arXiv subjects

Alexander Smith

Publications and source records attributed to Alexander Smith.

At least 19 recordsLinked to original sources

arXiv2026

Tamagawa ratios and unbounded Selmer moments

We develop a framework to predict whether a family of Selmer groups has average size that is bounded or unbounded. Applying this framework to certain geometric families of abelian varieties over $\mathbb{Q}$, we give a conjectural characterization of which such families have $\ell$-Selmer groups of unbounded average size for a given prime $\ell$. In the case that the $\ell$-torsion Galois module is constant across the family, we show that our characterization is correct. The key tool of our technique is the Greenberg--Wiles' formula, which expresses the ratio of the sizes of a Selmer group and the corresponding dual Selmer group as a product of local factors. This formula gives a purely local lower bound for the size of a Selmer group that we conjecture is close to sharp most of the time.

math.NT↗

arXiv2025

Lattice points in thickened parabolas and rational points near hypersurfaces

Among the nondegenerate C^4 hypersurfaces M in R^n, we characterize the rational quadrics as the hypersurfaces that are the least well approximated by rational points. Given M other than a rational quadric, we prove a heuristically sharp lower bound for the number of rational points very near M, improving the sensitivity of prior results of Beresnevich and Huang. Our methods are dynamical, and rely on an application of Ratner's theorems to 1-parameter unipotent subgroups U of SL_n(R) such that u - Id has rank at most 2 for every u in U. As part of our work, we study the algebraic subgroups of SL_n(Q) whose collection of real points can contain such a subgroup.

math.NT↗

arXiv2025

First Full Dalitz Plot Measurement in Neutron $\beta$-Decay using the Nab Spectrometer and Implications for New Physics

Precision measurements of observables in neutron $\beta$-decay are used to test the Standard Model description of the weak interaction and search for evidence of new physics. The Nab experiment at the Fundamental Neutron Physics Beamline at the Spallation Neutron Source was constructed to measure correlations in neutron decay by utilizing an asymmetric spectrometer and novel detection system to accurately reconstruct the proton momentum and electron energy for each $\beta$-decay. This work describes the detection of neutron $\beta$-decay products in the Nab spectrometer and presents the first full Dalitz plot representation of the phase space of neutron $\beta$-decay for all electrons >100 keV. In addition, new constraints are placed on a possible excited neutron state, hypothesized to explain the disagreement between the appearance and disappearance neutron lifetime techniques.

nucl-ex↗

arXiv2025

The Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture

Given an elliptic curve E/Q, we show that 50% of the quadratic twists of E have $2^{\infty}$-Selmer corank 0 and 50% have $2^{\infty}$-Selmer corank 1. As one consequence, we prove that the Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture. Previously, this result was known by work of the author for elliptic curves over Q satisfying certain technical conditions. As part of this work, we determine the distribution of 2-Selmer ranks in the quadratic twist family of E. In the cases where this distribution was not already known, it is distinct from the model for distributions of 2-Selmer groups constructed by Poonen and Rains.

math.NT↗

arXiv2025

Topological descriptors for the electron density of inorganic solids

Descriptors play an important role in data-driven materials design. While most descriptors of crystalline materials emphasize structure and composition, they often neglect the electron density - a complex yet fundamental quantity that governs material properties. Here, we introduce Betti curves as topological descriptors that compress electron densities into compact representations. Derived from persistent homology, Betti curves capture bonding characteristics by encoding components, cycles, and voids across varied electron density thresholds. Machine learning models trained on Betti curves outperform those trained on raw electron densities by an average of 33 percentage points in classifying structure prototypes, predicting thermodynamic stability, and distinguishing metals from non-metals. Shannon entropy calculations reveal that Betti curves retain comparable information content to electron density while requiring two orders of magnitude less data. By combining expressive power with compact representation, Betti curves highlight the potential of topological data analysis to advance materials design.

cond-mat.mtrl-sci↗

arXiv2024

Inferring the mass content of galaxy clusters with satellite kinematics and Jeans Anisotropic modeling

Satellite galaxies can be used to indicate the dynamical mass of galaxy groups and clusters. In this study, we apply the axis-symmetric Jeans Anisotropic Multi-Gaussian Expansion JAM modeling to satellite galaxies in 28 galaxy clusters selected from the TNG300-1 simulation with halo mass of $\log_{10}M_{200}/M_\odot>14.3$. If using true bound satellites as tracers, the best constrained total mass within the half-mass radius of satellites, $M(<r_\mathrm{half})$, and the virial mass, $M_{200}$, have average biases of -0.01 and $0.03$~dex, with average scatters of 0.11~dex and 0.15~dex. If selecting companions in redshift space with line-of-sight depth of 2,000~km/s, the biases are -0.06 and $0.01$~dex, while the scatters are 0.12 and 0.18~dex for $M(<r_\mathrm{half})$ and $M_{200}$. By comparing the best-fitting and actual density profiles, we find $\sim$29% of best-fitting density profiles show very good agreement with the truth, $\sim$32% display over or under estimates at most of the radial range with biased $M(<r_\mathrm{half})$, and 39% show under/over estimates in central regions and over/under estimates in the outskirts, with good constraints on $M(<r_\mathrm{half})$, yet most of the best constraints are still consistent with the true profiles within 1-$\sigma$ statistical uncertainties for the three circumstances. Using a mock DESI Bright Galaxy Survey catalog with the effect of fiber incompleteness, we find DESI fiber assignments and the choice of flux limits barely modify the velocity dispersion profiles and are thus unlikely to affect the dynamical modeling outcomes. Our results show that with current and future deep spectroscopic surveys, JAM can be a powerful tool to constrain the underlying density profiles of individual massive galaxy clusters.

