Showing 1–50 of 85 results for author: Best, A

Detector characterization for a new $^{12}$C+$^{12}$C reaction study at LUNA

Authors: R. M. Gesùè, S. Turkat, J. Skowroński, M. Aliotta, L. Barbieri, F. Barile, D. Bemmerer, A. Best, A. Boeltzig, C. Broggini, C. G. Bruno, A. Caciolli, M. Campostrini, F. Casaburo, F. Cavanna, T. Chillery, G. F. Ciani, P. Colombetti, A. Compagnucci, P. Corvisiero, L. Csedreki, T. Davinson, D. Dell'Aquila, R. Depalo, A. Di Leva , et al. (28 additional authors not shown)

Abstract: The $^{12}$C+$^{12}$C fusion reaction plays a crucial role in stellar evolution, including the occurrence of supernova explosions, and in the synthesis of the chemical elements. However, our understanding of its cross section remains severely deficient, particularly below $E_\textrm{cm}=2.5$\,MeV, the energy range of interest for astrophysics. To address these unresolved issues, the LUNA collabora… ▽ More The $^{12}$C+$^{12}$C fusion reaction plays a crucial role in stellar evolution, including the occurrence of supernova explosions, and in the synthesis of the chemical elements. However, our understanding of its cross section remains severely deficient, particularly below $E_\textrm{cm}=2.5$\,MeV, the energy range of interest for astrophysics. To address these unresolved issues, the LUNA collaboration will conduct a dedicated study of the $^{12}$C+$^{12}$C reaction at the Bellotti Ion Beam Facility (Bellotti IBF) located deep underground within the Gran Sasso National Laboratory (LNGS) in Italy. Based on the combination of passive and active shields, this campaign aims to achieve unprecedented sensitivity in measuring the cross sections of the two key reaction channels, $^{12}$C($^{12}$C,$α$)$^{20}$Ne and $^{12}$C($^{12}$C,$p$)$^{23}$Na in the low-energy regime via $γ$-ray detection. Here, we report on a sensitivity study for the upcoming campaign with a focus on the characterization of two detectors, namely a HPGe detector and a NaI(Tl) array. Furthermore, their intrinsic contamination is thoroughly investigated since this could potentially influence the overall sensitivity. Assuming typical beam intensities of the Bellotti IBF, we will be able to investigate reaction rates significantly below 100 counts per day. In case of the $^{12}$C+$^{12}$C reaction we therefore expect to acquire experimental data well below the current limit of $E_\textrm{cm}=2.1\,$MeV. The results are supported by simulations to highlight the advantageous low-background environment, essential for high-precision nuclear astrophysics studies. △ Less

Submitted 7 January, 2026; originally announced January 2026.

Journal ref: J. Phys. G: Nucl. Part. Phys. 52 105103 (2025)

Nuclear Physics Mid Term Plan at LNGS

Authors: R. Buompane, F. Cavanna, C. Curceanu, A. D'Onofrio, A. Di Leva, A. Formicola, L. Gialanella, C. Gustavino, G. Imbriani, M. Junker, A. Marcianò, F. Marzaioli, R. Nania, F. Napolitano, K. Piscicchia, O. Straniero, C. Abia, M. Aliotta, D. Bemmerer, A. Best, A. Boeltzig, C. Bruno, A. Caciolli, A. Chieffi, G. Ciani , et al. (37 additional authors not shown)

Abstract: The Istituto Nazionale di Fisica Nucleare-Laboratori Nazionali del Gran Sasso (LNGS) is one of the largest underground physics laboratory, a very peculiar environment suited for experiments in Astroparticle Physics, Nuclear Physics and Fundamental Symmetries. The newly established Bellotti Ion Beam facility represents a major advance in the possibilities of studying nuclear processes in an undergr… ▽ More The Istituto Nazionale di Fisica Nucleare-Laboratori Nazionali del Gran Sasso (LNGS) is one of the largest underground physics laboratory, a very peculiar environment suited for experiments in Astroparticle Physics, Nuclear Physics and Fundamental Symmetries. The newly established Bellotti Ion Beam facility represents a major advance in the possibilities of studying nuclear processes in an underground environment. A workshop was organized at LNGS in the framework of the Nuclear Physics Mid Term Plan in Italy, an initiative of the Nuclear Physics Division of the Instituto Nazionale di Fisica Nucleare to discuss the opportunities that will be possible to study in the near future by employing state-of-the-art detection systems. In this report, a detailed discussion of the outcome of the workshop is presented. △ Less

Submitted 22 December, 2025; originally announced December 2025.

A prototype neutron-detector array for future deep-underground s-process studies

Authors: Thomas Chillery, David Rapagnani, Chemseddine Ananna, Edoardo D'Amore, Gianluca Imbriani, Antonino di Leva, Daniela Mercogliano, Jakub Skowronski, Benjamin Brückner, Sophia Dellmann, Philipp Erbacher, Tanja Heftrich, René Reifarth, Mario Weigand, Andreas Best

Abstract: We report a novel neutron-detection approach employing an EJ-309 liquid scintillator surrounded by six 3He proportional counters. Tests were performed at the FRANZ facility of the Goethe-University Frankfurt using the 7Li(p,n0)7Be reaction, producing neutrons across energies 50-720 keV. The scintillator's neutron energy quenching is determined, and its neutron/gamma-ray discrimination performance… ▽ More We report a novel neutron-detection approach employing an EJ-309 liquid scintillator surrounded by six 3He proportional counters. Tests were performed at the FRANZ facility of the Goethe-University Frankfurt using the 7Li(p,n0)7Be reaction, producing neutrons across energies 50-720 keV. The scintillator's neutron energy quenching is determined, and its neutron/gamma-ray discrimination performance is evaluated. The lowest detectable neutron energy is 163 keV. EJ-309 - 3He counter neutron coincidences are compared with those from simulations. This array forms the prototype of a larger design, called scintillator-3He array for deep-underground experiments on the S-process, currently undergoing construction and testing for an upcoming deep-underground study of the 22Ne(alpha,n)25Mg reaction cross-section at the freshly-commissioned Bellotti Ion Beam facility of the INFN Laboratori Nazionali del Gran Sasso. This upcoming project is expected to achieve exceptionally low sensitivity for measuring the cross section at energies of interest for the astrophysical 'weak' and 'main' slow neutron-capture processes. △ Less

Submitted 6 November, 2025; originally announced November 2025.

Comments: 17 pages, 13 figures, 2 tables

Journal ref: J. Phys. G: Nucl. Part. Phys. 52 075202 (2025)

Towards a comprehensive study of the 14N(p,g)15O astrophysical key reaction: Description of the experimental technique including novel target preparation

Authors: A. Compagnucci, A. Formicola, M. Campostrini, J. Cruz, M. Aliotta, C. Ananna, L. Barbieri, F. Barile, D. Bemmerer, A. Best, A. Boeltzig, C. Broggini, C. G. Bruno, A. Caciolli, F. Casaburo, F. Cavanna, G. F. Ciani, P. Colombetti, P. Corvisiero, L. Csedreki, T. Davinson, R. Depalo, A. Di Leva, Z. Elekes, F. Ferraro , et al. (24 additional authors not shown)

Abstract: While the 14N(p,g)15O reaction plays a key role in the hydrogen-burning processes in various stellar conditions, its reaction rate is not known with sufficient precision. Therefore, the first scientific project at the recently launched Bellotti Ion Beam Facility of the Laboratori Nazionali del Gran Sasso was the measurement of the 14N(p,g)15O reaction cross section in the proton energy range betwe… ▽ More While the 14N(p,g)15O reaction plays a key role in the hydrogen-burning processes in various stellar conditions, its reaction rate is not known with sufficient precision. Therefore, the first scientific project at the recently launched Bellotti Ion Beam Facility of the Laboratori Nazionali del Gran Sasso was the measurement of the 14N(p,g)15O reaction cross section in the proton energy range between 250 and 1500 keV. In this paper, the experimental techniques are summarized with special emphasis on the description of solid state nitrogen target production and characterization. The first results of the reaction yield measured at 55 deg detection angle are also presented. △ Less

Submitted 27 October, 2025; originally announced October 2025.

