Four-Dimensional Mathematics Creates the Super Universe

In the common theory of the Universe, the redshift of the light wavelength from distant stars indicates the speed of the star. In this study, the model of the Universe is the surface volume of the four-dimensional sphere, and the shape of the Universe results in the most of the redshift of light wavelength. Therefore, there is no dark energy accelerating the Universe. The surface of the four-dimensional sphere is a volume, and this volume is a good model for the Universe. The surface volume of the four-dimensional sphere has been explained by a model of four-dimensional cube, within which the forming of surface volume can be easily shown. The model of four-dimensional cube containing six side cubes is ingenious for explaining the structure of the four-dimensional Universe, but it is not enough because the four-dimensional cube has not six side cubes, but eight side cubes. Therefore, in this study a better method has been created to construct the four-dimensional cube. Our three-dimensional Universe is the surface of the four-dimensional sphere Universe. The volume of our three-dimensional Universe is finite, and beneath it is the infinite volume four-dimensional Super Universe. Two important basic formulae have been derived: The surface volume of the four-dimensional sphere is π3R3 in which R is the radius of the sphere, and the fourth-power volume of the four-dimensional sphere is 1/4 π3R4. The volume of the Universe has been calculated π3R3 = 62 × 1030 ly3. Time as the fourth dimension of the space takes effect only near the speed of light, and therefore it has been ignored in this study.

Localized wave solutions and interactions of the (2+1)-dimensional Hirota-Satsuma-Ito equation

This paper studies the(2+1)-dimensional Hirota-Satsuma-Ito equation.Based on an associated Hirota bilinear form,lump-type solution,two types of interaction solutions,and breather wave solution of the(2+1)-dimensional Hirota-Satsuma-Ito equation are obtained,which are all related to the seed solution of the equation.It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of resonance stripe solitons,and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton.Furthermore,the breather wave solution is also obtained by reducing the two-soliton solutions.The trajectory and period of the one-order breather wave are analyzed.The corresponding dynamical characteristics are demonstrated by the graphs.

Abundant invariant solutions of extended(3+1)-dimensional KP-Boussinesq equation

Lie group analysis method is applied to the extended(3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq equation and the corresponding similarity reduction equations are obtained with various infinitesimal generators.By selecting suitable arbitrary functions in the similarity reduction solutions,we obtain abundant invariant solutions,including the trigonometric solution,the kink-lump interaction solution,the interaction solution between lump wave and triangular periodic wave,the two-kink solution,the lump solution,the interaction between a lump and two-kink and the periodic lump solution in different planes.These exact solutions are also given graphically to show the detailed structures of this high dimensional integrable system.

Dynamics of Nonlinear Waves in(2+1)-Dimensional Extended Boiti-Leon-Manna-Pempinelli Equation

Based on the Hirota bilinear method,this study derived N-soliton solutions,breather solutions,lump solutions and interaction solutions for the(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli equation.The dynamical characteristics of these solutions were displayed through graphical,particularly revealing fusion and ssion phenomena in the interaction of lump and the one-stripe soliton.

Hyper Catastrophe on 4-Dimensional Canards

When discovering the potential of canards flying in 4-dimensional slow-fast system with a bifurcation parameter, the key notion “symmetry” plays an important role. It is of one parameter on slow vector field. Then, it should be determined to introduce parameters to all slow/fast vectors. It is, however, there might be no way to explore for another potential in this system, because the geometrical structure is quite different from the system with one parameter. Even in this system, the “symmetry” is also useful to obtain the potentials classified by R. Thom. In this paper, via the coordinates changing, the possible way to explore for the potential will be shown. As it is analyzed on “hyper finite time line”, or done by using “non-standard analysis”, it is called “Hyper Catastrophe”. In the slow-fast system which includes a very small parameter , it is difficult to do precise analysis. Thus, it is useful to get the orbits as a singular limit. When trying to do simulations, it is also faced with difficulty due to singularity. Using very small time intervals corresponding small , we shall overcome the difficulty, because the difference equation on the small time interval adopts the standard differential equation. These small intervals are defined on hyper finite number N, which is nonstandard. As and the intervals are linked to use 1/N, the simulation should be done exactly.

