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← Mathematics and statistics this week · 26 October 2026 · 5 min read

Mathematics and statistics this week: The limits of control, codes and rare events

New preprints settle conjectures about steering dynamical systems, transmitting information and classifying waves, while revealing unexpected order in billiards, random graphs and quantum spectra.

How these papers were chosen among the week's 2055 papers

All 2055 papers of the week whose main category is in this field were read, title and start of the abstract, and 84 were kept. They were then ordered by what can be counted: how many research profiles of our readers they come close to, whether a rising topic of the field includes them, and how many categories they were posted in. The abstracts of the first 40 were read in full to choose these, no more than two from one category and one per team.

The 1 number of the text was checked against the abstracts.

Sources: arXiv.

When control cannot be certified by an algorithm #

Can a system be steered from any starting state to any desired destination? For systems governed by polynomial equations with rational coefficients, a new preprint proves that no algorithm can always answer that question. More surprisingly, there is not even an algorithm that could eventually list all the globally controllable systems, despite being allowed to run indefinitely. This confirms the undecidability part of Sontag’s conjecture while overturning its proposed escape route.

The proof translates a question about stability into a question about control. The authors explicitly construct a control system that is globally controllable exactly when the origin of an associated polynomial flow is not globally asymptotically stable. An existing result about the computational difficulty of stability then supplies the obstruction. The conclusion also holds when inputs enter through a constant matrix: even that familiar engineering form does not remove the fundamental barrier to universal certification.

Who works on it? Literature review Follow this subject

A sharp limit on error-correcting codes #

Error-correcting codes add redundancy so that damaged messages can still be recovered. Maximum distance separable codes achieve the best possible trade-off between message length, redundancy and error protection. A new preprint resolves the linear MDS conjecture over finite fields of characteristic two, establishing the conjectured ceiling on how long these optimal codes can be, including the exceptional dimensions where a longer code is possible.

The result is a mathematical bound, rather than a new encoding procedure: it proves that every nontrivial linear MDS code over these fields must obey the stated limits. Such impossibility results matter to information theory because they distinguish an engineering challenge from a mathematical dead end. Beyond the ceiling, a designer must change the requirements rather than simply search harder for a better linear code.

Who works on it? Literature review Follow this subject

Orderly boundaries, thoroughly mixing billiards #

A billiard ball following ideal reflections can generate dynamics far more complicated than its simple rules suggest. A new preprint presents the first known strictly convex ergodic billiards: sufficiently asymmetric lemon-shaped tables formed by intersecting circles. Ergodicity means, roughly, that a typical trajectory explores the available phase space according to its natural statistical distribution, rather than remaining trapped in a separate dynamical region.

The authors build on an earlier proof of uniformly hyperbolic returns to a suitable section of the motion. The new ingredient controls singularity curves, where the billiard dynamics become delicate. This establishes not only ergodicity but the stronger Bernoulli property, a precise form of statistical randomness. The result gives a concrete setting in which deterministic mechanics produces robust statistical behavior.

Who works on it? Literature review Follow this subject

How a random graph loses its triangles #

Suppose a random network contains far fewer triangles than expected. Has it simply become uniformly sparser, or has it reorganized? A new preprint answers this for sparse Erdős–Rényi graphs in the stated probability range. The typical conditioned graph resembles either another Erdős–Rényi graph or a symmetric block model with equally sized groups. A first order phase transition separates these alternatives at the threshold 0.009217… for the retained fraction of the expected triangle count.

Earlier work reduced the probability and structure of this rare event to an optimization problem over graphons, continuous objects that describe large networks. The new paper solves that problem throughout the relevant range, showing that every minimizer is constant or has the specified block structure. It resolves a conjecture and an open problem about triangles, illustrating a broader lesson in probability: an unusually low count can reflect a change in organization, not merely a reduction in overall density.

Who works on it? Literature review Follow this subject

Complex invasion waves from simpler building blocks #

The Fisher–KPP equation is a standard model for growth and spreading, including idealized biological invasions. Its solutions can describe fronts advancing through space, but classifying all solutions that exist for all time is a much larger challenge. A new preprint resolves the Hamel–Nadirashvili conjecture: under the stated assumptions on the reaction term, every bounded solution in the relevant interval can be approximated by solutions generated from finitely many planar traveling fronts, possibly accompanied by a spatially homogeneous solution.

The proof uncovers an ordered linear structure beneath the nonlinear equation. It separates solutions into critical and subcritical spectral components, combining linear reconstruction with a measure representation of critical solutions. The significance is not that every spreading pattern is itself a finite collection of flat fronts, but that these simpler ingredients suffice to approximate the entire solution space in the precise local sense established by the paper.

Who works on it? Literature review Follow this subject

An infinite spectral ladder inside a gap #

The Efimov effect concerns an infinite sequence of quantum energy levels accumulating at a spectral threshold. A new preprint establishes such a sequence inside an internal gap of a lattice Hamiltonian describing identical spinless fermions and a distinguishable particle, with repulsive pair interactions. The setting is carefully arranged: the mass ratio exceeds a critical value, the mixed interaction is tuned to a threshold resonance, and sufficiently strong repulsion between the fermions opens the relevant gap.

The authors prove logarithmic asymptotics for the number of accumulating eigenvalues. The coefficient matches that found without fermion–fermion coupling and also governs accumulation above the essential spectrum across the stated weak-coupling interval, including its critical endpoint. This connects apparently different spectral regimes and shows how repulsive interactions can reshape the spectrum while preserving a quantitative signature of Efimov accumulation.

Who works on it? Literature review Follow this subject

These preprints share a concern with the boundary between possibility and impossibility. Some establish limits that no algorithm or code design can overcome; others show that seemingly complicated behavior admits a precise classification or a simpler underlying structure. Together, they illustrate how mathematical progress can mean both discovering a new phenomenon and proving exactly which alternatives remain.

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Written by AI from the 2483 papers of the week in this field. Most are preprints, not yet peer reviewed: the text says what their authors report.

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