Journal ref: Eur. Phys. J. A 61 (2025) 191

On the generalized Fermat equation $x^{13} + y^{13} = z^n$

Authors: Alex J. Best, Sander R. Dahmen, Nuno Freitas

Abstract: Let $n \in \mathbb{Z}_{\geq 2}$. We study the generalized Fermat equation \[x^{13}+y^{13}=z^n, \quad x,y,z \in \mathbb{Z}, \quad \gcd(x,y,z)=1.\] Using a combination of techniques, including the modular method, classical descent, unit sieves, and Chabauty and Mordell--Weil sieve methods over number fields, we show that for $n=5$ all its solutions $(a,b,c)$ are trivial, i.e. satisfy $abc=0$. Under… ▽ More Let $n \in \mathbb{Z}_{\geq 2}$. We study the generalized Fermat equation \[x^{13}+y^{13}=z^n, \quad x,y,z \in \mathbb{Z}, \quad \gcd(x,y,z)=1.\] Using a combination of techniques, including the modular method, classical descent, unit sieves, and Chabauty and Mordell--Weil sieve methods over number fields, we show that for $n=5$ all its solutions $(a,b,c)$ are trivial, i.e. satisfy $abc=0$. Under the assumption of GRH, we also show that for $n=7$ there are only trivial solutions. Furthermore, we provide partial results towards solving the equation for general $n \in \mathbb{Z}_{\geq 2}$, in particular that any solution $(a,b,c)$ with $13\mid c$ is trivial. △ Less

Submitted 13 October, 2025; originally announced October 2025.

Report number: MPIM-Bonn-2025 MSC Class: Primary 11D41; Secondary 11F80; 11G05; 11G10; 11G30

SHADES -- $^{22}$Ne($α$,n)$^{25}$Mg reaction rate in the Gamow window

Authors: David Rapagnani, Chemseddine Ananna, Antonino Di Leva, Gianluca Imbriani, Matthias Junker, Marco Pignatari, Andreas Best

Abstract: Neutron capture reactions are the main contributors to the synthesis of the heavy elements through the s-process. Together with $^{13}$C($α$,n)16O, which has recently been measured by the LUNA collaboration in an energy region inside the Gamow peak, 22Ne(α,n)25Mg is the other main neutron source in stars. Its cross section is mostly unknown in the relevant stellar energy (450 keV < Ecm < 750 keV),… ▽ More Neutron capture reactions are the main contributors to the synthesis of the heavy elements through the s-process. Together with $^{13}$C($α$,n)16O, which has recently been measured by the LUNA collaboration in an energy region inside the Gamow peak, 22Ne(α,n)25Mg is the other main neutron source in stars. Its cross section is mostly unknown in the relevant stellar energy (450 keV < Ecm < 750 keV), where only upper limits from direct experiments and highly uncertain estimates from indirect sources exist. The ERC project SHADES (UniNa/INFN) aims to provide for the first time direct cross section data in this region and to reduce the uncertainties of higher energy resonance parameters. High sensitivity measurements will be performed with the new LUNA-MV accelerator at the INFN-LNGS laboratory in Italy: the energy sensitivity of the SHADES hybrid neutron detector, together with the low background environment of the LNGS and the high beam current of the new accelerator promises to improve the sensitivity by over 2 orders of magnitude over the state of the art, allowing to finally probe the unexplored low-energy cross section. Here we present an overview of the project and first results on the setup characterization. △ Less

Submitted 6 October, 2025; originally announced October 2025.

Comments: 6 pages, 3 figures, NIC-VI conference proceeding

Journal ref: EPJ Web of Conferences 260, 11031 (2022)

Advances in Radiative Capture Studies at LUNA with a Segmented BGO Detector

Authors: Jakub Skowronski, Riccardo Maria Gesuè, Axel Boeltzig, Giovanni Francesco Ciani, Denise Piatti, David Rapagnani, Marialuisa Aliotta, Chemsedinne Ananna, Francesco Barile, Daniel Bemmerer, Andreas Best, Carlo Broggini, Carlo Giovanni Bruno, Antonio Caciolli, Matteo Campostrini, Francesca Cavanna, Paolo Colombetti, Alessandro Compagnucci, Piero Corvisiero, Laszlo Csedreki, Tom Davinson, Rosanna Depalo, Antonino Di Leva, Zoltan Elekes, Federico Ferraro , et al. (21 additional authors not shown)

Abstract: Studies of charged-particle reactions for low-energy nuclear astrophysics require high sensitivity, which can be achieved by means of detection setups with high efficiency and low backgrounds, to obtain precise measurements in the energy region of interest for stellar scenarios. High-efficiency total absorption spectroscopy is an established and powerful tool for studying radiative capture reactio… ▽ More Studies of charged-particle reactions for low-energy nuclear astrophysics require high sensitivity, which can be achieved by means of detection setups with high efficiency and low backgrounds, to obtain precise measurements in the energy region of interest for stellar scenarios. High-efficiency total absorption spectroscopy is an established and powerful tool for studying radiative capture reactions, particularly if combined with the cosmic background reduction by several orders of magnitude obtained at the Laboratory for Underground Nuclear Astrophysics (LUNA). We present recent improvements in the detection setup with the Bismuth Germanium Oxide (BGO) detector at LUNA, aiming to reduce high-energy backgrounds and to increase the summing detection efficiency. The new design results in enhanced sensitivity of the BGO setup, as we demonstrate and discuss in the context of the first direct measurement of the 65 keV resonance ($E_{x} = 5672$ keV) of the $^{17}$O($p,γ$)$^{18}$F reaction. Moreover, we show two applications of the BGO detector, which exploit its segmentation. In case of complex $γ$-ray cascades, e.g. the de-excitation of $E_{x} = 5672$ keV in $^{18}$F, the BGO segmentation allows to identify and suppress the beam-induced background signals that mimic the sum peak of interest. We demonstrate another new application for such a detector in form of in-situ activation measurements of a reaction with $β^{+}$ unstable product nuclei, e.g., the $^{14}$N($p,γ$)$^{15}$O reaction. △ Less

Submitted 7 October, 2025; originally announced October 2025.

Journal ref: J. Phys. G: Nucl. Part. Phys. 50 045201 (2023)

Status and future directions for direct cross-section measurements of the 13C(a,n)16O reaction for astrophysics

Authors: L. Csedreki, Gy. Gyürky, D. Rapagnani, G. F. Ciani, M. Aliotta, C. Anannad, L. Barbieri, F. Barile, D. Bemmerer, A. Best, A. Boeltzig, C. Broggini, C. G. Bruno, A. Caciolli, F. Casaburom, F. Cavannak, P. Colombetti, A. Compagnucci, P. Corvisiero, T. Davinson, R. Depalo, A. Di Leva, Z. Elekes, F. Ferraro, A. Formicola , et al. (23 additional authors not shown)

Abstract: The 13C(a,n)16O reaction is the main neutron source of the s-process taking place in thermally pulsing AGB stars and it is one of the main candidate sources of neutrons for the i-process in the astrophysical sites proposed so far. Therefore, its rate is crucial to understand the production of the nuclei heavier than iron in the Universe. For the first time, the LUNA collaboration was able to measu… ▽ More The 13C(a,n)16O reaction is the main neutron source of the s-process taking place in thermally pulsing AGB stars and it is one of the main candidate sources of neutrons for the i-process in the astrophysical sites proposed so far. Therefore, its rate is crucial to understand the production of the nuclei heavier than iron in the Universe. For the first time, the LUNA collaboration was able to measure the 13C(a,n)16O cross section at Ec.m.=0.23-0.3 MeV drastically reducing the uncertainty of the S(E)-factor in the astrophysically relevant energy range. In this paper, we provide details and critical thoughts about the LUNA measurement and compare them with the current understanding of the 13C(a,n)16O reaction in view of future prospect for higher energy measurements. The two very recent results (from the University of Notre Dame and the JUNA collaboration) published after the LUNA data represent an important step forward. There is, however, still room for a lot of improvement in the experimental study of the 13C(a,n)16O reaction, as emphasized in the present manuscript. We conclude that to provide significantly better constraints on the low-energy extrapolation, experimental data need to be provided over a wide energy range, which overlaps with the energy range of current measurements. Furthermore, future experiments need to focus on the proper target characterisation, the determination of neutron detection efficiency having more nuclear physics input, such as angular distribution of the 13C(a,n)16O reaction below Ea<0.8 MeV and study of nuclear properties of monoenergetic neutron sources and/or via the study of sharp resonances of 13C(a,n)16O. Moreover, comprehensive, multichannel R-matrix analysis with a proper estimate of uncertainty budget of experimental data are still required. △ Less

Submitted 2 October, 2025; originally announced October 2025.