Multiverse/Hyperverse Models: (4 + 1)-Dimensional Landscape (Black Saturns, Bousso-Hawking Nucleation, Gogberashvili Multiverses, Schwarzschild-De Sitter Nurseries) and a (3 + 1)-Dimensional Model for Dark Energy

We consider the Hyperverse as a collection of multiverses in a (4 + 1)-dimensional spacetime with gravitational constant G. Multiverses in our model are bouquets of thin shells (with synchronized intrinsic times). If gis the gravitational constant of a shell Sand εits thickness, then G~εg. The physical universe is supposed to be one of those thin shells inside the local bouquet called Local Multiverse. Other remarkable objects of the Hyperverse are supposed to be black holes, black lenses, black rings and (generalized) Black Saturns. In addition, Schwarzschild-de Sitter multiversal nurseries can be hidden inside those Black Saturns, leading to their Bousso-Hawking nucleation. It also suggests that black holes in our physical universe might harbor embedded (2 + 1)-dimensional multiverses. This is compatible with outstanding ideas and results of Bekenstein, Hawking-Vaz and Corda about “black holes as atoms” and the condensation of matter on “apparent horizons”. It allows us to formulate conjecture 12.1 about the origin of the Local Multiverse. As an alternative model, we examine spacetime warping of our universe by external universes. It gives data for the accelerated expansion and the cosmological constant Λ, which are in agreement with observation, thus opening a possibility for verification of the multiverse model.

Bifurcations, Analytical and Non-Analytical Traveling Wave Solutions of (2 + 1)-Dimensional Nonlinear Dispersive Boussinesq Equation

For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.

The Past and Future of Aesthetics in Mathematics

This paper explores the connotations of mathematical aesthetics and its connections with art,facilitated by collaboration with Ester,an individual with an artistic professional background.It begins by tracing the historical evolution of aesthetics from the classical pursuit of authenticity to the modern shift toward self-expression in art.The discussion then highlights the similarities in the pursuit of truth between mathematics and art,despite their methodological differences.Through an analysis of aesthetic elements in mathematics,such as lines and function graphs,the article illustrates that the beauty of mathematics is not only manifested in cognitive processes but can also be intuitively expressed through visual arts.The paper further examines the influence of mathematics on the development of art,particularly how Leonardo da Vinci applied mathematical principles to his artworks.Additionally,the article addresses art students’perceptions of mathematics,proposes the customization of math courses for art students,and discusses future trends in the integration of mathematics and art,emphasizing the significance of art therapy and the altruistic direction of art.Lastly,the authors use a poster to visually convey the idea that the beauty of mathematics can be experienced through the senses.

Integration of Mathematics History into Junior Middle School Education: A Pedagogical Approach to Enhance Mathematical Literacy

The integration of the history of mathematics into junior middle school mathematics education represents a significant focus of international research in mathematics and education.The mathematics curriculum standards for compulsory education have emphasized the importance of incorporating the history of mathematics,aiming to gradually integrate it into the mathematics classroom.However,in the practical implementation of junior middle school mathematics education,the effective combination of the history of mathematics with teaching methodologies remains largely unexplored.This article explores the integration of junior middle school mathematics education and the history of mathematics,aiming to provide classroom teaching recommendations for teachers and promote the formation of students’mathematical literacy.

Enhancing Learners’Performance in Grade 7 Mathematics Through 50-30-20 Exercise

Assessment exercises constitute a crucial component of the teaching and learning process,serving the purpose of gauging the degree to which learning objectives have been accomplished.This study aims to assess the mathematics performance of Grade 7 learners using the 50-30-20 exercise.Specifically,this study seeks to determine the learners’pre-test and post-test mean scores,identify significant differences between the pre-test and post-test results,evaluate learners’exercises,and propose enhanced exercises.The research employs a quasi-experimental design,with 40 Grade 7 learners in the school year 2023-2024 as participants,selected through purposive non-random sampling.Statistical data analysis involves the use of mean,standard deviation,paired t-test,and Cohen’s D effect size.Ethical considerations were paramount,as evidenced by a letter of authorization from the school head outlining the strict adherence to voluntary participation,informed parental consent,anonymity,confidentiality,risk mitigation,results-sharing protocols,and the commitment to keeping research data confidential.The data yielded a remarkable outcome:the experimental group exhibited improvement in both the pre-test and post-test.This result substantiates the initial objective of the study,showcasing a noteworthy and favorable performance among the participants.Consequently,it suggests that a majority of the participants strongly agree that the 50-30-20 exercises contribute to enhancing their understanding and problem-solving skills,as well as their ability to grasp mathematical concepts and improve their overall performance in mathematics.Therefore,the 50-30-20 exercises not only facilitated students in understanding mathematics lessons but were also aligned with the Department of Education’s development plan.

