Language development is frequently described as a process of discourse expansion in which learners gradually acquire new lexical and grammatical resources. However, observations from academic and specialized communication suggest that language growth involves more than the accumulation of new expressions. Learners also eliminate inaccurate usages, refine semantic distinctions, and develop sensitivity to stylistic and rhetorical conventions. For this reason, the notion of linguistic enrichment provides a more comprehensive description of language development than the notion of simple expansion. In this paper, we propose a formal framework for modeling linguistic enrichment in academic discourse. The framework views language development as a sequence of enrichment processes that may simultaneously introduce, modify, and remove linguistic elements. Particular attention is paid to the emergence of stable discourse repertoires, representing relatively balanced states in which further enrichment no longer produces substantial changes. The approach also allows for the existence of multiple stable outcomes, reflecting the diversity of discourse communities and communicative goals. The proposed framework is illustrated through examples drawn from everyday language and mathematical academic writing. These examples demonstrate how learners gradually move from broad and often undifferentiated language use toward more precise and context-sensitive discourse practices. The analysis highlights the distinction between vocabulary growth and discourse refinement and emphasizes the role of semantic, stylistic, and rhetorical differentiation in advanced language development. The study contributes to the growing dialogue between formal language modeling and discourse-oriented approaches to language learning by providing a conceptual framework in which language development is interpreted as a process of enrichment rather than mere expansion.

This paper establishes a local fixed point theorem for multivalued mappings in zero-complete strong partial (b)-metric spaces and develops a framework for modelling multistage processes with several admissible terminal states. The contractive condition is formulated using a Bianchini–Grandolfi gauge function and the associated excess functional. Unlike global principles, the theorem requires contractive assumptions only within a prescribed closed ball. A localization condition keeps the successive approximations inside this ball, while an adapted chain estimate proves that the iterative sequence is zero-Cauchy. The proof does not require the family of partial (b)-metric balls to form a topological basis; the ball serves only as a localization set, and the limiting argument relies on zero-completeness and the zero-closedness of the mapping values. The main theorem guarantees a fixed point for a multivalued mapping and, under an additional condition, uniqueness in the single-valued case. Its consequences include local and global principles for linear set-valued contractions, a partial metric version, and a local Banach-type theorem. A further contribution is a method for constructing strong partial (b)-metrics from bounded metric spaces and prescribed nonempty target families. A nonlinear transformation of the original metric is combined with the transformed distances from the target family. Thus, the self-distance of each point is determined by its position relative to that family and vanishes precisely on it. This provides a natural model for processes with several distinct but equally stable terminal states. The theory is illustrated through a finite model of linguistic enrichment. Twenty formulations of the same mathematical statement are arranged into successive levels of grammatical, terminological, logical, and stylistic refinement. A weighted revision graph generates the generalized distance, while the self-distance represents the remaining effort required to reach a stable formulation. A multivalued revision mapping allows several admissible improvements at every stage. The model admits two distinct stable formulations: a concise formal version and a more explanatory, pedagogically oriented version. Both require no further essential revision but remain distinct. This demonstrates that stabilization need not imply uniqueness and that different enrichment trajectories may lead to different acceptable terminal texts. More generally, the example shows how generalized fixed point methods can describe local, nonunique, and multistage stabilization processes in language dynamics and related nonlinear systems.

This paper introduces and studies alternating points of two set-valued mappings acting between two 0-complete strong partial b-metric spaces. An alternating point is a mutually compatible pair of states, each of which belongs to the image of the other under the corresponding mapping. Such a pair simultaneously generates fixed points of the two set-valued compositions. We establish a local existence theorem under nonlinear contractive conditions expressed by one-sided excess functionals on prescribed closed balls. The two successive transitions are controlled by different comparison functions, while the Bianchini–Grandolfi summability condition is imposed only on their composition, which represents one complete alternating cycle. The proof constructs two interdependent sequences, shows that they remain within their respective localization regions, and establishes their zero-Cauchy convergence by means of chain estimates adapted to strong partial b-metrics. A finite discrete model is developed to verify the assumptions of the theorem explicitly and to demonstrate that several alternating points may coexist. The same abstract model is then interpreted in two different settings: as a set-valued price–demand adjustment process and as an alternating process of lexical enrichment and grammatical stabilization. These applications illustrate how the proposed framework can describe nonunique adjustment and refinement processes with multiple mutually compatible terminal configurations.

