Navigation & Index
Site Directory & Sitemap
A complete catalog of interactive visualization tools, theory wiki articles, mathematical surveys, and academic profile pages.
Computational Tools
Interactive Solvers & Visualizers
- NIEP Research Hub Home Master dashboard, tool showcases, open problems, and literature catalog.
- About the Hub & Contributing Project mission, getting involved for mathematicians, developers, and students, and contribution guidelines.
- Karpelevič Region Interactive Viewer Live HTML5 Canvas renderer of stochastic eigenvalue regions $n=2$ to $n=8$ with Farey arcs.
- Trace Polytope 3D Visualizer 3D WebGL dataset browser for order-5 symmetric and real trace polytopes.
- Numerical Spectra Realizer Alternating projection solver with exact Suleimanova and Soules synthesis.
- Research Blog & Visual Explorations In-depth mathematical blog posts and interactive 2D/3D explorations on open problems in the NIEP.
Theory Wiki
Mathematical Theory Articles
- Theoretical Overview Comprehensive foundational survey covering Perron-Frobenius theory and semi-algebraic geometry.
- The NIEP (Core Formulation) Dimensional solvability progress, trace moment obstructions, and open frontiers.
- Symmetric NIEP (SNIEP) Orthogonal eigenspaces, Soules bases, Fiedler theorems, and trace polyhedra.
- Real NIEP (RNIEP) Non-orthogonal frames, general real spectra, and the Laffey-Loewy separation gap.
- Karpelevič Region & Theorem Kolmogorov's problem, stochastic boundaries, Farey sequence, and Ito's algebraic polynomials.
- Suleĭmanova Spectra Single positive eigenvalue spectra, trace sufficiency, and Fiedler's symmetric realization.
- Boyle-Handelman Theorem Symbolic dynamics, shifts of finite type, and realization with auxiliary zero eigenvalues.
- Perron Similarities & Ideal Spectracones Johnson-Paparella theory, spectracones, and Gershnik-Lewis-Paparella's theorem on group character tables.
- Character Table Cones ($n=5$ SNIEP) Interactive 2D & 3D exploration of $D_8, Q_8, S_4, A_5, \mathbb{Z}_5$, and $V_4 \oplus [1]$ character cones for $(1,a,a,b,b)$ and $(1,a,a,b,c)$.
- Comprehensive Bibliography & Literature Archive Comprehensive catalog of 30+ papers across Perron-Frobenius, stochastic matrices, SNIEP, and spectracones with interactive search and BibTeX export.
Research Blog & Monograph
Inside Karpelevič’s Proof (9-Stop Series)
- Roadmap: Inside Karpelevič’s Proof Master architectural map, Theorem A & Theorem B statements, unified notation table, and proof ledger.
- Stop 1 of 9 · Stochastic Matrices as Contracting Polygons Dmitriev–Dynkin duality $\lambda M \subseteq M$, row normalization, and why boundary eigenvalues cannot hide inside a polygon.
- Stop 2 of 9 · Boundary Eigenvalues and Edge Contacts Minimal order $n$, vertex vs. interior-edge contacts, and why every transformed vertex $\lambda z_i$ must lie on $\partial M$.
- Stop 3 of 9 · The Circular-Board Game Behind Karpelevič’s Proof Elementary edge slides $D_1(\gamma)$ and $D_2(\gamma)$ translated into a circular-board chip game that concentrates interior-edge contacts.
- Stop 4 of 9 · Return Relations and the Characteristic Equation Collapsing $n-g$ pure vertex transitions into an $m \times m$ return matrix $R(\lambda)$ and extracting the characteristic product.
- Stop 5 of 9 · Why Every Contact Parameter Must Be Equal Single-loop cycle decomposition and strict concavity of $\theta \mapsto \log|v(\theta)|$, forcing all contact parameters to coincide.
- Stop 6 of 9 · The Two Canonical Boundary Equations Deriving Family I ($\lambda^\tau(\lambda^\ell - t)^m = (1-t)^m$) and Family II ($(\lambda^\ell - t)^m = (1-t)^m\lambda^\sigma$) and connecting them to Ito’s unified equation.
- Stop 7 of 9 · Farey Fractions and the Two-Step Law Sector bounds, Bézout recurrence $qk - pf = 1$, the two-step restriction $s_i \in \{\ell, \ell+h\}$, and the Farey exponent dictionary.
- Stop 8 of 9 · Cyclic Polygons and Explicit Boundary Matrices Cyclic polygons $C(\mu, h)$, monotone polynomial envelopes, and explicit $n \times n$ stochastic companion realizations.
- Stop 9 of 9 · Induction on Order and the Final Region $K_n$ Farey mediant surgery from $K_{n-1}$ to $K_n$, eliminating non-adjacent arcs via $n_* \le n-1$, and closing the proof of Theorem A and Theorem B.
Academic & Research
Profile & Publications
- About Benjamin J. Clark Biography, research background in nonnegative matrix theory, and professional software engineering.
- Curriculum Vitae Full academic and professional CV with downloadable PDF and embedded viewer.
- Publications & Preprints Peer-reviewed articles, arXiv preprints, DOI links, and abstract summaries.
- Invited Talks & Presentations Conference lectures, seminar presentations, and slides in spectral graph theory and matrix analysis.
- Teaching & Course Archives University courses taught, syllabi, lecture notes, and educational materials.
- Publications Section Index Direct archive redirect for publication listings.
- Talks Section Index Direct archive redirect for presentations.
- Teaching Section Index Direct archive redirect for instructional materials.