arXiv subjects

Nathan Jones

Publications and source records attributed to Nathan Jones.

At least 19 recordsLinked to original sources

arXiv2026

Creeping flows through confined arrays of cylinders

Hair-covered appendages serve a variety of purposes in Nature, from chemical sensing and particle capture on the antennae of a crustacean to drag generation on bristled wings. At low to intermediate Reynolds numbers, these finite porous media experience three flow regimes. At low Reynolds numbers, the flow goes around the porous structure; this is the paddle or rake regime. As the Reynolds number increases, so does the relative flow rate through the array. If the fluid exits the structure mostly laterally, the flow is in the deflection regime. If the fluid exits downstream, the flow is in the sieve regime. Confining structures, such as the animal body or larger hairs, have been hypothesized to focus the flow on the hair-covered region. We investigate the influence of confinement on the flow through and around an array of cylinders, using a combination of experiments and numerical simulations. Experimentally, we vary the cylinder spacing, channel dimension, and flow rate and measure the velocity field using Particle Image Velocimetry. After comparing the results of finite element analysis with the experimental data, we numerically investigate a broader range of system geometries and flow parameters. Our results show that the confinement focuses the flow in the array and shifts the domains of existence of the three regimes. We present an analytical model that relies on the permeability of rectangular slits to predict the relative flow rate through and around the array. The model is in quantitative agreement with the numerical results, demonstrating that the flow through the array increases with increasing confinement, while the flow angle decreases. These results should provide insight into the morphology of hairy surfaces and have implications in the design of bio-inspired flow sensors and filters.

physics.flu-dyn↗

arXiv2026

Fast Object Removal Attacks on Safety-Critical Video-based Perception Systems

By leveraging data from video-based perception systems, intelligent transportation systems (ITS) support safety-critical applications that improve road safety. However, adversaries may manipulate video frames to compromise downstream perception modules, causing failures in safety-critical functions and increasing risks to vulnerable road users. This paper presents a novel attack model and an end-to-end framework for near-real-time targeted object removal attack on a video-based safety-critical system. The end-to-end attack pipeline consists of four stages: localizing targets in each frame, retrieving coherent patches from earlier frames, blending them using context-aware alpha compositing, and reconstructing attacked frames. Experiments at an intersection on the South Carolina Connected Vehicle Testbed (SC-CVT) show that reconstructed frames have high global similarity to the originals, with frame-level Peak Signal to Noise Ratio (PSNR) above 40 dB and Structural Similarity Index Measure (SSIM) above 0.996. Using the YOLO-based detector, the attack reduces object detections by up to 97.59% and achieves a frame-level attack success rate of 94.48%. Across the evaluated detectors and frame resolutions, the mean execution time ranges from 0.074 to 0.172 seconds per frame on GPU hardware, indicating near-real-time performance in testing. The forensic evaluation using several pretrained tamper-detection models shows limited ability to distinguish reconstructed from authentic frames. The findings suggest that video-based perception is vulnerable to stealthy object removal attacks that can degrade the performance of safety-critical applications by reducing object detectability. These findings can help develop mitigation strategies against adversarial object removal attacks that threaten safety-critical applications, such as vision-based pedestrian safety systems.

cs.CR↗

arXiv2026

MEDA: Measurement-Efficient Disorder-Aware Majorana Zero Mode Detection in Realistic Devices

Fault-tolerant topological quantum computing relies on identifying Majorana zero modes (MZMs), but reliable detection in realistic devices remains challenging. Conventional topological indicators are inherently biased in finite, disordered systems, blurring the distinction between true MZMs and trivial states. Furthermore, attempts to map these indicators to real observables via machine learning require dense, expensive conductance measurements, creating a severe scaling bottleneck. To simultaneously address topological bias and measurement limitations, we present MEDA: a Measurement-Efficient, Disorder-Aware framework for MZM detection in realistic devices. MEDA maps sparse, practically obtainable observables directly to the robust periodic disorder invariant (PDI). Using a novel sparse parameter regime, MEDA reduces measurement volume by 10x while maintaining predictive quality, even in moderate to strong disorder regimes that limit conventional methods. Furthermore, MEDA naturally prioritizes input features consistent with the topological gap protocol, demonstrating strong physical interpretability.

