1 citations · 3 across the 11 of their papers we have counts for
11 papers
Pushout Attachments and Conditional Complexity of Executable Models
Alexander Kolpakov
We represent the modular enlargement of an executable model by a pushout of finite typed presentations. Adhesivity preserves the original model and recovers the shared interface, w…
Random Knots via Stiefel manifolds
Alexander Kolpakov, Igor Rivin
A fixed simplex, randomly projected into three dimensions and joined in Hamiltonian order, produces a rich and unusually tractable model of random stick knots. We prove that Gaussi…
Discriminant Varieties for Stick Knots and Links
Alexander Kolpakov, Igor Rivin
How many knot types can be built from a fixed budget of straight sticks? We prove that the answer has factorial-scale growth, settling its order for the first time. No previously p…
DiRe-RAPIDS: Topology-faithful dimensionality reduction at scale
Alexander Kolpakov, Igor Rivin
Dimensionality reduction methods such as UMAP and t-SNE are central tools for visualising high-dimensional data, but their local-neighborhood objectives can preserve sampling noise…
Multiplicative Turing Ensembles, Pareto's Law, and Creativity
Alexander Kolpakov, Aidan Rocke
We study integer-valued multiplicative dynamics driven by i.i.d. prime multipliers and connect their macroscopic statistics to universal codelengths. We introduce the Multiplicativ…
Lagrangians, Renormalization, and Quantization in Prefix Coding
Alexander Kolpakov, Aidan Rocke
We develop a statistical mechanics framework for prefix coding based on variational principles, renormalization, and quantization. A Lagrangian formulation of entropy-optimal encod…
Fast Geometric Embedding for Node Influence Maximization
Alexander Kolpakov, Igor Rivin
Computing classical centrality measures such as betweenness and closeness is computationally expensive on large-scale graphs. In this work, we introduce an efficient force layout a…
Dimensionality reduction for homological stability and global structure preservation
Alexander Kolpakov, Igor Rivin
We propose DiRe, a force-directed dimensionality reduction framework designed to preserve global structure and homological features while remaining practical on modern hardware. Th…
Benford's Law from Turing Ensembles and Integer Partitions
Alexander Kolpakov, Aidan Rocke
We develop two complementary generative mechanisms that explain when and why Benford's first-digit law arises. First, a probabilistic Turing machine (PTM) ensemble induces a geomet…
A ripple in time: a discontinuity in American history
Alexander Kolpakov, Igor Rivin
In this technical note we suggest a novel approach to discover temporal (related and unrelated to language dilation) and personality (authorship attribution) aspects in historical…
The Information Geometry of UMAP
Alexander Kolpakov, Aidan Rocke
In this note we highlight some connections of UMAP to the basic principles of Information Geometry. Originally, UMAP was derived from Category Theory observations. However, we posi…