A Temporal Neutrosophic Nano–Interior Ideal Framework for  Performance Boundary Optimization in University Volleyball  Courses

Xuelian Liu; Chaolumen Ge

Authors

Xuelian Liu School of Physical Education, Inner Mongolia Medical University, 010000, China
Chaolumen Ge Mongolia Medical University, Huhehaote City, 010000, China

Keywords:

Neutrosophic sets; nano semi-frontier; ℵ–interior ideal; temporal modeling; performance boundaries; volleyball training optimization; sports course evaluation

Abstract

University volleyball courses demand continuous assessment and adaptation to address
the varying skill levels, uncertainties, and development rates of students. Traditional
evaluation systems often fail to account for indeterminacy and transitional performance
states. This paper proposes a novel Temporal Neutrosophic Nano–Interior Ideal
Framework that integrates the precision of neutrosophic nano semi-frontier theory with
the structural rigor of ℵ–interior ideals. The model represents each volleyball skill as a
dynamic neutrosophic triplet over time, capturing truth, indeterminacy, and falsity
components for multiple skill dimensions. The nano semi-frontier component identifies
players whose performance lies near the boundary between mastery and non-mastery,
while the ℵ–interior ideal structure ensures the stability of performance progression
within the training system. A temporal extension tracks weekly changes and optimizes
training schedules to maximize overall skill advancement by the end of the semester. The
framework is demonstrated through a case study using a complete semester of
hypothetical volleyball course data. The proposed approach offers a mathematically
rigorous, uncertainty-aware methodology for performance optimization in sports
education.