A Linear Scale‑Mixing Operator with Power‑Law Eigenfunctions and a Continuous Flow
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2026
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| _version_ | 1866901238316007424 |
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| author | Scales, Raymond |
| author_facet | Scales, Raymond |
| contents | <p class="p1"><span class="s1">Overview</span></p> <p class="p2"><span class="s1">This paper introduces and analyzes a linear scale-mixing operator acting on the Banach space of bounded continuous functions over the positive real axis. The framework formalizes the dynamics of scale-dependent mixing by combining local evaluation with scaled evaluation to describe how information or fields evolve across scales.</span><span class="s2"> </span></p> <p class="p1"><span class="s1">Key Mathematical Results</span></p> <p class="p3"><span class="s1">• Operator Formalism: Defines the operator </span><span class="s2">\bm{(\mathbb{R}A)(x) = \lambda A(x) + \mu A(sx)}</span><span class="s1"> and proves its boundedness on </span><span class="s2">\bm{C_b(\mathbb{R}_+)}</span><span class="s1">.</span><span class="s2"> </span></p> <p class="p4"><span class="s1">• Exact Spectral Decomposition: Derives the eigenvalue map </span><span class="s2">\bm{\Lambda(\beta) = \lambda + \mu s^\beta}</span><span class="s1"> for power-law eigenfunctions </span><span class="s2">\bm{A(x) = x^\beta}</span><span class="s1">.</span><span class="s2"> </span></p> <p class="p4"><span class="s1">• Neutral Stability: Identifies a unique scale-invariant exponent </span><span class="s2">\bm{\alpha}</span><span class="s1"> where the eigenfunction remains stable under iteration, providing a "stable ground" for scale-mixing trajectories.</span><span class="s2"> </span></p> <p class="p4"><span class="s1">• Continuous Scaling Limit: Establishes a rigorous transition from discrete iterations to a first-order transport-type partial differential equation: </span><span class="s2">\bm{\partial_t A = -\kappa A + \mu_0 x \partial_x A}</span><span class="s1">.</span><span class="s2"> </span></p> <p class="p4"><span class="s1">• Nonlinear Dynamics: Extends the continuous flow to include a quadratic saturation term, identifying the trivial vacuum and a nontrivial equilibrium </span><span class="s2">\bm{A = \kappa/\gamma}</span><span class="s1">.</span><span class="s2"> </span></p> <p class="p1"><span class="s1">Significance</span></p> <p class="p2"><span class="s1">The model serves as an analytically tractable dynamical system for studying infinitesimal scale evolution. It bridges the gap between discrete scaling operations and continuous transport-type PDEs, offering a foundation for generalizations into nonlinear, stochastic, or higher-dimensional mixing processes.</span><span class="s2"> </span></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19813859 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Linear Scale‑Mixing Operator with Power‑Law Eigenfunctions and a Continuous Flow Scales, Raymond Operator Theory Banach Spac Spectral Decomposition Power-Law Eigenfunctions Functional Equations Scale-Mixing Renormalization Dynamics Scale Invariance Marginal Fixed Point Continuous Flow Limit Transport Equations Infinitesimal Scaling Nonlinear Stability Analysis Mathematical Physics Discrete Dynamical Systems Asymptotic Analysis <p class="p1"><span class="s1">Overview</span></p> <p class="p2"><span class="s1">This paper introduces and analyzes a linear scale-mixing operator acting on the Banach space of bounded continuous functions over the positive real axis. The framework formalizes the dynamics of scale-dependent mixing by combining local evaluation with scaled evaluation to describe how information or fields evolve across scales.</span><span class="s2"> </span></p> <p class="p1"><span class="s1">Key Mathematical Results</span></p> <p class="p3"><span class="s1">• Operator Formalism: Defines the operator </span><span class="s2">\bm{(\mathbb{R}A)(x) = \lambda A(x) + \mu A(sx)}</span><span class="s1"> and proves its boundedness on </span><span class="s2">\bm{C_b(\mathbb{R}_+)}</span><span class="s1">.</span><span class="s2"> </span></p> <p class="p4"><span class="s1">• Exact Spectral Decomposition: Derives the eigenvalue map </span><span class="s2">\bm{\Lambda(\beta) = \lambda + \mu s^\beta}</span><span class="s1"> for power-law eigenfunctions </span><span class="s2">\bm{A(x) = x^\beta}</span><span class="s1">.</span><span class="s2"> </span></p> <p class="p4"><span class="s1">• Neutral Stability: Identifies a unique scale-invariant exponent </span><span class="s2">\bm{\alpha}</span><span class="s1"> where the eigenfunction remains stable under iteration, providing a "stable ground" for scale-mixing trajectories.</span><span class="s2"> </span></p> <p class="p4"><span class="s1">• Continuous Scaling Limit: Establishes a rigorous transition from discrete iterations to a first-order transport-type partial differential equation: </span><span class="s2">\bm{\partial_t A = -\kappa A + \mu_0 x \partial_x A}</span><span class="s1">.</span><span class="s2"> </span></p> <p class="p4"><span class="s1">• Nonlinear Dynamics: Extends the continuous flow to include a quadratic saturation term, identifying the trivial vacuum and a nontrivial equilibrium </span><span class="s2">\bm{A = \kappa/\gamma}</span><span class="s1">.</span><span class="s2"> </span></p> <p class="p1"><span class="s1">Significance</span></p> <p class="p2"><span class="s1">The model serves as an analytically tractable dynamical system for studying infinitesimal scale evolution. It bridges the gap between discrete scaling operations and continuous transport-type PDEs, offering a foundation for generalizations into nonlinear, stochastic, or higher-dimensional mixing processes.</span><span class="s2"> </span></p> |
| title | A Linear Scale‑Mixing Operator with Power‑Law Eigenfunctions and a Continuous Flow |
| topic | Operator Theory Banach Spac Spectral Decomposition Power-Law Eigenfunctions Functional Equations Scale-Mixing Renormalization Dynamics Scale Invariance Marginal Fixed Point Continuous Flow Limit Transport Equations Infinitesimal Scaling Nonlinear Stability Analysis Mathematical Physics Discrete Dynamical Systems Asymptotic Analysis |
| url | https://doi.org/10.5281/zenodo.19813859 |