Comments: 19 pages, 10 figures

Journal ref: Journal of Physics G: Nuclear and Particle Physics, 2024, Volume 51, Number 10

Aristotle: IMO-level Automated Theorem Proving

Authors: Tudor Achim, Alex Best, Alberto Bietti, Kevin Der, Mathïs Fédérico, Sergei Gukov, Daniel Halpern-Leistner, Kirsten Henningsgard, Yury Kudryashov, Alexander Meiburg, Martin Michelsen, Riley Patterson, Eric Rodriguez, Laura Scharff, Vikram Shanker, Vladmir Sicca, Hari Sowrirajan, Aidan Swope, Matyas Tamas, Vlad Tenev, Jonathan Thomm, Harold Williams, Lawrence Wu

Abstract: We introduce Aristotle, an AI system that combines formal verification with informal reasoning, achieving gold-medal-equivalent performance on the 2025 International Mathematical Olympiad problems. Aristotle integrates three main components: a Lean proof search system, an informal reasoning system that generates and formalizes lemmas, and a dedicated geometry solver. Our system demonstrates state-… ▽ More We introduce Aristotle, an AI system that combines formal verification with informal reasoning, achieving gold-medal-equivalent performance on the 2025 International Mathematical Olympiad problems. Aristotle integrates three main components: a Lean proof search system, an informal reasoning system that generates and formalizes lemmas, and a dedicated geometry solver. Our system demonstrates state-of-the-art performance with favorable scaling properties for automated theorem proving. △ Less

Submitted 10 October, 2025; v1 submitted 1 October, 2025; originally announced October 2025.

Trans-Fe elements from Type Ia Supernovae. I. Heavy element nucleosynthesis during the formation of near-Chandrasekhar white dwarfs

Authors: Umberto Battino, James Keegans, Megan Allen, Friedrich Röpke Röpke, Falk Herwig, Andreas Best, Raphael Hirschi, Luciano Piersanti, Oscar Straniero, Stuart Sim, Claudia Travaglio, Pavel Denissenkov

Abstract: Type Ia supernovae (SNIa) are thermonuclear explosions of white dwarfs in binary systems. They are central to galactic chemical evolution and serve as standardizable candles in cosmology, yet their progenitors remain uncertain. In this work, we present a grid of five models detailing the evolution and nucleosynthesis of slowly merging carbon-oxygen white dwarfs approaching the Chandrasekhar mass.… ▽ More Type Ia supernovae (SNIa) are thermonuclear explosions of white dwarfs in binary systems. They are central to galactic chemical evolution and serve as standardizable candles in cosmology, yet their progenitors remain uncertain. In this work, we present a grid of five models detailing the evolution and nucleosynthesis of slowly merging carbon-oxygen white dwarfs approaching the Chandrasekhar mass. These models test a variety of physics input settings, including accretion rates, nuclear reaction rates, convection parameters, and the composition of the accreted material. During the merger process, as the mass of the primary white dwarf approaches the Chandrasekhar limit, carbon burning is initiated first on the surface before eventually igniting explosively at the center. As a consequence, the 22Ne(a,n)25Mg reaction activates in the outer layers of all models. The neutrons released in this way produce a weak s-process-like abundance distribution peaking at Kr, which is overproduced by more than a factor of 1000 compared to solar. The trans-Fe elements-enriched outer layer mass varies from 0.04 Msun to 0.11 Msun, depending on the accretion rate. Our explosion simulation of these progenitor models ejects significant amount of first-peak elements (e.g., Kr, Sr) as well as of some long-lived radioactive species, such as 60Fe. In a previous theoretical study, we found that a similar nucleosynthesis process during the progenitor phase may also occur on the surface of near-Chandrasekhar white dwarfs formed through the accretion of H-rich material via the single-degenerate scenario. Therefore, these results suggest trans-Fe enrichment might be a hallmark of near-Chandrasekhar SNIa ejecta, regardless of the specific progenitor channel, and could provide a new spectral signature distinguishing them from sub-Chandrasekhar explosions. △ Less

Submitted 20 August, 2025; originally announced August 2025.

Comments: Accepted for publication in Astronomy & Astrophysics

Journal ref: A&A 703, A44 (2025)

Improved direct measurement of low-energy resonances in the $^{21}$Ne(p,$γ$)$^{22}$Na reaction

Authors: R. S. Sidhu, F. Casaburo, E. Masha, M. Aliotta, C. Ananna, L. Barbieri, F. Barile, C. Baron, D. Bemmerer, A. Best, R. Biassisi, A. Boeltzig, R. Bonnel, C. Broggini, C. G. Bruno, A. Caciolli, M. Campostrini, F. Cavanna, T. Chillery, G. F. Ciani, P. Colombetti, A. Compagnucci, P. Corvisiero, L. Csedreki, T. Davinson , et al. (35 additional authors not shown)

Abstract: In the nova temperature range, 0.1 GK $< T <$ 0.4 GK, several low-energy resonances dominate the $^{21}$Ne(p,$γ$)$^{22}$Na reaction rate, which is currently affected by large uncertainties. We present a high-precision study of the resonances at $E^{\rm{lab}}_{\rm{r}}$ = 127.3, 271.4, 272.3, 291.5, and 352.6 keV, measured directly at the Laboratory for Underground Nuclear Astrophysics in Italy. The… ▽ More In the nova temperature range, 0.1 GK $< T <$ 0.4 GK, several low-energy resonances dominate the $^{21}$Ne(p,$γ$)$^{22}$Na reaction rate, which is currently affected by large uncertainties. We present a high-precision study of the resonances at $E^{\rm{lab}}_{\rm{r}}$ = 127.3, 271.4, 272.3, 291.5, and 352.6 keV, measured directly at the Laboratory for Underground Nuclear Astrophysics in Italy. The strengths of the 127.3, 271.4, and 291.5 keV resonances are consistent with previous measurements within 1$σ$. However, for the 272.3 keV and 352.6 keV resonances, we report strength values of (129.9 $\pm$ 5.8) meV and (14.9 $\pm$ 0.8) meV, respectively, more than a factor of 1.5 higher than literature values. In addition, we report on new branching ratios for the 127.3, 272.3, and 352.6 keV resonances, leading to updated decay schemes. Finally, we present a revised thermonuclear reaction rate and investigate its impact on the NeNa nucleosynthesis. △ Less

Submitted 28 May, 2025; originally announced May 2025.