cs.ET↗

arXiv2024

Pattern Formation of Freezing Infiltration in Porous Media

Gravity-driven infiltration of liquid water into unsaturated porous media can be a spatially heterogeneous process due to the gravity fingering instability. When such infiltration occurs in a subfreezing porous medium, liquid water can readily freeze, leading to both the removal of liquid water available for transport and a reduction in local permeability. As a result of the coupling between gravity fingering and freezing, macroscopic frozen structures can form that record the shape and history of the wetting front. These structures have been observed in the field in terrestrial snowpack and glacial firn layers and are believed to have profound impacts on how liquid water and its accompanying thermal content distribute during infiltration. However, a more detailed physics-based understanding of freezing infiltration has been missing. In this work, we use a thermodynamic nonequilibrium infiltration model to investigate the emergence of refrozen structures during water infiltration into an initially homogeneous and subfreezing porous medium. From scaling analysis, we recover the relevant nondimensional groups that govern the physics of the freezing infiltration process. We identify two key mechanisms caused by freezing that reduce the effective infiltration rate, calculated as the maximum depth of infiltration per elapsed time. In the first mechanism, the effective infiltrate rate decreases because a portion of the liquid water is consumed due to freezing, and such effect can be well quantified by the freezing Damk\"ohler number. For the second mechanism, we report on a new phenomenon termed secondary fingering, where new flow paths are established in between the primary infiltration channels. We find that secondary fingering reduces the degree of flow channelization and thus weakens the effective rate of infiltration via flow field homogenization.

physics.flu-dyn↗

arXiv2024

The Promises and Pitfalls of Using Language Models to Measure Instruction Quality in Education

Assessing instruction quality is a fundamental component of any improvement efforts in the education system. However, traditional manual assessments are expensive, subjective, and heavily dependent on observers' expertise and idiosyncratic factors, preventing teachers from getting timely and frequent feedback. Different from prior research that mostly focuses on low-inference instructional practices on a singular basis, this paper presents the first study that leverages Natural Language Processing (NLP) techniques to assess multiple high-inference instructional practices in two distinct educational settings: in-person K-12 classrooms and simulated performance tasks for pre-service teachers. This is also the first study that applies NLP to measure a teaching practice that is widely acknowledged to be particularly effective for students with special needs. We confront two challenges inherent in NLP-based instructional analysis, including noisy and long input data and highly skewed distributions of human ratings. Our results suggest that pretrained Language Models (PLMs) demonstrate performances comparable to the agreement level of human raters for variables that are more discrete and require lower inference, but their efficacy diminishes with more complex teaching practices. Interestingly, using only teachers' utterances as input yields strong results for student-centered variables, alleviating common concerns over the difficulty of collecting and transcribing high-quality student speech data in in-person teaching settings. Our findings highlight both the potential and the limitations of current NLP techniques in the education domain, opening avenues for further exploration.

cs.CL↗

arXiv2023

Locally imprimitive points on elliptic curves

Under GRH, any element in the multiplicative group of a number field $K$ that is globally primitive (i.e., not a perfect power in $K^*$) is a primitive root modulo a set of primes of $K$ of positive density. For elliptic curves $E/K$ that are known to have infinitely many primes $\mathfrak p$ of cyclic reduction, possibly under GRH, a globally primitive point $P\in E(K)$ may fail to generate any of the point groups $E(k_{\mathfrak p})$. We describe this phenomenon in terms of an associated Galois representation $\rho_{E/K, P}:G_K\to\mathrm{GL}_3(\hat{\mathbf Z})$, and use it to construct non-trivial examples of global points on elliptic curves that are locally imprimitive.

math.NT↗

arXiv2023

CM elliptic curves and vertically entangled 2-adic groups

Consider the elliptic curve $E$ given by the Weierstrass equation $y^2 = x^3 - 11x - 14$, which has complex multiplication by the order of conductor $2$ inside $\mathbb{Z}[i]$. It was recently observed in a paper of Daniels and Lozano-Robledo that, for each $n \geq 2$, $\mathbb{Q}(\mu_{2^{n+1}}) \subseteq \mathbb{Q}(E[2^n])$. In this note, we prove that this (a priori surprising) ``tower of vertical entanglements'' is actually more a feature than a bug: it holds for any elliptic curve $E$ over $\mathbb{Q}$ with complex multiplication by any order of even discriminant.

math.NT↗

arXiv2022

On the acyclicity of reductions of elliptic curves modulo primes in arithmetic progressions

