Hyperbolic-Valued Multi-Norms: Theory, Examples, and Emerging Applications
Author(s): Neetu Singh
Publication #: 2608007
Date of Publication: 12.07.2025
Country: India
Pages: 1-9
Published In: Volume 11 Issue 4 July-2025
DOI: https://doi.org/10.62970/IJIRCT.v11.i4.2608007
Abstract
A two-component method for measuring finite families in modules over the split-complex algebra is provided by hyperbolic-valued multi-norms. Their usefulness is contingent upon the availability of concrete constructions whose cross-level axioms can be verified, rather than on formal notation. This paper establishes a theory that emphasises the use of examples. It constructs minimum, Hilbert-projection, function-space, sequence-space, pullback, and operator-family multi-norms by idempotent recombination after reviewing the classical multi-norm axioms and the positive hyperbolic cone. Permutation invariance, coordinatewise scalar domination, stability under adjoining zero, repetition consistency, and compatibility with the underlying norm are all verified for each proposed construction. In a common hierarchy, comparison estimates arrange maximum, sum, and Euclidean tuple gauges. The latter two are demonstrated to be genuine norms on each finite power, but they are not multi-norms per the standard repetition axiom. The proof of fixed-level equivalence in finite dimensions is presented, and the level-dependent nature of equivalence constants is underscored. Null-cone values and asymmetric component geometry are illustrated in the worked calculations. Interpretive applications are created to address the stability of coupled systems, split-channel optimisation, uncertain models, operator equations, and hypercomplex machine-learning representations. The discussion does not assert empirical superiority; rather, it identifies mathematically testable responsibilities for multi-level control. The following are open problems: uniform equivalence, computational approximations of Hilbert multi-norms, coupled nondecomposable constructions, and maximum multi-norms.
Keywords: hyperbolic multi-norm; minimum multi-norm; Hilbert multi-norm; function space; sequence space; split-complex model; stability
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