The $^{22}$Ne($α$,n)$^{25}$Mg reaction -- state of the art, astrophysics, and perspectives

Authors: Andreas Best, Philip Adsley, Ryan Amberger, Umberto Battino, Thomas Chillery, Marco La Cognata, Richard James deBoer, Daniela Mercogliano, Shuya Ota, David Rapagnani, Ragandeep Singh Sidhu, Roberta Spartà, Aurora Tumino, Michael Wiescher

Abstract: One of the most important stellar neutron sources is the $^{22}$Ne($α$,n)$^{25}$Mg reaction, which gets activated both during the helium intershell burning in asymptotic giant branch stars and in core helium and shell carbon burning in massive stars. The $^{22}$Ne($α$,n)$^{25}$Mg reaction serves as the main neutron producer for the weak s-process and provides a short but strong neutron exposure du… ▽ More One of the most important stellar neutron sources is the $^{22}$Ne($α$,n)$^{25}$Mg reaction, which gets activated both during the helium intershell burning in asymptotic giant branch stars and in core helium and shell carbon burning in massive stars. The $^{22}$Ne($α$,n)$^{25}$Mg reaction serves as the main neutron producer for the weak s-process and provides a short but strong neutron exposure during the helium flash phase of the main s-process, significantly affecting the abundances at the s-process branch points. The cross section needs to be known at very low energies, as close as possible to the neutron threshold at $E_α= 562$ keV ($Q = - 478$ keV), but both direct and indirect measurements have turned out to be very challenging, leading to significant uncertainties. Here we discuss the current status of the reaction, including recent and upcoming measurements, and provide a discussion on the astrophysical implications as well as an outlook into the near future. △ Less

Submitted 17 May, 2025; originally announced May 2025.

Journal ref: Eur. Phys. J. A (2025) 61:99

Total cross section of $^{14}$N+$n$ from 0.1 to 12~MeV

Authors: R. J. deBoer, R. Arquette, D. Bemmerer, A. Best, R. Beyer, A. Boeltzig, G. Clarke, J. Görres, T. Hensel, A. R. Junghans, M. Matney, S. E. Müller, D. Rapagnani, A. Roberts, K. Römer, S. Turkat, K. Schmidt, J. Skowronski, A. Wagner, M. Wiescher, A. Yadav

Abstract: The reaction $^{14}$N$(n,p)^{14}$C is one of the main neutron poisons during $s$-process nucleosynthesis. In addition, the reaction provides insight into the yields of atmospheric nuclear weapon testing. Because of their high level of sensitivity, total neutron cross sections provide a great deal of constraint on the modeling of reaction cross sections through the $R$-matrix analyses used for nucl… ▽ More The reaction $^{14}$N$(n,p)^{14}$C is one of the main neutron poisons during $s$-process nucleosynthesis. In addition, the reaction provides insight into the yields of atmospheric nuclear weapon testing. Because of their high level of sensitivity, total neutron cross sections provide a great deal of constraint on the modeling of reaction cross sections through the $R$-matrix analyses used for nuclear data evaluations. Yet for $^{14}$N+$n$, only one high sensitivity measurement is available and it lacks detailed information about its experimental conditions and uncertainties. With these motivations in mind, a new measurement of the $^{14}$N+$n$ total cross section has been performed at the nELBE facility. The cross sections were found to be in good agreement with previous data over much of the energy range with the key exception of the lowest energy resonance at a neutron energy of 433 keV. △ Less

Submitted 8 May, 2025; originally announced May 2025.

Comments: 12 pages, 13 figures

Journal ref: Physical Review C 112, 025805 (2025)

A Comparative Analysis of R-Matrix Fitting: ${}^{12}$C$(p,γ)$${}^{13}$N as Test Case

Authors: J. Skowronski, D. Piatti, D. Rapagnani, M. Aliotta, C. Ananna, L. Barbieri, F. Barile, D. Bemmerer, A. Best, A. Boeltzig, C. Broggini, C. G. Bruno, A. Caciolli, M. Campostrini, F. Casaburo, F. Cavanna, G. F. Ciani, P. Colombetti, A. Compagnucci, P. Corvisiero, L. Csedreki, T. Davinson, R. Depalo, A. Di Leva, Z. Elekes , et al. (27 additional authors not shown)

Abstract: In nuclear astrophysics, the accurate determination of nuclear reaction cross sections at astrophysical energies is critical for understanding stellar evolution and nucleosynthesis. This study focuses on the $^{12}$C($p, γ$)$^{13}$N reaction, which takes part in the CNO cycle and is significant for determining the $^{12}$C/$^{13}$C ratio in stellar interiors. Data from various studies, including r… ▽ More In nuclear astrophysics, the accurate determination of nuclear reaction cross sections at astrophysical energies is critical for understanding stellar evolution and nucleosynthesis. This study focuses on the $^{12}$C($p, γ$)$^{13}$N reaction, which takes part in the CNO cycle and is significant for determining the $^{12}$C/$^{13}$C ratio in stellar interiors. Data from various studies, including recent LUNA measurements, reveal high discrepancies in cross section values, underscoring the need for robust fitting approaches. Utilizing the R-matrix theory, we compare different frequentist and Bayesian methodologies for estimating reaction cross sections and their uncertainties. The analysis evaluates the strengths and weaknesses of different statistical techniques, highlighting the importance of systematic uncertainty treatment and the estimate of covariance matrix estimation to enhance the reliability and reproducibility of uncertainty estimates in nuclear astrophysics. △ Less

Submitted 22 February, 2025; v1 submitted 19 February, 2025; originally announced February 2025.

Edit distance in substitution systems

Authors: Andrew Best, Yuval Peres

Abstract: Let $σ$ be a primitive substitution on an alphabet $\mathcal{A}$, and let $\mathcal{W}_n$ be the set of words of length $n$ determined by $σ$ (i.e., $w \in \mathcal{W}_n$ if $w$ is a subword of $σ^k(a)$ for some $a \in \mathcal{A}$ and $k \geq 1$). It is known that the corresponding substitution dynamical system is loosely Kronecker (also known as zero-entropy loosely Bernoulli), so the diameter o… ▽ More Let $σ$ be a primitive substitution on an alphabet $\mathcal{A}$, and let $\mathcal{W}_n$ be the set of words of length $n$ determined by $σ$ (i.e., $w \in \mathcal{W}_n$ if $w$ is a subword of $σ^k(a)$ for some $a \in \mathcal{A}$ and $k \geq 1$). It is known that the corresponding substitution dynamical system is loosely Kronecker (also known as zero-entropy loosely Bernoulli), so the diameter of $\mathcal{W}_n$ in the edit distance is $o(n)$. We improve this upper bound to $O(n/\sqrt{\log n})$. The main challenge is handling the case where $σ$ is non-uniform; a better bound is available for the uniform case. Finally, we show that for the Thue--Morse substitution, the diameter of $\mathcal{W}_n$ is at least $\sqrt {n/6} - 1$. △ Less

Submitted 17 September, 2025; v1 submitted 31 December, 2024; originally announced January 2025.

Comments: 25 pages, 4 figures; updated introduction

MSC Class: 37B10 (Primary) 68R15 (Secondary)

Characterization of the LUNA neutron detector array for the measurement of the 13C(a,n)16O reaction

Authors: L. Csedreki, G. F. Ciani, J. Balibrea-Correa, A. Best, M. Aliotta, F. Barile, D. Bemmerer, A. Boeltzig, C. Broggini, C. G. Bruno, A. Caciolli, F. Cavanna, T. Chillery, P. Colombetti, P. Corvisiero, T. Davinson, R. Depalo, A. Di Leva, Z. Elekes, F. Ferraro, E. M. Fiore, A. Formicola, Zs. Fulop, G. Gervino, A. Guglielmetti , et al. (24 additional authors not shown)

Abstract: We introduce the LUNA neutron detector array developed for the investigation of the 13C(a,n)16O reaction towards its astrophysical s-process Gamow peak in the low-background environment of the Laboratori Nazionali del Gran Sasso (LNGS). Eighteen 3He counters are arranged in two different configurations (in a vertical and a horizontal orientation) to optimize neutron detection effciency, target han… ▽ More We introduce the LUNA neutron detector array developed for the investigation of the 13C(a,n)16O reaction towards its astrophysical s-process Gamow peak in the low-background environment of the Laboratori Nazionali del Gran Sasso (LNGS). Eighteen 3He counters are arranged in two different configurations (in a vertical and a horizontal orientation) to optimize neutron detection effciency, target handling and target cooling over the investigated energy range Ea;lab = 300 - 400 keV (En = 2.2 - 2.6 MeV in emitted neutron energy). As a result of the deep underground location, the passive shielding of the setup and active background suppression using pulse shape discrimination, we reached a total background rate of 1.23 +- 0.12 counts/hour. This resulted in an improvement of two orders of magnitude over the state of the art allowing a direct measurement of the 13C(a,n)16O cross-section down to Ea;lab = 300 keV. The absolute neutron detection efficiency of the setup was determined using the 51V(p,n)51Cr reaction and an AmBe radioactive source, and completed with a Geant4 simulation. We determined a (34+-3) % and (38+-3) % detection efficiency for the vertical and horizontal configurations, respectively, for En = 2.4 MeV neutrons. △ Less

Submitted 7 November, 2024; originally announced November 2024.