astro-ph.CO↗

arXiv2024

Sums of rational cubes and the $3$-Selmer group

Recently, Alp\"oge-Bhargava-Shnidman determined the average size of the $2$-Selmer group in the cubic twist family of any elliptic curve over $\mathbb{Q}$ with $j$-invariant $0$. We obtain the distribution of the $3$-Selmer groups in the same family. As a consequence, we improve their upper bound on the density of integers expressible as a sum of two rational cubes. Assuming a $3$-converse theorem, we also improve their lower bound on this density. The $\sqrt{-3}$-Selmer group in this cubic twist family is well-known to be large, which poses significant challenges to the methods previously developed by the second author. We overcome this problem by strengthening the analytic core of these methods. Specifically, we prove a "trilinear large sieve" for an appropriate generalization of the classical R\'edei symbol, then use this to control the restriction of the Cassels-Tate pairing to the $\sqrt{-3}$-Selmer groups in these twist families.

math.NT↗

arXiv2024

Faithful Artin induction and the Chebotarev density theorem

Given a finite group G, we prove that the vector space spanned by the faithful irreducible characters of G is generated by the monomial characters in the vector space. As a consequence, we show that in any family of G-extensions of a fixed number field F, almost all are subject to a strong effective version of the Chebotarev density theorem. We use this version of the Chebotarev density theorem to deduce several consequences for class groups in families of number fields.

math.NT↗

arXiv2024

Topological Data Analysis for Particulate Gels

Soft gels, formed via the self-assembly of particulate organic materials, exhibit intricate multi-scale structures that provides them with flexibility and resilience when subjected to external stresses. This work combines molecular simulations and topological data analysis (TDA) to characterize the complex multi-scale structure of soft gels. Our TDA analysis focuses on the use of the Euler characteristic, which is an interpretable and computationally-scalable topological descriptor that is combined with filtration operations to obtain information on the geometric (local) and topological (global) structure of soft gels. We reduce the topological information obtained with TDA using principal component analysis (PCA) and show that this provides an informative low-dimensional representation of gel structure. We use the proposed computational framework to investigate the influence of gel preparation (e.g., quench rate, volume fraction) on soft gel structure and to explore dynamic deformations that emerge under oscillatory shear in various response regimes (linear, nonlinear, and flow). Our analysis identifies specific scales and extents at which hierarchical structures in soft gels are affected; moreover, correlations between structural deformations and mechanical phenomena (such as shear stiffening) are explored. In summary, we show that TDA facilitates the mathematical representation, quantification, and analysis of soft gel structures, extending traditional network analysis methods to capture both local and global organization.

cond-mat.soft↗

arXiv2024

New Lower Bounds for the Schur-Siegel-Smyth Trace Problem

We derive and implement a new way to find lower bounds on the smallest limiting trace-to-degree ratio of totally positive algebraic integers and improve the previously best known bound to 1.80203. Our method adds new constraints to Smyth's linear programming method to decrease the number of variables required in the new problem of interest. This allows for faster convergence recovering Schur's bound in the simplest case and Siegel's bound in the second simplest case of our new family of bounds. We also prove the existence of a unique optimal solution to our newly phrased problem and express the optimal solution in terms of polynomials. Lastly, we solve this new problem numerically with a gradient descent algorithm to attain the new bound 1.80203.

math.NT↗

arXiv2023

A Fully Explicit Integrator for Modeling Astrophysical Reactive Flows

Simulating complex astrophysical reacting flows is computationally expensive -- reactions are stiff and typically require implicit integration methods. The reaction update is often the most expensive part of a simulation, which motivates the exploration of more economical methods. In this research note, we investigate how the explicit Runge--Kutta--Chebyshev (RKC) method performs compared to an implicit method when applied to astrophysical reactive flows. These integrators are applied to simulations of X-ray bursts arising from unstable thermonuclear burning of accreted fuel on the surface of neutron stars. We show that the RKC method performs with similar accuracy to our traditional implicit integrator, but is more computationally efficient when run on CPUs.

astro-ph.IM↗

arXiv2023

Automated Characterization and Monitoring of Material Shape using Riemannian Geometry