This article examines how mathematical concepts find their way into figurative language in English and Bulgarian. The aim is to identify and classify idiomatic expressions that use ideas such as numbers, operations, measurement, geometry, proportion, and abstract notions like probability or infinity. The data were gathered from dictionaries, corpora, and other reference sources, and grouped into thematic categories. The analysis shows that idioms with numbers are the most widespread, often with close parallels across the two languages, while others highlight cultural differences and unique linguistic creativity. Beyond numbers, idioms related to calculation, measurement, and shape illustrate how mathematical reasoning subtly shapes everyday speech and thought. The study provides both a structured overview of these expressions and a basis for further exploration of the links between mathematics, language, and culture.

This paper establishes a local fixed point theorem for multivalued mappings in zero-complete strong partial b-metric spaces and develops a framework for modelling multistage processes with several admissible terminal states. The contractive condition is formulated using a Bianchini–Grandolfi gauge function and the associated excess functional. Unlike global principles, the theorem requires contractive assumptions only within a prescribed closed ball. A localization condition keeps the successive approximations inside this ball, while an adapted chain estimate proves that the iterative sequence is zero-Cauchy. The proof does not require the family of partial b-metric balls to form a topological basis; the ball serves only as a localization set, and the limiting argument relies on zero-completeness and the zero-closedness of the mapping values. The main theorem guarantees a fixed point for a multivalued mapping and, under an additional condition, uniqueness in the single-valued case. Its consequences include local and global principles for linear set-valued contractions, a partial metric version, and a local Banach-type theorem. A further contribution is a method for constructing strong partial b-metrics from bounded metric spaces and prescribed nonempty target families. A nonlinear transformation of the original metric is combined with the transformed distances from the target family. Thus, the self-distance of each point is determined by its position relative to that family and vanishes precisely on it. This provides a natural model for processes with several distinct but equally stable terminal states. The theory is illustrated through a finite model of linguistic enrichment. Twenty formulations of the same mathematical statement are arranged into successive levels of grammatical, terminological, logical, and stylistic refinement. A weighted revision graph generates the generalized distance, while the self-distance represents the remaining effort required to reach a stable formulation. A multivalued revision mapping allows several admissible improvements at every stage. The model admits two distinct stable formulations: a concise formal version and a more explanatory, pedagogically oriented version. Both require no further essential revision but remain distinct. This demonstrates that stabilization need not imply uniqueness and that different enrichment trajectories may lead to different acceptable terminal texts. More generally, the example shows how generalized fixed point methods can describe local, nonunique, and multistage stabilization processes in language dynamics and related nonlinear systems.

This paper proposes a formal framework for the study of mathematical language at the intersection of fixed point theory, formal language theory, and academic discourse analysis. Mathematical texts are modeled as languages over a finite alphabet of discourse tokens, combining natural-language expressions with symbolic content. To suppress irrelevant symbolic variation, we introduce a normalization procedure in which concrete mathematical expressions are replaced by an abstract placeholder while the surrounding linguistic structure is preserved. Within this framework, we define enrichment operators on phrases and the induced operators on languages, which model admissible stylistic and structural transformations of mathematical discourse. The collection of all languages over a fixed alphabet, ordered by inclusion, is shown to form a complete lattice, allowing the application of the Knaster–Tarski fixed point theorem. As a consequence, stable linguistic configurations can be interpreted as fixed points of the induced enrichment operator. We further show that different initial languages may lead to different fixed points under the same operator, reflecting the existence of multiple stable forms of mathematical expression. In addition, we introduce a notion of lexical distance based on frequency distributions of discourse units, which provides a quantitative tool for comparing languages. The illustrative analysis suggests a saturation effect: while enrichment increases the overall distance from the initial language, the incremental changes between successive stages remain bounded, indicating a tendency towards stabilization. A concrete illustrative example based on a classical theorem from mathematical analysis demonstrates how a proof evolves through successive levels of enrichment, from a minimal linguistic core to more elaborate stylistic realizations. The proposed framework thus provides a bridge between formal language models and the linguistic structure of mathematical discourse, offering a new perspective on the organization, stability, and variation of mathematical language.