arXiv subjects

Scott Neville

Publications and source records attributed to Scott Neville.

10 recordsLinked to original sources

arXiv2026

Eventual sign coherence

The sign coherence of $c$-vectors is one of the fundamental theorems of cluster algebras with principal coefficients. In 2019, Gekhtman and Nakanishi posed the asymptotic sign coherence conjecture for arbitrary cluster algebras of geometric type, which says sign coherence should eventually hold in any sufficiently generic infinite mutation sequence. We prove that their conjecture holds almost always for skew-symmetric cluster algebras of arbitrary rank. That is, we prove that with probability $1$, the sequence of $c$-vectors obtained by random mutation of an arbitrary quiver eventually becomes sign-coherent. Our results also establish the conjecture in full generality for many families of quivers by studying a new class of brog quivers.

math.CO↗

arXiv2026

Mysterious points in keys but not trees

The deep locus of a cluster variety is defined to be the set of its points that do not belong to any cluster torus. We show that, if the cluster variety has a seed whose mutable part is a tree without multiple edges, then the deep locus can be characterized as the set of points whose stabilizer under a certain group action is nontrivial. Deep points without a stabilizer are called mysterious. We establish that many other classes of acyclic quivers (including keys) often have mysterious points. This refutes Conjecture 1.1 of arXiv:2402.16970, but establishes it in many important cases.

math.CO↗

arXiv2025

Mutation Cycles from Reddening Sequences

Given two quivers, each with a reddening sequence, we show how to construct a plethora of mutation cycles. We give several examples, including a generalization of the construction of long mutation cycles in earlier work by the second author. We also give new results on the reddening sequences of certain mutation-acyclic quivers and forks, classifying them in some cases.

math.CO↗

arXiv2024

Mutation-acyclic quivers are totally proper

Totally proper quivers, introduced by S.~Fomin and the author arXiv:2406.03604, have many useful properties including powerful mutation invariants. We show that every mutation-acyclic quiver (i.e., a quiver that is mutation equivalent to an acyclic one) is totally proper. This yields new necessary conditions for a quiver to be mutation-acyclic. In particular, we show that a generalization of the Markov invariant for $3$-vertex quivers applies to all mutation-acyclic quivers. Only finitely many acyclic quivers share the same Markov invariant.

math.RT↗

arXiv2024

Cyclically ordered quivers

A cyclically ordered quiver is a quiver endowed with an additional structure of a cyclic ordering of its vertices. This structure, which naturally arises in many important applications, gives rise to new powerful mutation invariants.

math.RT↗

arXiv2024

On the phase diagram of the polymer model

In dimensions 3 or larger, it is a classical fact that the directed polymer model has two phases: Brownian behavior at high temperature, and non-Brownian behavior at low temperature. We consider the response of the polymer to an external field or tilt, and show that at fixed temperature, the polymer has Brownian behavior for some fields and non-Brownian behavior for others. In other words, the external field can induce the phase transition in the directed polymer model.

math.PR↗

arXiv2023

Long mutation cycles

A mutation cycle is a cycle in a graph whose vertices are labeled by the quivers in a given mutation class and whose edges correspond to single mutations. For any fixed $n\ge 4$, we describe arbitrarily long mutation cycles involving $n$-vertex quivers. Each of these mutation cycles allows for an arbitrary choice of $n \choose 2$ positive integer parameters. None of the mutation cycles we construct can be paved by short mutation cycles.

math.CO↗

arXiv2021

Deep Multilabel CNN for Forensic Footwear Impression Descriptor Identification

In recent years deep neural networks have become the workhorse of computer vision. In this paper, we employ a deep learning approach to classify footwear impression's features known as \emph{descriptors} for forensic use cases. Within this process, we develop and evaluate an effective technique for feeding downsampled greyscale impressions to a neural network pre-trained on data from a different domain. Our approach relies on learnable preprocessing layer paired with multiple interpolation methods used in parallel. We empirically show that this technique outperforms using a single type of interpolated image without learnable preprocessing, and can help to avoid the computational penalty related to using high resolution inputs, by making more efficient use of the low resolution inputs. We also investigate the effect of preserving the aspect ratio of the inputs, which leads to considerable boost in accuracy without increasing the computational budget with respect to squished rectangular images. Finally, we formulate a set of best practices for transfer learning with greyscale inputs, potentially widely applicable in computer vision tasks ranging from footwear impression classification to medical imaging.

cs.CV↗

arXiv2020

Kostka Numbers and Longest Increasing Subsequences

A classical bijection relates certain Kostka numbers, the Catalan numbers, and permutations of length $n$ with longest increasing subsequence (LIS) of length at most $2.$ We generalize this bijection and find Kostka numbers which count the number of permutations of $n$ with LIS length at most $w,$ the number of permutations with $(1, \cdots, w)$ as a LIS, and other similar subsets of permutations.

math.CO↗

arXiv2017

Runaway Feedback Loops in Predictive Policing

Predictive policing systems are increasingly used to determine how to allocate police across a city in order to best prevent crime. Discovered crime data (e.g., arrest counts) are used to help update the model, and the process is repeated. Such systems have been empirically shown to be susceptible to runaway feedback loops, where police are repeatedly sent back to the same neighborhoods regardless of the true crime rate. In response, we develop a mathematical model of predictive policing that proves why this feedback loop occurs, show empirically that this model exhibits such problems, and demonstrate how to change the inputs to a predictive policing system (in a black-box manner) so the runaway feedback loop does not occur, allowing the true crime rate to be learned. Our results are quantitative: we can establish a link (in our model) between the degree to which runaway feedback causes problems and the disparity in crime rates between areas. Moreover, we can also demonstrate the way in which \emph{reported} incidents of crime (those reported by residents) and \emph{discovered} incidents of crime (i.e. those directly observed by police officers dispatched as a result of the predictive policing algorithm) interact: in brief, while reported incidents can attenuate the degree of runaway feedback, they cannot entirely remove it without the interventions we suggest.

cs.CY↗