arXiv subjects

Tom Verhoeff

Publications and source records attributed to Tom Verhoeff.

5 recordsLinked to original sources

arXiv2026

The $k$-out-of-$n$ picture-hanging puzzle: shorter solutions for small $k$ and $n-k$

The picture-hanging puzzle, popularized by Demaine et al. (2014), asks for a way to wrap a wire around $n$ nails such that the picture hangs as long as fewer than $k$ nails are removed, but falls as soon as any $k$ are removed. Solutions correspond to words in the free group $F_n$. We give explicit, deterministic, polynomial-length constructions for two regimes: $2$-out-of-$n$ with word length at most $\tfrac{8}{3}n^{\log_2 6} - 4n^2$, and $(n-2)$-out-of-$n$ with word length $6n\log_2(n/2)$, both for $n$ a power of two. These improve on W\"astlund's quasi-polynomial deterministic construction in their respective regimes. We also report, via exhaustive computer search, the exact minimum length of $16$ for the $2$-out-of-$4$ puzzle, attained by two structurally distinct solutions. As an additional contribution, we observe that the natural workshop realization with carabiners on a flat board introduces an over/under ambiguity at every wire crossing; a wrong choice can produce a Whitehead link, which is topologically distinct from the intended commutator.

math.CO↗

arXiv2026

AI-Friendly LaTeX: Using LaTeX Code as a Knowledge Source for Retrieval-Augmented Generation

Large language models can answer questions about textbooks, lecture notes, and programming exercises more reliably when their answers are grounded in an explicit knowledge source. Retrieval-augmented generation (RAG) is a common approach: relevant fragments of a document are retrieved and inserted into the model context before answering. For mathematical and technical material, the original LaTeX source can be a better starting point than a PDF, because it contains structural information, labels, sectioning commands, macros, and authorial intent that are often lost or distorted in PDF extraction. However, LaTeX source is not automatically AI-friendly. Cross-references must be resolved, custom macros must be interpreted, exercises and examples must be identified, and author-supplied semantic metadata may be needed. This article describes a focused preprocessing approach for turning LaTeX source, together with its compiled auxiliary files and optional author annotations, into Markdown and JSONL chunks suitable for indexing in a vector database.

cs.IR↗

arXiv2026

A Minimal Mathematical Model for Conducting Patterns

We present a minimal mathematical model for conducting patterns that separates geometric trajectory from temporal parametrization. The model is based on a cyclic sequence of preparation and ictus points connected by cubic Hermite segments with constrained horizontal tangents, combined with a quintic timing law controlling acceleration and deceleration. A single parameter governs the balance between uniform motion and expressive emphasis. The model provides a compact yet expressive representation of conducting gestures. It is implemented as the interactive Wolfram Demonstration "Conducting Patterns" and is used in the Crusis web app.

math.HO↗

arXiv2025

Comparison of Methods for Rotating a Point in $\mathbb{R}^3$: From Vector Algebra to Geometric Algebra

This article starts by presenting and comparing four methods for computing the rotation of a point about an axis by an angle in $\mathbb{R}^3$. We illustrate these methods by computing, by hand, the rotation of point $P=(1,0,1)^T$ about axis $\mathbf{a}=(1,1,1)^T$ by angle $\theta=60^\circ$ (following the right-hand rule). The four methods considered are: (1) an ad hoc geometric method exploiting a symmetry in the situation; (2) a projection method that sets up a new coordinate system using the dot and cross products; (3) a matrix method which rotates the standard basis and uses matrix-vector multiplication; (4) a Geometric (Clifford) Algebra method that represents the rotation as a double reflection via a rotor. We provide a brief introduction to Geometric Algebra. Subsequently, we also address rotations in other dimensions and explain how these can be handled with Geometric Algebra to provide deeper insights.

math.MG↗

arXiv2022

Handling Sub-symmetry in Integer Programming using Activation Handlers

Symmetry in integer programs (IPs) can be exploited in order to reduce solving times. Usually only symmetries of the original IP are handled, but new symmetries may arise at some nodes of the branch-and-bound tree. While symmetry-handling inequalities (SHIs) can easily be used to handle original symmetries, handling sub-symmetries arising later on is more intricate. To handle sub-symmetries, it has recently been proposed to add SHIs that are activated by auxiliary variables. This, however, may increase the size of the IP substantially as all sub-symmetries need to be modeled explicitly. As an alternative, we propose a new framework for generically activating SHIs, so-called activation handlers. This framework allows for a direct implementation of routines that check for active sub-symmetries, eliminating the need for auxiliary variables. In particular, activation handlers can activate symmetry-handling techniques that are more powerful than SHIs. We show that our approach is flexible, with applications in the multiple-knapsack, unit commitment, and graph coloring problems. Numerical results show a substantial performance improvement on the existing sub-symmetry-handling methods.

math.OC↗