Measurement Analysis and Evaluation of Twenty-Five Years of Mathematics Education Research in China:Visual Econometric Analysis Based on CiteSpace Knowledge Mapping(1999-2024)

Since the new century,China’s mathematics curriculum reform in basic education has continued to move forward in attempts and explorations,presenting many new changes,trends,movements,and developments.Sorting out,analyzing,and summarizing the achievements,experiences,problems,and challenges in this journey are conducive to providing insights for the reform and development of the Chinese basic education mathematics curriculum in the new era.This paper analyses the research on mathematics education in China(1999-2024)using the visual measurement of CiteSpace knowledge mapping,hoping to provide directions for the future of mathematics education in China.

A Study on the Application of Task-Driven Teaching Method in High School Mathematics Teaching: Taking “The Determination of Vertical Line and Plane” as an Example

The“Ordinary High School Mathematics Curriculum Standards(2017 Edition,2020 Revision)”clearly stated in“Teaching Suggestions”that“teaching activities based on the core literacy of mathematics should grasp the essence of mathematics,create appropriate teaching situations,put forward appropriate mathematical questions,stimulate students to think and communicate,and form and develop the core literacy of mathematics.”The task-driven teaching model is a new type of teaching method that takes tasks as the main line,teachers as the guide,and students as the main body,which can enable students to engage deeply in classroom discussions and think actively.Based on the characteristics and principles of the task-driven teaching method,this paper designs a high school mathematics classroom teaching based on the task-driven teaching method,hoping to provide a reference for the majority of front-line teachers.

Measurement Analysis and Evaluation of Twenty- Five Years of Chinese Mathematics Textbooks: Visual Analysis Based on CiteSpace Knowledge Graph (1999-2024)

At present,textbooks based on core literacy have become the inevitable demands of China’s curriculum reform,and the literacy of textbook goal construction is the key to the implementation of core literacy requirements,which is a huge challenge for textbook compilers.In this paper,we use the visual metrology of the CiteSpace knowledge graph to analyze Chinese mathematics textbooks(1999-2024),hoping to guide the future direction of Chinese mathematics textbook research.

Curriculum Reform of Higher Vocational Mathematics Under the Background of Digital Transformation

Under the background of digital transformation,the reform of the higher vocational mathematics curriculum faces urgent challenges and opportunities.This article explores the impact of digital transformation on the reform of higher vocational mathematics curriculum and emphasizes the importance of improving teaching methods centered on learners.The article proposes specific reform methods and discusses the practical application of digital technology in the reform process.By combining digital technology with specific reform methods,further conducting innovative practice research,and continuously exploring the path of reform,we can effectively improve the quality of higher vocational mathematics classroom teaching and provide strong support for the cultivation of comprehensive qualities and employment abilities.

Discussion on the Skillful Use of Mind Maps in High School Mathematics Teaching

With the continuous development of China’s education,the social requirements for high school teaching are constantly improving.The teaching of high school mathematics is a key point in the high school curriculum,but also a major difficulty.Due to the strong logic and abstraction of the content of high school mathematics,some students find it very difficult to learn.In order to solve this problem,high school mathematics teachers can make use of mind maps to teach,so that students can exercise their thinking ability,and realize the improvement of comprehensive ability in mathematics.This paper analyzes the shortcomings of high school mathematics classrooms under the background of new curriculum reform and discusses the significance and methods of applying mind maps in high school mathematics classrooms,so as to put forward reasonable suggestions for realizing the efficient teaching of high school mathematics.