Comments: 8 pages, 5 figures, to be published in NIMA

Journal ref: Nuclear Inst. and Methods in Physics Research, A 994 (2021) 165081

A complete formalization of Fermat's Last Theorem for regular primes in Lean

Authors: Alex Best, Christopher Birkbeck, Riccardo Brasca, Eric Rodriguez Boidi, Ruben van De Velde, Andrew Yang

Abstract: We formalize a complete proof of the regular case of Fermat's Last Theorem in the Lean4 theorem prover. Our formalization includes a proof of Kummer's lemma, that is the main obstruction to Fermat's Last Theorem for regular primes. Rather than following the modern proof of Kummer's lemma via class field theory, we prove it by using Hilbert's Theorems 90-94 in a way that is more amenable to formali… ▽ More We formalize a complete proof of the regular case of Fermat's Last Theorem in the Lean4 theorem prover. Our formalization includes a proof of Kummer's lemma, that is the main obstruction to Fermat's Last Theorem for regular primes. Rather than following the modern proof of Kummer's lemma via class field theory, we prove it by using Hilbert's Theorems 90-94 in a way that is more amenable to formalization. △ Less

Submitted 13 June, 2025; v1 submitted 2 October, 2024; originally announced October 2024.

First direct measurement of the 64.5 keV resonance strength in $^{17}$O(p,$γ$)$^{18}$F reaction

Authors: R. M. Gesuè, G. F. Ciani, D. Piatti, A. Boeltzig, D. Rapagnani, M. Aliotta, C. Ananna, L. Barbieri, F. Barile, D. Bemmerer, A. Best, C. Broggini, C. G. Bruno, A. Caciolli, M. Campostrini, F. Casaburo, F. Cavanna, P. Colombetti, A. Compagnucci, P. Corvisiero, L. Csedreki, T. Davinson, G. M. De Gregorio, D. Dell'Aquila, R. Depalo , et al. (28 additional authors not shown)

Abstract: The CNO cycle is one of the most important nuclear energy sources in stars. At temperatures of hydrostatic H-burning (20 MK $<$ T $<$ 80 MK) the $^{17}$O(p,$γ$)$^{18}$F reaction rate is dominated by the poorly constrained 64.5~keV resonance. Here we report on the first direct measurements of its resonance strength and of the direct capture contribution at 142 keV, performed with a new high sensiti… ▽ More The CNO cycle is one of the most important nuclear energy sources in stars. At temperatures of hydrostatic H-burning (20 MK $<$ T $<$ 80 MK) the $^{17}$O(p,$γ$)$^{18}$F reaction rate is dominated by the poorly constrained 64.5~keV resonance. Here we report on the first direct measurements of its resonance strength and of the direct capture contribution at 142 keV, performed with a new high sensitivity setup at LUNA. The present resonance strength of $ωγ_{(p, γ)}$\textsuperscript{bare} = (30 $\pm$ 6\textsubscript{stat} $\pm$ 2\textsubscript{syst})~peV is about a factor of 2 higher than the values in literature, leading to a $Γ$\textsubscript{p}\textsuperscript{bare} = (34 $\pm$ 7\textsubscript{stat} $\pm$ 3\textsubscript{syst})~neV, in agreement with LUNA result from the (p,$α$) channel. Such agreement strengthen our understanding of the oxygen isotopic ratios measured in red giant stars and in O-rich presolar grains. △ Less

Submitted 6 August, 2024; originally announced August 2024.

Solar fusion III: New data and theory for hydrogen-burning stars

Authors: B. Acharya, M. Aliotta, A. B. Balantekin, D. Bemmerer, C. A. Bertulani, A. Best, C. R. Brune, R. Buompane, F. Cavanna, J. W. Chen, J. Colgan, A. Czarnecki, B. Davids, R. J. deBoer, F. Delahaye, R. Depalo, A. García, M. Gatu Johnson, D. Gazit, L. Gialanella, U. Greife, D. Guffanti, A. Guglielmetti, K. Hambleton, W. C. Haxton , et al. (25 additional authors not shown)

Abstract: In stars that lie on the main sequence in the Hertzsprung-Russel diagram, like our sun, hydrogen is fused to helium in a number of nuclear reaction chains and series, such as the proton-proton chain and the carbon-nitrogen-oxygen cycles. Precisely determined thermonuclear rates of these reactions lie at the foundation of the standard solar model. This review, the third decadal evaluation of the nu… ▽ More In stars that lie on the main sequence in the Hertzsprung-Russel diagram, like our sun, hydrogen is fused to helium in a number of nuclear reaction chains and series, such as the proton-proton chain and the carbon-nitrogen-oxygen cycles. Precisely determined thermonuclear rates of these reactions lie at the foundation of the standard solar model. This review, the third decadal evaluation of the nuclear physics of hydrogen-burning stars, is motivated by the great advances made in recent years by solar neutrino observatories, putting experimental knowledge of the proton-proton chain neutrino fluxes in the few-percent precision range. The basis of the review is a one-week community meeting held in July 2022 in Berkeley, California, and many subsequent digital meetings and exchanges. The relevant reactions of solar and stellar hydrogen burning are reviewed here, from both theoretical and experimental perspectives. Recommendations for the state of the art of the astrophysical S-factor and its uncertainty are formulated for each of them. Several other topics of paramount importance for the solar model are reviewed, as well: recent and future neutrino experiments, electron screening, radiative opacities, and current and upcoming experimental facilities. In addition to reaction-specific recommendations, also general recommendations are formed. △ Less

Submitted 15 September, 2025; v1 submitted 10 May, 2024; originally announced May 2024.

Comments: 101 pages, 25 figures, including supplementary material. Final version published in Reviews of Modern Physics

Report number: N3AS-24-016

Journal ref: Rev. Mod. Phys. 97, 035002 (2025)

Validation of the VUV-reflective coating for next-generation liquid xenon detectors

Authors: D. Bajpai, A. Best, I. Ostrovskiy, D. Poitras, W. Wang

Abstract: Coating detector materials with films highly reflective in the ultraviolet region improves sensitivity of the rare-event detectors that use liquid xenon. In this work, we investigate the MgF$_2$-Al-MgF$_2$ coating designed to achieve high reflectance at 175 nm, the mean wavelength of liquid xenon (LXe) scintillation. The coating was applied to an unpolished, passivated copper substrate mimicking a… ▽ More Coating detector materials with films highly reflective in the ultraviolet region improves sensitivity of the rare-event detectors that use liquid xenon. In this work, we investigate the MgF$_2$-Al-MgF$_2$ coating designed to achieve high reflectance at 175 nm, the mean wavelength of liquid xenon (LXe) scintillation. The coating was applied to an unpolished, passivated copper substrate mimicking a realistic detector component of the proposed nEXO experiment, as well as to two unpassivated substrates with "high" and "average" levels of polishing. After confirming the composition and morphology of the thin-film coating using TEM and EDS, the samples underwent reflectance measurements in LXe and gaseous nitrogen (GN2). Measurements in LXe exposed the coated samples to -100 $°$C for several hours. No peeling of the coatings was observed after several thermal cycles. Polishing is found to strongly correlate with the measured specular reflectance ($R_{\mathrm{spec}}$). In particular, 5.8(5)% specular spike reflectance in LXe was measured for the realistic sample at 20$°$ of incidence, while the values for similar angles of incidence on the high and average polish samples are 62.3(1.3)% and 27.4(7)%, respectively. At large angles (66°--75$°$), the $R_{\mathrm{spec}}$ in LXe for the three samples increases to 23(5)%, 80(8)%, and 84(18)%, respectively. The $R_{\mathrm{spec}}$ at around 45$°$ was measured in both GN2 and LXe for average polish sample and shows a reasonable agreement. Importantly, the total reflectance of the samples is comparable and estimated to be 92(8)%, 85(8)%, and 83(8)% in GN2 for the realistic, average, and high polish samples, respectively. This is considered satisfactory for the next-generation LXe experiments that could benefit from using reflective films, such as nEXO and DARWIN, thus validating the design of the coating. △ Less

Submitted 14 May, 2024; v1 submitted 23 February, 2024; originally announced February 2024.