Shape affects both the physical and chemical properties of a material. Characterizing the roughness, convexity, and general geometry of a material can yield information on its catalytic efficiency, solubility, elasticity, porosity, and overall effectiveness in the application of interest. However, material shape can be defined in a multitude of conflicting ways where different aspects of a material's geometry are emphasized over others, leading to bespoke measures of shape that are not easily generalizable. In this paper, we explore the use of Riemannian geometry in the analysis of shape and show that a Riemannian geometric framework for shape analysis is generalizable, computationally scalable, and can be directly integrated into common data analysis methods. In this framework, material shapes are abstracted as points on a Riemannian manifold. This information can be used to construct statistical moments (e.g., means, variances) and perform tasks such as dimensionality reduction and statistical process control. We provide a practical introduction to the mathematics of shape analysis through Riemannian geometry and illustrate its application on a manufactured/mined granular material dataset provided by Covia Corp. We show that the Riemannian framework can be used to automatically extract and quantify the shape of granular materials in a statistically rigorous manner.

cs.CE↗

arXiv2022

The distribution of $\ell^\infty$-Selmer groups in degree $\ell$ twist families I

In this paper and its sequel, we develop a technique for finding the distribution of $\ell^{\infty}$-Selmer groups in degree $\ell$ twist families of Galois modules over number fields. Given an elliptic curve E over a number field satisfying certain technical conditions, this technique can be used to show that 100% of the quadratic twists of E have rank at most 1. Given a prime $\ell$ and a number field F not containing $\mu_{2\ell}$, this method also shows that the $\ell^{\infty}$-class groups in the family of degree $\ell$ cyclic extensions of F have a distribution consistent with the Cohen-Lenstra-Gerth heuristics. For this work, we develop the theory of the fixed point Selmer group, which serves as the base layer of the $\ell^{\infty}$-Selmer group. This first paper gives a technique for finding the distribution of $\ell^{\infty}$-Selmer groups in certain families of twists where the fixed point Selmer group is stable. In the sequel paper, we will give a technique for controlling fixed point Selmer groups.

math.NT↗

arXiv2022

The distribution of $\ell^{\infty}$-Selmer groups in degree $\ell$ twist families II

We continue the investigation of the distribution of $\ell^{\infty}$-Selmer groups in degree $\ell$ twist families of Galois modules over number fields begun in the previous paper. Building off the work on higher Selmer groups in that part, we find conditions under which we can compute the distribution of the $\ell^{\infty}$-Selmer groups for a given degree $\ell$ twist family. Along the way, we show that the average rank in the quadratic twist family of any given abelian variety over a number field is bounded.

math.NT↗

arXiv2022

Field change for the Cassels-Tate pairing and applications to class groups

In previous work, the authors defined a category $SMod_F$ of finite Galois modules decorated with local conditions for each global field $F$. In this paper, given an extension $K/F$ of global fields, we define a restriction of scalars functor from $SMod_K$ to $SMod_F$ and show that it behaves well with respect to the Cassels-Tate pairing. We apply this work to study the class groups of global fields in the context of the Cohen-Lenstra heuristics.

math.NT↗

arXiv2022

Data Analysis using Riemannian Geometry and Applications to Chemical Engineering

We explore the use of tools from Riemannian geometry for the analysis of symmetric positive definite matrices (SPD). An SPD matrix is a versatile data representation that is commonly used in chemical engineering (e.g., covariance/correlation/Hessian matrices and images) and powerful techniques are available for its analysis (e.g., principal component analysis). A key observation that motivates this work is that SPD matrices live on a Riemannian manifold and that implementing techniques that exploit this basic property can yield significant benefits in data-centric tasks such classification and dimensionality reduction. We demonstrate this via a couple of case studies that conduct anomaly detection in the context of process monitoring and image analysis.

stat.AP↗

arXiv2022

Automated Heart and Lung Auscultation in Robotic Physical Examinations

This paper presents the first implementation of autonomous robotic auscultation of heart and lung sounds. To select auscultation locations that generate high-quality sounds, a Bayesian Optimization (BO) formulation leverages visual anatomical cues to predict where high-quality sounds might be located, while using auditory feedback to adapt to patient-specific anatomical qualities. Sound quality is estimated online using machine learning models trained on a database of heart and lung stethoscope recordings. Experiments on 4 human subjects show that our system autonomously captures heart and lung sounds of similar quality compared to tele-operation by a human trained in clinical auscultation. Surprisingly, one of the subjects exhibited a previously unknown cardiac pathology that was first identified using our robot, which demonstrates the potential utility of autonomous robotic auscultation for health screening.

cs.RO↗

arXiv2021

Algebraic integers with conjugates in a prescribed distribution

Given a compact subset $\Sigma$ of the real numbers obeying some technical conditions, we consider the set of algebraic integers whose conjugates all lie in $\Sigma$. The distribution of conjugates of such an integer defines a probability measure on $\Sigma$; our main result gives a necessary and sufficient condition for a given probability measure on $\Sigma$ to be the limit of some sequence of distributions of conjugates. As one consequence, we show there are infinitely many totally positive algebraic integers $\alpha$ with $tr(\alpha) < 1.89831\cdot deg(\alpha)$. We also show how this work can be applied to find simple abelian varieties over finite fields with extreme point counts.

math.NT↗