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December 11, 2025 (v1)

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"The 9th Point Theorem: Emergence of Pulsation via Hopf Bifurcation in Reduced 2D Navier–Stokes Models"

Description:

This work presents a complete analytical and numerical proof of the "9th Point Theorem", which describes the emergence of a stable pulsating central mode in two-dimensional incompressible fluid flows. Starting from the full Navier–Stokes equations, we construct a rigorous reduced-order model using a Galerkin approximation with NN symmetric, localized vortex modes and a single central mode.

We prove that for this symmetric configuration, there exists a critical coupling strength kcrit=−(μ+ν)/(N−1)kcrit=−(μ+ν)/(N−1) at which the system undergoes a supercritical Hopf bifurcation. This bifurcation gives rise to sustained oscillations in the amplitude of the central mode, termed the "9th Point". The proof includes the derivation of the reduced ordinary differential equation system, linear stability analysis, calculation of the first Lyapunov coefficient to confirm the supercritical nature of the bifurcation, and validation via direct numerical simulation of the full 2D Navier–Stokes equations using a pseudo-spectral method.

The theorem establishes a fundamental mechanism by which nonlinear interactions between discrete vortex structures and a background flow can generate coherent pulsations, a phenomenon relevant to geophysical fluid dynamics, vortex dynamics, and low-dimensional modeling of complex flows.

Keywords: 9th Point Theorem, Hopf bifurcation, Navier–Stokes equations, reduced-order model, Galerkin approximation, vortex dynamics, pulsating flow, fluid dynamics, dynamical systems, stability analysis.

Resource Type: Preprint / Theoretical Work
License: Creative Commons Attribution 4.0 International
Contributors: Hristo N. (Author), AI Assistant (Methodology & Validation)
Related Publications: (Optional: Link to arXiv or journal submission if applicable)

 

The Prime Phase Transform: A Geometric Framework for Visualizing Prime Numbers

Description:

This work introduces the Prime Phase Transform (PPT), a novel mathematical construction that provides a geometric representation of prime numbers. Defined as PPT_N(p) = e^(2πi·p/N) / log(p) for even integers N ≥ 4 and primes p ≤ N, the transform maps each prime to a point in the complex plane with radius 1/log(p) and argument 2π·p/N.

The Prime Phase Transform reveals several remarkable properties:

1. Geometric Interpretation: Creates visual patterns where primes appear as points on concentric circles

2. Product Formula: For p+q=N, PPT(p)·PPT(q) = 1/(log p log q) (always real and positive)

3. Pattern Classification: Different N values (primes, composites, powers of 2) produce distinct geometric configurations

4. Numerical Observations: Systematic behavior in |T(N)| = |Σ PPT(p)| and S(N) = Σ_{p+q=N} PPT(p)PPT(q)

Key Features:

• Educational Tool: Makes prime number theory accessible through visualization

• Research Framework: Provides new approach to studying additive properties of primes

• Algorithmic Implementation: Complete Python code for computation and visualization

• Hypothesis Generation: Numerical patterns suggest new mathematical questions

Included Files:

• Full paper in PDF format

• LaTeX source code

• Python implementation with examples

• Visualizations of prime patterns

Potential Applications:

• Mathematics education and visualization

• Pattern recognition in number theory

• Algorithm development for prime analysis

• Bridge between analytic and geometric number theory

Keywords: Prime numbers, geometric visualization, complex analysis, number theory, mathematical transforms, prime distribution, computational mathematics, educational mathematics, Python implementation

Note: This is a preprint/work in progress. The Prime Phase Transform represents a new tool for exploring prime numbers geometrically, not a solution to existing open problems.

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https://zenodo.org/records/17903207

https://zenodo.org/records/17912132

 

_PRIME_PHASE_TRANSFORM_.pdf _PRIME_PHASE_TRANSFORM_.tex The_9th_Point_Theorem_.pdf The_9th_Point_Theorem_.tex