Comments: As accepted to JINST

Journal ref: 2024_JINST_19_P06008

Personalizing Driver Safety Interfaces via Driver Cognitive Factors Inference

Authors: Emily S Sumner, Jonathan DeCastro, Jean Costa, Deepak E Gopinath, Everlyne Kimani, Shabnam Hakimi, Allison Morgan, Andrew Best, Hieu Nguyen, Daniel J Brooks, Bassam ul Haq, Andrew Patrikalakis, Hiroshi Yasuda, Kate Sieck, Avinash Balachandran, Tiffany Chen, Guy Rosman

Abstract: Recent advances in AI and intelligent vehicle technology hold promise to revolutionize mobility and transportation, in the form of advanced driving assistance (ADAS) interfaces. Although it is widely recognized that certain cognitive factors, such as impulsivity and inhibitory control, are related to risky driving behavior, play a significant role in on-road risk-taking, existing systems fail to l… ▽ More Recent advances in AI and intelligent vehicle technology hold promise to revolutionize mobility and transportation, in the form of advanced driving assistance (ADAS) interfaces. Although it is widely recognized that certain cognitive factors, such as impulsivity and inhibitory control, are related to risky driving behavior, play a significant role in on-road risk-taking, existing systems fail to leverage such factors. Varying levels of these cognitive factors could influence the effectiveness and acceptance of driver safety interfaces. We demonstrate an approach for personalizing driver interaction via driver safety interfaces that are triggered based on a learned recurrent neural network. The network is trained from a population of human drivers to infer impulsivity and inhibitory control from recent driving behavior. Using a high-fidelity vehicle motion simulator, we demonstrate the ability to deduce these factors from driver behavior. We then use these inferred factors to make instantaneous determinations on whether or not to engage a driver safety interface. This interface aims to decrease a driver's speed during yellow lights and reduce their inclination to run through them. △ Less

Submitted 8 February, 2024; originally announced February 2024.

Comments: 12 pages, 7 figures

Clarifying the Radiative Decay of the Hoyle State with Charged-Particle Spectroscopy

Authors: D. Dell'Aquila, I. Lombardo, L. Redigolo, M. Vigilante, F. Angelini, L. Baldesi, S. Barlini, A. Best, A. Camaiani, G. Casini, C. Ciampi, M. Cicerchia, M. D'Andrea, J. Diklić, D. Fabris, B. Gongora Servin, A. Gottardo, F. Gramegna, G. Imbriani, T. Marchi, A. Massara, D. Mengoni, A. Ordine, L. Palada, G. Pasquali , et al. (11 additional authors not shown)

Abstract: A detailed knowledge of the decay properties of the so called Hoyle state in the $^{12}$C nucleus ($E_x=7.654$ MeV, $0^+$) is required to calculate the rate at which carbon is forged in typical red-giant stars. This paper reports on a new almost background-free measurement of the radiative decay branching ratio of the Hoyle state using advanced charged particle coincidence techniques. The exploita… ▽ More A detailed knowledge of the decay properties of the so called Hoyle state in the $^{12}$C nucleus ($E_x=7.654$ MeV, $0^+$) is required to calculate the rate at which carbon is forged in typical red-giant stars. This paper reports on a new almost background-free measurement of the radiative decay branching ratio of the Hoyle state using advanced charged particle coincidence techniques. The exploitation, for the first time in a similar experiment, of a bidimensional map of the coincidence efficiency allows to reach an unitary value and, consequently, to strongly reduce sources of systematic uncertainties. The present results suggest a value of the radiative branching ratio of $Γ_{rad}/Γ_{tot}=4.2(6)\cdot10^{-4}$. This finding helps to resolve the tension between recent data published in the literature. △ Less

Submitted 19 August, 2024; v1 submitted 31 January, 2024; originally announced January 2024.

First measurement of the low-energy direct capture in 20Ne(p, γ)21Na and improved energy and strength of the Ecm = 368 keV resonance

Authors: E. Masha, L. Barbieri, J. Skowronski, M. Aliotta, C. Ananna, F. Barile, D. Bemmerer, A. Best, A. Boeltzig, C. Broggini, C. G. Bruno, A. Caciolli, M. Campostrini, F. Casaburo, F. Cavanna, G. F. Ciani, A. Ciapponi, P. Colombetti, A. Compagnucci, P. Corvisiero, L. Csedreki, T. Davinson, R. Depalo, A. Di Leva, Z. Elekes , et al. (26 additional authors not shown)

Abstract: The $\mathrm{^{20}Ne(p, γ)^{21}Na}$ reaction is the slowest in the NeNa cycle and directly affects the abundances of the Ne and Na isotopes in a variety of astrophysical sites. Here we report the measurement of its direct capture contribution, for the first time below $E\rm_{cm} = 352$~keV, and of the contribution from the $E^{\rm }_{cm} = 368$~keV resonance, which dominates the reaction rate at… ▽ More The $\mathrm{^{20}Ne(p, γ)^{21}Na}$ reaction is the slowest in the NeNa cycle and directly affects the abundances of the Ne and Na isotopes in a variety of astrophysical sites. Here we report the measurement of its direct capture contribution, for the first time below $E\rm_{cm} = 352$~keV, and of the contribution from the $E^{\rm }_{cm} = 368$~keV resonance, which dominates the reaction rate at $T=0.03-1.00$~GK. The experiment was performed deep underground at the Laboratory for Underground Nuclear Astrophysics, using a high-intensity proton beam and a windowless neon gas target. Prompt $γ$ rays from the reaction were detected with two high-purity germanium detectors. We obtain a resonance strength $ωγ~=~(0.112 \pm 0.002_{\rm stat}~\pm~0.005_{\rm sys})$~meV, with an uncertainty a factor of $3$ smaller than previous values. Our revised reaction rate is 20\% lower than previously adopted at $T < 0.1$~GK and agrees with previous estimates at temperatures $T \geq 0.1$~GK. Initial astrophysical implications are presented. △ Less

Submitted 7 November, 2023; originally announced November 2023.

Comments: 8 pages, 4 figures, accepted to PRC (letter)

A Safe Preference Learning Approach for Personalization with Applications to Autonomous Vehicles

Authors: Ruya Karagulle, Nikos Arechiga, Andrew Best, Jonathan DeCastro, Necmiye Ozay

Abstract: This work introduces a preference learning method that ensures adherence to given specifications, with an application to autonomous vehicles. Our approach incorporates the priority ordering of Signal Temporal Logic (STL) formulas describing traffic rules into a learning framework. By leveraging Parametric Weighted Signal Temporal Logic (PWSTL), we formulate the problem of safety-guaranteed prefere… ▽ More This work introduces a preference learning method that ensures adherence to given specifications, with an application to autonomous vehicles. Our approach incorporates the priority ordering of Signal Temporal Logic (STL) formulas describing traffic rules into a learning framework. By leveraging Parametric Weighted Signal Temporal Logic (PWSTL), we formulate the problem of safety-guaranteed preference learning based on pairwise comparisons and propose an approach to solve this learning problem. Our approach finds a feasible valuation for the weights of the given PWSTL formula such that, with these weights, preferred signals have weighted quantitative satisfaction measures greater than their non-preferred counterparts. The feasible valuation of weights given by our approach leads to a weighted STL formula that can be used in correct-and-custom-by-construction controller synthesis. We demonstrate the performance of our method with a pilot human subject study in two different simulated driving scenarios involving a stop sign and a pedestrian crossing. Our approach yields competitive results compared to existing preference learning methods in terms of capturing preferences and notably outperforms them when safety is considered. △ Less

Submitted 26 March, 2024; v1 submitted 30 October, 2023; originally announced November 2023.

