ANALYSIS OF THE IMPACT OF MATHEMATICAL OPERATION SIGNS IN MATHEMATICAL FORMULA PLAGIARISM AND THEIR TRANSFORMATIONS

Abstract

 The research is devoted to addressing the pressing issue of plagiarism in mathematical formulas within scientific publications, which holds particular importance in contexts where syntactic changes can obscure copyright violations. Special attention is given to the comprehensive analysis of the impact of mathematical operations and their symbolic representations on plagiarism detection processes. The primary objective of the study is to develop an effective approach for identifying syntactically modified formulas, whose structure has been altered to conceal the borrowing of mathematical concepts without changing their semantic meaning. 
 The specific features of mathematical language, such as rigid semantics and structured expressions, complicate the adaptation of traditional text plagiarism detection tools to the peculiarities of mathematical texts. The analysis revealed that standard systems fail to detect up to 62% of plagiarism cases in texts that include mathematical expressions. With the increasing volume of digital publications, researchers emphasized the necessity of developing interdisciplinary approaches that integrate methods of mathematical analysis, computational linguistics, and theoretical computation. 
 The research encompasses several key components. Syntactic analysis: Decomposition of formulas into distinct components (tokens) and the construction of logical operation trees to determine their structure. Semantic analysis: Algebraic simplification and canonical representation of formulas that allow the comparison of mathematically equivalent expressions, even with varying syntactic forms. Statistical approach: Formation of operational signatures of formulas based on frequency distributions of operators and symbols to quickly detect similarities. Vector models: Vectorization of mathematical expressions, enabling the application of machine learning algorithms to analyze formulas.

 Particular attention is paid to the effects of mathematical operations on the efficiency of plagiarism detection. For example, arithmetic operations exhibit high usage frequency and are relatively straightforward to analyze due to their semantic stability. In contrast, exponential, functional, and integral-differential operations pose significant challenges to automated systems due to their intricate semantics and numerous transformation rules. During the experiments, the impact of operations was analyzed on several levels. Normalization of formula representations to reduce syntactic variability. Assessment of structural similarity using operational trees to identify equivalences despite concealed modifications. Comparison of semantic similarity coefficients, based on algebraic transformations. 
 A significant result of the study was the creation of a table and evaluation of the influence of different types of operations. It demonstrated that functional and integral-differential operators are critical for plagiarism detection due to their complexity and variability in representation. For example, formulas involving logarithmic or trigonometric functions can be expressed in multiple equivalent forms with substantial visual differences. 
 The research also highlights the necessity of improving existing algorithms for tokenization, normalization, and semantic analysis. The conclusions underscore that the integration of machine learning and the development of hybrid approaches will be pivotal in creating robust systems for verifying mathematical content. The study offers prospects for further development of academic monitoring tools aimed at ensuring integrity in science and enhancing the overall level of transparency in research activities.