Steps
Step 1
Problem Selection and Background Research
The core task of this step is to select a mathematical problem with theoretical significance or application value, and conduct sufficient background research. Choose specific research questions from the core content of the course, which can be generalization of classical theorems, counterexample construction, algorithm design, or mathematical modeling of practical problems. Understand the research history and current status of the problem through literature review, and clarify the starting point and innovation points of the research.
• Select specific problems of interest from course content, clarify the mathematical formulation and research objectives, ensuring the problem has a certain depth and challenge
• Review relevant textbooks and academic literature (MathSciNet, arXiv, CNKI) to understand the research history, existing results, and unsolved problems of the issue
• Write literature review and research plan: organize relevant theoretical foundations, propose research ideas and technical routes, and develop phased research objectives and schedule
Deliverable: Topic selection report and literature review (problem statement, research significance, literature review, research plan) | Quality standard: Meaningful and in-depth topic selection, comprehensive literature review, feasible research plan
Step 2
Theoretical Analysis and Theorem Derivation
The core task of this step is to conduct in-depth theoretical analysis and rigorous mathematical derivation of the selected problem using mathematical theories and methods learned in the course. Starting from definitions and axioms, establish lemmas and theorems through logical reasoning, and gradually approach the solution to the problem. Theoretical derivation is the core of mathematical research, requiring rigor, logic, and systematization.
• Establish theoretical framework: clarify basic concepts and definitions involved in the problem, organize required prerequisite knowledge and existing theorems, and build the theoretical foundation for research
• Theorem derivation and proof: through rigorous logical reasoning, gradually derive and prove related lemmas and theorems, using proof methods such as proof by contradiction, mathematical induction, and construction method
• Verification and testing: verify the correctness of the derivation process through special case verification, numerical testing, or logical checking, and promptly correct loopholes and errors in reasoning
Deliverable: Theoretical derivation report (definitions and lemmas, theorem proofs, derivation process, verification examples) | Quality standard: Rigorous derivation process, clear logic, correct and error-free proof
Step 3
Example Construction and Numerical Verification
The core task of this step is to verify the correctness and effectiveness of theoretical results by constructing specific examples or performing numerical calculations. Mathematical research requires not only abstract theoretical derivation but also concrete examples to illustrate concepts, verify theorems, and discover patterns. Through examples, theoretical results can be understood more intuitively, and the scope and limitations of theory can be tested.
• Construct typical examples: design representative examples and counterexamples to verify theorems, illustrate concepts, and test conclusions, including trivial and extreme cases
• Numerical calculation and simulation: use software such as MATLAB, Mathematica, Python (NumPy, SciPy, SymPy) for numerical and symbolic computation to verify theoretical results
• Visualization: use tools like Matplotlib and MATLAB plotting to visualize mathematical results, draw function graphs, geometric figures, data charts, etc., to intuitively display results
Deliverable: Examples and computation report (typical examples, numerical results, visualization charts, verification analysis) | Quality standard: Representative examples, sufficient numerical verification, good visualization effects
Step 4
Generalization, Application and In-depth Discussion
The core task of this step is to generalize and deepen on the basis of obtained results, explore more general cases of the problem, or apply theoretical results to solve practical problems. Mathematical research emphasizes generalization from special to general, and the combination of theory and application. Through in-depth discussion, explore more connotations and extensions of the problem.
• Result generalization: attempt to generalize obtained results to more general cases, such as weakening conditions, strengthening conclusions, generalizing to higher dimensions or more abstract spaces
• Application exploration: find application scenarios for theoretical results, such as applications in other branches of mathematics, or practical applications in physics, engineering, economics and other fields
• In-depth discussion: analyze the scope and limitations of results, compare advantages and disadvantages of different methods, and propose unsolved problems and future research directions
Deliverable: Generalization and application report (generalized theorems, application cases, in-depth discussion, outlook) | Quality standard: Innovative generalization, valuable applications, in-depth discussion
Step 5
Paper Writing and Outcome Summary
The core task of this step is to organize the entire research process and results into a standardized mathematical academic paper, and comprehensively summarize the research work. Mathematical papers require accurate expression, rigorous argumentation, clear structure, and standardized citations. By writing the paper, systematically organize research ideas, extract main contributions, and lay the foundation for subsequent learning and research.
• Write mathematical paper: organize content according to academic paper standards, including abstract, introduction, preliminaries, main results, proofs, applications, conclusions, references, typeset with LaTeX
• Mathematical writing standards: correct use of mathematical symbols and terminology, standardized formula numbering, clear charts, unified citation format (AMS or GB/T 7714 format)
• Research summary: summarize the main achievements and innovations of the research, reflect on experiences and lessons in the research process, and propose insights on mathematics learning and research
Deliverable: Mathematical paper and summary (complete paper, LaTeX source code, research summary, references) | Quality standard: Complete paper structure, rigorous argumentation, standardized writing, academic value