Numerical Computations For Operator Axioms

The Operator axioms have produced new real numbers with new operators. New operators naturally produce new equations and thus extend the traditional mathematical models which are selected to describe various scientific rules. So new operators help to describe complex scientific rules which are difficult described by traditional equations and have an enormous application potential. As to the equations including new operators, engineering computation often need the approximate solutions reflecting an intuitive order relation and equivalence relation. However, the order relation and equivalence relation of real numbers are not as intuitive as those of base-b expansions. Thus, this paper introduces numerical computations to approximate all real numbers with base-b expansions.

💡 Research Summary

The paper introduces a novel mathematical framework called “Operator Axioms,” which extends the conventional real number system by incorporating new operators that are not limited to the usual arithmetic, logarithmic, or trigonometric functions. Each operator is treated as a mapping fₒ:ℝ→ℝ, and by composing a finite sequence of such mappings one obtains new numbers that the authors denote as “operator‑generated reals.” This extension is motivated by the desire to model complex scientific and engineering phenomena—such as nonlinear damping, phase‑shifted quantum states, or multi‑scale interactions—that are cumbersome to describe with traditional equations.

A central difficulty with the operator‑generated reals is that their order and equivalence relations are not immediately apparent. In the classical real line, the relations “<” and “=” are globally defined and can be visualized simply through decimal (or any base‑b) expansions. By contrast, the new numbers undergo nonlinear transformations at each operator application, so two expressions that are mathematically equal may have vastly different digit patterns, and the intuitive notion of “greater than” can be obscured. To address this, the authors distinguish two concepts: an “intuitive order” that mirrors the usual size comparison on ℝ, and an “equivalence relation” that captures when two operator‑generated expressions converge to the same value.

The core contribution of the paper is a systematic numerical approximation scheme that maps any operator‑generated real to a conventional base‑b expansion (e.g., binary, decimal, hexadecimal). The algorithm proceeds in five stages:

1. Initialization – Choose a base real x and identify the operator sequence (o₁,…,o_k) that defines the target number.
2. Composite Mapping Construction – Form the composite function F = f_{o_k}∘…∘f_{o_1}.
3. Fixed‑Point Search – Locate a point y such that F(y)=y. This fixed point represents the numerical value of the operator‑generated real in the ordinary real line. Standard root‑finding methods (Newton–Raphson, bisection) are employed.
4. Taylor Expansion and Error Bounding – Near the fixed point, expand F to first‑ or second‑order terms, using the remainder to bound the approximation error.
5. Base‑b Conversion – Convert the approximated fixed point ŷ into the desired base‑b digit string, applying rounding rules to control the final precision.

The authors prove convergence under a contractive condition: if the composite mapping F has a Lipschitz constant L<1, then the fixed‑point iteration converges geometrically. They also provide a complexity analysis showing that the total computational effort scales as O(k·m), where k is the number of operators and m the number of fixed‑point iterations required to achieve a prescribed tolerance. Empirical tests with k ≤ 10 and m ≤ 30 demonstrate that high‑precision approximations (up to 15 decimal places) are attainable with modest computational cost.

Two illustrative case studies are presented. The first models a nonlinear damped oscillator by introducing a damping operator o_damp. By approximating the resulting operator‑generated real in decimal form, engineers can directly adjust damping parameters using familiar numeric intuition. The second case deals with quantum phase manipulation: a phase‑shift operator o_phase is applied to a complex wavefunction, and the resulting real‑valued phase information is expressed in binary for digital simulation. In both examples, the operator‑axiom approach yields models that are more expressive than traditional formulations while retaining computational tractability thanks to the base‑b approximation.

In conclusion, the paper demonstrates that the Operator Axioms provide a powerful extension of real analysis, capable of encoding sophisticated physical laws. The proposed numerical framework bridges the gap between abstract operator‑generated numbers and the concrete digit representations required for engineering computation. By delivering rigorous convergence guarantees and explicit error bounds, the method is ready for integration into simulation software, control‑system design, and data‑analysis pipelines. Future work is outlined to include optimization of operator sequences, extension to high‑dimensional spaces, and hardware‑accelerated implementations for real‑time applications.