Comments: 9 pages, 3 figures, 2 tables. This work has been published at IEEE Robotics and Automation Letters

New proton-capture rates on carbon isotopes and their impact on the astrophysical $^{12}\mathrm{C}/{}^{13}\mathrm{C}$ ratio

Authors: J. Skowronski, A. Boeltzig, G. F. Ciani, L. Csedreki, D. Piatti, M. Aliotta, C. Ananna, F. Barile, D. Bemmerer, A. Best, C. Broggini, C. G. Bruno, A. Caciolli, M. Campostrini, F. Cavanna, P. Colombetti, A. Compagnucci, P. Corvisiero, T. Davinson, R. Depalo, A. Di Leva, Z. Elekes, F. Ferraro, A. Formicola, Zs. Fülöp , et al. (21 additional authors not shown)

Abstract: The ${}^{12}\mathrm{C}/{}^{13}\mathrm{C}$ ratio is a significant indicator of nucleosynthesis and mixing processes during hydrogen burning in stars. Its value mainly depends on the relative rates of the ${}^{12}\mathrm{C}(p,γ){}^{13}\mathrm{N}$ and ${}^{13}\mathrm{C}(p,γ){}^{14}\mathrm{N}$ reactions. Both reactions have been studied at the Laboratory for Underground Nuclear Astrophysics (LUNA) in… ▽ More The ${}^{12}\mathrm{C}/{}^{13}\mathrm{C}$ ratio is a significant indicator of nucleosynthesis and mixing processes during hydrogen burning in stars. Its value mainly depends on the relative rates of the ${}^{12}\mathrm{C}(p,γ){}^{13}\mathrm{N}$ and ${}^{13}\mathrm{C}(p,γ){}^{14}\mathrm{N}$ reactions. Both reactions have been studied at the Laboratory for Underground Nuclear Astrophysics (LUNA) in Italy down to the lowest energies to date ($E_\mathrm{c.m.} = 60\,\mathrm{keV}$) reaching for the first time the high energy tail of hydrogen burning in the shell of giant stars. Our cross sections, obtained with both prompt $γ$-ray detection and activation measurements, are the most precise to date with overall systematic uncertainties of $7-8\%$. Compared with most of the literature, our results are systematically lower, by $25\%$ for the ${}^{12}\mathrm{C}(p,γ){}^{13}\mathrm{N}$ reaction and by $30\%$ for ${}^{13}\mathrm{C}(p,γ){}^{14}\mathrm{N}$. We provide the most precise value up to now of $(3.6 \pm 0.4)$ in the $20-140\,\mathrm{MK}$ range for the lowest possible ${}^{12}\mathrm{C}/{}^{13}\mathrm{C}$ ratio that can be produced during H burning in giant stars. △ Less

Submitted 30 August, 2023; originally announced August 2023.

Comments: Submitted to Phys. Rev. Lett

Journal ref: Phys. Rev. Lett. 131, 162701 (2023)

Comparing intervention measures in a model of a disease outbreak on a university campus

Authors: Alex Best, Prerna Singh

Abstract: A number of theoretical models have been developed in recent years modelling epidemic spread in educational settings such as universities to help inform re-opening strategies during the Covid-19 pandemic. However, these studies have had differing conclusions as to the most effective non-pharmaceutical interventions. They also largely assumed permanent acquired immunity, meaning we have less unders… ▽ More A number of theoretical models have been developed in recent years modelling epidemic spread in educational settings such as universities to help inform re-opening strategies during the Covid-19 pandemic. However, these studies have had differing conclusions as to the most effective non-pharmaceutical interventions. They also largely assumed permanent acquired immunity, meaning we have less understanding of how disease dynamics will play out when immunity wanes. Here we complement these studies by developing and analysing a stochastic simulation model of disease spread on a university campus where we allow immunity to wane, expoloring the effectiveness of different interventions. We find that the two most effective interventions to limit the severity of a disease outbreak are reducing extra-household mixing and surveillance testing backed-up by a moderate isolation period. We find that contact tracing only has a limited effect, while reducing class sizes only has much effect if extra-household mixing is already low. We identify a range of measures that can not only limit an outbreak but prevent it entirely, and also comment on the variation in measures of severity that emerge from our stochastic simulations. We hope that our model may help in designing effective strategies for universities in future disease outbreaks. △ Less

Submitted 16 June, 2023; originally announced June 2023.

Fermat's Last Theorem for regular primes

Authors: Alex J. Best, Christopher Birkbeck, Riccardo Brasca, Eric Rodriguez Boidi

Abstract: We formalise the proof of the first case of Fermat's Last Theorem for regular primes using the \emph{Lean} theorem prover and its mathematical library \emph{mathlib}. This is an important 19th century result that motivated the development of modern algebraic number theory. Besides explaining the mathematics behind this result, we analyze in this paper the difficulties we faced in the formalisation… ▽ More We formalise the proof of the first case of Fermat's Last Theorem for regular primes using the \emph{Lean} theorem prover and its mathematical library \emph{mathlib}. This is an important 19th century result that motivated the development of modern algebraic number theory. Besides explaining the mathematics behind this result, we analyze in this paper the difficulties we faced in the formalisation process and how we solved them. For example, we had to deal with a diamond about characteristic zero fields and problems arising from multiple nested coercions related to number fields. We also explain how we integrated our work to \emph{mathlib}. △ Less

Submitted 21 May, 2023; v1 submitted 15 May, 2023; originally announced May 2023.

Comments: Accepted for the 14th International Conference on Interactive Theorem Proving (ITP 2023)

MSC Class: 68V20; 03B70 ACM Class: F.4.1

On the Polynomial Szemerédi Theorem in Finite Commutative Rings

Authors: Vitaly Bergelson, Andrew Best

Abstract: The polynomial Szemerédi theorem implies that, for any $δ\in (0,1)$, any family $\{P_1,\ldots, P_m\} \subset \mathbb{Z}[y]$ of nonconstant polynomials with constant term zero, and any sufficiently large $N$, every subset of $\{1,\ldots, N\}$ of cardinality at least $δN$ contains a nontrivial configuration of the form $\{x,x+P_1(y),\ldots, x+P_m(y)\}$. When the polynomials are assumed independent,… ▽ More The polynomial Szemerédi theorem implies that, for any $δ\in (0,1)$, any family $\{P_1,\ldots, P_m\} \subset \mathbb{Z}[y]$ of nonconstant polynomials with constant term zero, and any sufficiently large $N$, every subset of $\{1,\ldots, N\}$ of cardinality at least $δN$ contains a nontrivial configuration of the form $\{x,x+P_1(y),\ldots, x+P_m(y)\}$. When the polynomials are assumed independent, one can expect a sharper result to hold over finite fields, special cases of which were proven recently, culminating with arXiv:1802.02200, which deals with the general case of independent polynomials. One goal of this article is to explain these theorems as the result of joint ergodicity in the presence of asymptotic total ergodicity. Guided by this concept, we establish, over general finite commutative rings, a version of the polynomial Szemerédi theorem for independent polynomials $\{P_1,\ldots, P_m\} \subset \mathbb{Z}[y_1,\ldots, y_n]$, deriving new combinatorial consequences, such as the following. Let $\mathcal R$ be a collection of finite commutative rings subject to a mild condition on their torsion. There exists $γ\in (0,1)$ such that, for every $R \in \mathcal R$, every subset $A \subset R$ of cardinality at least $|R|^{1-γ}$ contains a nontrivial configuration $\{x,x+P_1(y),\ldots, x+P_m(y)\}$ for some $(x,y) \in R \times R^n$, and, moreover, for any subsets $A_0,\ldots, A_m \subset R$ such that $|A_0|\cdots |A_m| \geq |R|^{(m+1)(1-γ)}$, there is a nontrivial configuration $(x, x+P_1(y), \ldots, x+P_m(y)) \in A_0\times \cdots \times A_m$. The fact that general rings have zero divisors is the source of many obstacles, which we overcome; for example, by studying character sums, we develop a bound on the number of roots of an integer polynomial over a general finite commutative ring, a result which is of independent interest. △ Less

Submitted 19 March, 2025; v1 submitted 31 March, 2023; originally announced March 2023.

