the problem (thematic, conceptual and competence layer).

Analysis of Difficulties of Junior High School Students in Solving Closed and Open Mathematical Problems

This study aims to describe the difficulties of junior high school students in solving closed and open mathematical problems and to describe the causes of junior high school students experiencing these difficulties. As many as 25 students of class VIII-D from one of the State Junior High Schools in Bandung Regency were used as research samples. Data was collected by using test techniques, interviews, and document studies. The stages of data analysis carried out are data collection, data reduction, data presentation, and drawing conclusions. The results of the study showed that students had difficulty in converting the problem into a mathematical model, determining the required decision variables, calculating multiplication and division operations, and determining more than one answer in solving open problems. The difficulties experienced by these students are caused by the following: students feel that the problem given is something new, students are less careful in understanding the problems given, students still do not master the material, students are less careful in performing arithmetic operations, and students not accustomed to solving problems in the form of problem solving and open questions.