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and, for a prime $p$ of good reduction for $E$ let $\tilde{E}_p$ denote the reduction of $E$ modulo $p$. Inspired by an elliptic curve analogue of Artin's primitive root conjecture posed by S. Lang and H. Trotter in 1977, J-P. Serre adapted methods of C. Hooley to prove a GRH-conditional asymptotic formula for the number of primes $p \leq x$ for which the group $\tilde{E}_p(\mathbb{F}_p)$ is cyclic. More recently, Akbal and G\"{u}lo$\breve{\text{g}}$lu considered the question of cyclicity of $\tilde{E}_p(\mathbb{F}_p)$ under the additional restriction that $p$ lie in an arithmetic progression. In this note, we study the issue of which arithmetic progressions $a \bmod n$ have the property that, for all but finitely many primes $p \equiv a \bmod n$, the group $\tilde{E}_p(\mathbb{F}_p)$ is not cyclic, answering a question of Akbal and G\"{u}lo$\breve{\text{g}}$lu on this issue.

math.NT↗

arXiv2021

Elliptic curves with missing Frobenius traces

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In 1976, Lang and Trotter conjectured an asymptotic formula for the number $\pi_{E,r}(X)$ of primes $p \leq X$ of good reduction for which the Frobenius trace at $p$ associated to $E$ is equal to a given fixed integer $r$. We investigate elliptic curves $E$ over $\mathbb{Q}$ that have a missing Frobenius trace, i.e. for which the counting function $\pi_{E,r}(X)$ remains bounded as $X \rightarrow \infty$, for some $r \in \mathbb{Z}$. In particular, we classify all elliptic curves $E$ over $\mathbb{Q}(t)$ that have a missing Frobenius trace.

math.NT↗

arXiv2020

Elliptic curves with non-abelian entanglements

We consider the problem of classifying quadruples $(K,E,m_1,m_2)$ where $K$ is a number field, $E$ is an elliptic curve defined over $K$ and $(m_1,m_2)$ is a pair of relatively prime positive integers for which the intersection $K(E[m_1]) \cap K(E[m_2])$ is a non-abelian extension of $K$. There is an infinite set $\mathcal{S}$ of modular curves whose $K$-rational points capture all elliptic curves over $K$ without complex multiplication that have this property. Our main theorem explicitly describes the (finite) subset of $\mathcal{S}$ consisting of those modular curves having genus zero. In the case $K = \mathbb{Q}$, this has applications to the problem of determining when the Galois representation on the torsion of $E$ is as large as possible modulo a prescribed obstruction; we illustrate this application with a specific example.

math.NT↗

arXiv2020

The Lang-Trotter Conjecture for products of non-CM elliptic curves

Inspired by the work of Lang-Trotter on the densities of primes with fixed Frobenius traces for elliptic curves defined over $\mathbb{Q}$ and by the subsequent generalization of Cojocaru-Davis-Silverberg-Stange to generic abelian varieties, we study the analogous question for abelian surfaces isogenous to products of non-CM elliptic curves over $\mathbb{Q}$. We formulate the corresponding conjectural asymptotic, provide upper bounds, and explicitly compute (when the elliptic curves lie outside a thin set) the arithmetically significant constants appearing in the asymptotic. This allows us to provide computational evidence for the conjecture.

math.NT↗

arXiv2020

Degree bounds for projective division fields associated to elliptic modules with a trivial endomorphism ring

Let $k$ be a global field, let $A$ be a Dedekind domain with $\text{Quot}(A) = k$, and let $K$ be a finitely generated field. Using a unified approach for both elliptic curves and Drinfeld modules $M$ defined over $K$ and having a trivial endomorphism ring, with $k= \mathbb{Q}$, $A = \mathbb{Z}$ in the former case and $k$ a global function field, $A$ its ring of functions regular away from a fixed prime in the latter case, for any nonzero ideal $\mathfrak{a} \lhd A$ we prove best possible estimates in the norm $|\mathfrak{a}|$ for the degrees over $K$ of the subfields of the $\mathfrak{a}$-division fields of $M$ fixed by scalars.

math.NT↗

arXiv2019

A bound for the conductor of an open subgroup of GL2 associated to an elliptic curve

Given an elliptic curve $E$ without complex multiplication defined over a number field $K$, consider the image of the Galois representation defined by letting Galois act on the torsion of $E$. Serre's open image theorem implies that there is a positive integer $m$ for which the Galois image is completely determined by its reduction modulo $m$. In this note, we prove a bound on the smallest such $m$ in terms of standard invariants associated with $E$. The bound is sharp and improves upon previous results.