Enhancement of Innovation Competence of College Students Majoring in Mathematics Education through Curriculum Optimization

The innovation competence of K-12 education teachers undoubtedly plays a crucial role in fostering the innovation abilities of their students.K-12 mathematics education equips students with the critical thinking and problem-solving skills essential for their future studies in colleges and universities,helping them grasp complex techniques to address challenges in everyday life and their careers.Therefore,it is of great significance to study strategies for improving the innovation competence of college students majoring in Mathematics Education,as they will likely become K-12 education mathematics teachers directly after graduating from colleges or universities.In this paper,we study strategies for enhancing the innovation competence of college students majoring in Mathematics Education through curriculum optimization.We analyze and explain in detail the importance of innovation competence for college students majoring in Mathematics Education and the difficulties encountered in enhancing college students’innovation competence.With the help of the analysis of the importance and challenges of enhancing college students’innovation competence,we propose several strategies to improve the innovation competence of college students majoring in Mathematics Education based on curriculum optimization.The findings presented in this paper can be applied to develop strategies for college students majoring in Physics and Chemistry Education.

The Integration Strategy of Class Teacher Management and Mathematics Teaching

The professional and moral education of high school mathematics teachers will make classroom management work better,but their work pressure will also lead to classroom management problems.To do a good job in high school class teacher management and organically integrate it with mathematics teaching,we need to start from two aspects:mathematics teaching class teachers and class teacher work teaching,and penetrate mathematical thinking into daily classroom management,moral education,and classroom culture construction.Based on the attributes of the subject,we guide high school students to reflect after class to stimulate their self-management initiative through the cultivation of qualified class representatives.In addition,it is necessary to skillfully resolve classroom generative problems,change the roles of teachers and students,and integrate classroom management with mathematics teaching.

A Study on High School Mathematics Teaching Design Based on Teaching for Robust Understanding: Taking the Cosine Theorem as an Example

Teachers’teaching behavior plays a crucial role in students’development,and there are problems in the current teaching behavior of mathematics teachers such as ignoring students’cognitive needs,lack of equal opportunities for students’classroom performance as well as lack of formative evaluation of students.In order to solve the phenomenon,this paper analyzes and explains how to promote teaching based on the Teaching for Robust Understanding(TRU)evaluation framework with the goal of focusing on the development of all students,taking the teaching design of The Cosine Theorem as an example,and provides ideas and methods for first-line high school mathematics teachers.

Effects of QD2R Numeracy Enhancement Strategy on the Mathematics Performance of Grade 10 Students

Learning mathematics requires an effective and strategic teaching approach.This study aimed to assess the mathematics performance of the learners with the implementation of the numeracy enhancement strategy QD2R(Questions,Drills,Repetition,and Recitation)and to propose a strategy implementation plan to elevate their performances.This study employed the use of a quasi-experimental research design,purposive sampling with 70 Grade 10 students of Lian National High School who were distributed equally to control and treatment groups.The pre-test and post-test results were statistically analyzed using independent and paired sample t-tests,and a survey questionnaire was examined by getting the mean and standard deviation.The results indicated that better performance was achieved by the students from the treatment group compared to the students from the control group,as revealed by the Mean Percentage Score(MPS)results,mean scores,and P values of their pre-test and post-test scores.The learners’perception of the implementation of this strategy was to a great extent,wherein it was perceived to be more helpful in concepts related to understanding the lesson compared to concepts related to developing their attitude and skills.Moreover,the proposed implementation plan of numeracy enhancement strategy QD2R had three expected outcomes:elevated understanding and performance in mathematics lessons;modified strategy to focus on the development of attitude and skills towards mathematics;and refined and well-implemented QD2R strategy in teaching mathematics.Relative to these expected outcomes,appropriate measures,timeframe,and resources of each were comprehensively formulated.

Research on the Teaching Design of Advanced Mathematics Courses Based on Outcome-Based Education Concept

This paper first analyzes the characteristics and current situation of the Advanced Mathematics course;secondly,it proposes a teaching model that integrates the outcome-based education(OBE)philosophy and blended teaching method,reorganizing the teaching objectives,teaching content,and assessment evaluation process of the Advanced Mathematics course;lastly,through practice,it is proved that this approach can effectively improve students’mastery of course content,enhance students’ability to apply mathematical knowledge,and strengthen teaching effectiveness.