Things built to find out.

1. Ongoing

   Interplanetary Vehicle Routing Problem

   A vehicle routing problem set in the solar system, where the cost of an arc is not distance but the delta-v required to transfer between two bodies, and where the destinations move while the route is being flown.

   Transfers are priced by solving Lambert’s problem between orbital positions, which makes the cost matrix both time-dependent and asymmetric. That breaks the assumptions most classical VRP heuristics are built on, and the interesting question is how much of the standard toolbox survives the change. The work so far covers a formal problem statement, a constructive heuristic, and benchmark instances generated from real solar-system ephemerides.

   • Orbital mechanics
   • Lambert transfers
   • Time-dependent routing
   • Constructive heuristics

   ivrp.zenus.systems ↗ Preprint, DOI 10.5281/zenodo.22710075 ↗

2. Public

   Multidimensional Ideological Model

   Political ideologies represented as unit vectors in a three-dimensional space, with historical events acting as forces that displace them over four centuries.

   Each ideology is a point on the sphere under economic, authority and identity axes. Historical events apply a force within a bounded temporal window, ideologies attract and repel one another through a dot-product interaction, and spherical k-means recovers the resulting coalitions. It is a modelling exercise rather than a political claim: the point is what a continuous geometry buys you over the usual two-axis grid.

   • Vector models
   • Spherical k-means
   • Force-directed dynamics

   mim.zenus.systems ↗

3. Paper in preparation

   Holographic Route Actualization

   A population-based metaheuristic for the capacitated vehicle routing and travelling salesman problems, built on a distributed representation of the search history rather than a tabular one.

   The nearest relatives are estimation-of-distribution algorithms and ant colony optimisation: like both, it maintains a model of what good solutions have looked like and samples from it. What differs is how that model is stored. Evaluated against TSPLIB and CVRPLIB instances with known optima. The method and the results are being written up, so they are not described here.

   • Metaheuristics
   • Distributed representations
   • CVRP
   • TSP

4. Concluded, negative result

   Zero distributions of the permanent polynomial

   An empirical study asking whether the zeros of perm(A − λI) in the complex plane carry any signature of computational hardness that the eigenvalues of the same matrix do not.

   The motivation was a geometric separation between the permanent and the determinant. The answer, established over a series of experiments, is that they do not: for Gaussian inputs both follow the same circular law, and the spacing statistic that first looked promising turned out to be an artefact of the permanent’s coefficient magnitudes rather than of its complexity. Reported here as a null result. Two incidental findings survived it, a parity law for the real zeros and a positivity conjecture for even dimensions, and those are worth a separate write-up.

   • Algebraic complexity
   • Random matrices
   • Complex analysis
   • Numerical experiments

Unpublished projects are listed by problem and method only. Where a result is still being written up, the write-up is the place it will appear first.