math.NT↗

arXiv2017

Cohen-Lenstra-Gerth Heuristics via Automorphism Counts

For a finite abelian 2-group $G$, we study the frequency with which quadratic imaginary number fields $K$ have 2-part of their class group $K$ isomorphic to $G$. A philosophy enunciated by Gerth extends the Cohen-Lenstra heuristics for imaginary quadratic number fields to the case $p=2$, by referencing both the 2-rank and the 4-rank of the group in question. A recent paper by Smith provides relative density statements about the $2^{k+1}$-rank of such a class group given its $2^1$- through $2^k$-ranks, for $k \geq 2$. We deduce from Smith's results an explicit automorphism-count-theoretic statement of the Cohen-Lenstra-Gerth heuristics, also describing connections to "higher R\'{e}dei matrices" introduced by Kolster to study the $2^k$-ranks of the class group of $K$.

math.NT↗

arXiv2017

Constants in Titchmarsh divisor problems for elliptic curves

Inspired by the analogy between the group of units $\mathbb{F}_p^{\times}$ of the finite field with $p$ elements and the group of points $E(\mathbb{F}_p)$ of an elliptic curve $E/\mathbb{F}_p$, E. Kowalski, A. Akbary & D. Ghioca, and T. Freiberg & P. Kurlberg investigated the asymptotic behaviour of elliptic curve sums analogous to the Titchmarsh divisor sum $\sum_{p \leq x} \tau(p + a) \sim C x$. In this paper, we present a comprehensive study of the constants $C(E)$ emerging in the asymptotic study of these elliptic curve divisor sums. Specifically, by analyzing the division fields of an elliptic curve $E/\mathbb{Q}$, we prove upper bounds for the constants $C(E)$ and, in the generic case of a Serre curve, we prove explicit closed formulae for $C(E)$ amenable to concrete computations. Moreover, we compute the moments of the constants $C(E)$ over two-parameter families of elliptic curves $E/\mathbb{Q}$. Our methods and results complement recent studies of average constants occurring in other conjectures about reductions of elliptic curves by addressing not only the average behaviour, but also the individual behaviour of these constants, and by providing explicit tools towards the computational verifications of the expected asymptotics.

math.NT↗

arXiv2015

Missing class groups and class number statistics for imaginary quadratic fields

The number F(h) of imaginary quadratic fields with a given class number h is of classical interest: Gauss' class number problem asks for a determination of those fields counted by F(h). The unconditional computation of F(h) for h up to 100 was completed by M. Watkins, using ideas of Goldfeld and Gross-Zagier; Soundararajan has more recently made conjectures about the order of magnitude of F(h) as h increases without bound, and determined its average order. In the present paper, we refine Soundararajan's conjecture to a conjectural asymptotic formula and also consider the subtler problem of determining the number F(G) of imaginary quadratic fields with class group isomorphic to a given finite abelian group G. Using Watkins' tables, one can show that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance the elementary abelian group of order 27 does not). This observation is explained in part by the Cohen-Lenstra heuristics, which have often been used to study the distribution of the p-part of an imaginary quadratic class group. We combine heuristics of Cohen-Lenstra together with our refinement of Soundararajan's conjecture to make precise predictions about the asymptotic nature of the entire imaginary quadratic class group, in particular addressing the above-mentioned phenomenon of "missing" class groups, for the case of p-groups as p tends to infinity. Furthermore, conditionally on the Generalized Riemann Hypothesis, we extend Watkins' data, tabulating F(h) for odd h up to 10^6 and F(G) for G a p-group of odd order with |G| up to 10^6. The numerical evidence matches quite well with our conjectures.

math.NT↗

arXiv2014

A local-global principle for power maps

Let f be a function from the set of rational numbers into itself. We call f a global power map if f(n) = n^k for some integer exponent k. We call f a local power map at the prime number p if f induces a well-defined group homomorphism on the multiplicative group of integers modulo p. We conjecture that if f is a local power map at an infinite number of primes p, then f must be a global power map. Our main theorem implies that if f is a local power map at every prime p in a set with positive upper density relative to the set of all primes, then f must be a global power map. In particular, this represents progress towards a conjecture of Fabrykowski and Subbarao.

math.NT↗

arXiv2014

Elliptic curves with 2-torsion contained in the 3-torsion field

There is a modular curve X'(6) of level 6 defined over Q whose Q-rational points correspond to j-invariants of elliptic curves E over Q for which Q(E[2]) is a subfield of Q(E[3]). In this note we characterize the j-invariants of elliptic curves with this property by exhibiting an explicit model of X'(6). Our motivation is two-fold: on the one hand, X'(6) belongs to the list of modular curves which parametrize non-Serre curves (and is not well-known), and on the other hand, X'(6)(Q) gives an infinite family of examples of elliptic curves with non-abelian "entanglement fields," which is relevant to the systematic study of correction factors of various conjectural constants for elliptic curves over Q.

math.NT↗