Comments: 105 pages. This version implements many, but not all, referee comments. A much shorter version of this article has been submitted to a journal, and this is the expanded version to which it refers

MSC Class: 11B30 (Primary); 11T23; 37A25 (Secondary)

MAVERIC: A Data-Driven Approach to Personalized Autonomous Driving

Authors: Mariah L. Schrum, Emily Sumner, Matthew C. Gombolay, Andrew Best

Abstract: Personalization of autonomous vehicles (AV) may significantly increase trust, use, and acceptance. In particular, we hypothesize that the similarity of an AV's driving style compared to the end-user's driving style will have a major impact on end-user's willingness to use the AV. To investigate the impact of driving style on user acceptance, we 1) develop a data-driven approach to personalize driv… ▽ More Personalization of autonomous vehicles (AV) may significantly increase trust, use, and acceptance. In particular, we hypothesize that the similarity of an AV's driving style compared to the end-user's driving style will have a major impact on end-user's willingness to use the AV. To investigate the impact of driving style on user acceptance, we 1) develop a data-driven approach to personalize driving style and 2) demonstrate that personalization significantly impacts attitudes towards AVs. Our approach learns a high-level model that tunes low-level controllers to ensure safe and personalized control of the AV. The key to our approach is learning an informative, personalized embedding that represents a user's driving style. Our framework is capable of calibrating the level of aggression so as to optimize driving style based upon driver preference. Across two human subject studies (n = 54), we first demonstrate our approach mimics the driving styles of end-users and can tune attributes of style (e.g., aggressiveness). Second, we investigate the factors (e.g., trust, personality etc.) that impact homophily, i.e. an individual's preference for a driving style similar to their own. We find that our approach generates driving styles consistent with end-user styles (p<.001) and participants rate our approach as more similar to their level of aggressiveness (p=.002). We find that personality (p<.001), perceived similarity (p<.001), and high-velocity driving style (p=.0031) significantly modulate the effect of homophily. △ Less

Submitted 27 February, 2023; v1 submitted 20 January, 2023; originally announced January 2023.

Formalized Class Group Computations and Integral Points on Mordell Elliptic Curves

Authors: Anne Baanen, Alex J. Best, Nirvana Coppola, Sander R. Dahmen

Abstract: Diophantine equations are a popular and active area of research in number theory. In this paper we consider Mordell equations, which are of the form $y^2=x^3+d$, where $d$ is a (given) nonzero integer number and all solutions in integers $x$ and $y$ have to be determined. One non-elementary approach for this problem is the resolution via descent and class groups. Along these lines we formalized in… ▽ More Diophantine equations are a popular and active area of research in number theory. In this paper we consider Mordell equations, which are of the form $y^2=x^3+d$, where $d$ is a (given) nonzero integer number and all solutions in integers $x$ and $y$ have to be determined. One non-elementary approach for this problem is the resolution via descent and class groups. Along these lines we formalized in Lean 3 the resolution of Mordell equations for several instances of $d<0$. In order to achieve this, we needed to formalize several other theories from number theory that are interesting on their own as well, such as ideal norms, quadratic fields and rings, and explicit computations of the class number. Moreover we introduced new computational tactics in order to carry out efficiently computations in quadratic rings and beyond. △ Less

Submitted 23 December, 2022; v1 submitted 30 September, 2022; originally announced September 2022.

Comments: 14 pages. Accepted for CPP '23. Source code available at https://github.com/lean-forward/class-group-and-mordell-equation

MSC Class: 68V20; 11D25; 11R11; 11R29 ACM Class: F.4.1

First direct limit on the 334 keV resonance strength in the $^{22}$Ne(α,γ)$^{26}$Mg reaction

Authors: D. Piatti, E. Masha, M. Aliotta, J. Balibrea-Correa, F. Barile, D. Bemmerer, A. Best, A. Boeltzig, C. Broggini, C. G. Bruno, A. Caciolli, F. Cavanna, T. Chillery, G. F. Ciani, A. Compagnucci, P. Corvisiero, L. Csedreki, T. Davinson, R. Depalo, A. di Leva, Z. Elekes, F. Ferraro, E. M. Fiore, A. Formicola, Zs. Fülöp , et al. (22 additional authors not shown)

Abstract: In stars, the fusion of $^{22}$Ne and $^4$He may produce either $^{25}$Mg, with the emission of a neutron, or $^{26}$Mg and a $γ$ ray. At high temperature, the ($α,n$) channel dominates, while at low temperature, it is energetically hampered. The rate of its competitor, the $^{22}$Ne($α$,$γ$)$^{26}$Mg reaction, and, hence, the minimum temperature for the ($α,n$) dominance, are controlled by many n… ▽ More In stars, the fusion of $^{22}$Ne and $^4$He may produce either $^{25}$Mg, with the emission of a neutron, or $^{26}$Mg and a $γ$ ray. At high temperature, the ($α,n$) channel dominates, while at low temperature, it is energetically hampered. The rate of its competitor, the $^{22}$Ne($α$,$γ$)$^{26}$Mg reaction, and, hence, the minimum temperature for the ($α,n$) dominance, are controlled by many nuclear resonances. The strengths of these resonances have hitherto been studied only indirectly. The present work aims to directly measure the total strength of the resonance at $E$_{r}$\,=\,$334$\,$keV (corresponding to $E$_{x}$\,=\,$10949$\,$keV in $^{26}$Mg). The data reported here have been obtained using high intensity $^4$He$^+$ beam from the INFN LUNA 400 kV underground accelerator, a windowless, recirculating, 99.9% isotopically enriched $^{22}$Ne gas target, and a 4$π$ bismuth germanate summing $γ$-ray detector. The ultra-low background rate of less than 0.5 counts/day was determined using 67 days of no-beam data and 7 days of $^4$He$^+$ beam on an inert argon target. The new high-sensitivity setup allowed to determine the first direct upper limit of 4.0$\,\times\,$10$^{-11}$ eV (at 90% confidence level) for the resonance strength. Finally, the sensitivity of this setup paves the way to study further $^{22}$Ne($α$,$γ$)$^{26}$Mg resonances at higher energy. △ Less

Submitted 7 September, 2022; originally announced September 2022.

Comments: Submitted to Eur. Phys. J. A

Journal ref: Eur. Phys. J. A 58, 194 (2022)

Horizons: Nuclear Astrophysics in the 2020s and Beyond

Authors: H. Schatz, A. D. Becerril Reyes, A. Best, E. F. Brown, K. Chatziioannou, K. A. Chipps, C. M. Deibel, R. Ezzeddine, D. K. Galloway, C. J. Hansen, F. Herwig, A. P. Ji, M. Lugaro, Z. Meisel, D. Norman, J. S. Read, L. F. Roberts, A. Spyrou, I. Tews, F. X. Timmes, C. Travaglio, N. Vassh, C. Abia, P. Adsley, S. Agarwal , et al. (140 additional authors not shown)

Abstract: Nuclear Astrophysics is a field at the intersection of nuclear physics and astrophysics, which seeks to understand the nuclear engines of astronomical objects and the origin of the chemical elements. This white paper summarizes progress and status of the field, the new open questions that have emerged, and the tremendous scientific opportunities that have opened up with major advances in capabilit… ▽ More Nuclear Astrophysics is a field at the intersection of nuclear physics and astrophysics, which seeks to understand the nuclear engines of astronomical objects and the origin of the chemical elements. This white paper summarizes progress and status of the field, the new open questions that have emerged, and the tremendous scientific opportunities that have opened up with major advances in capabilities across an ever growing number of disciplines and subfields that need to be integrated. We take a holistic view of the field discussing the unique challenges and opportunities in nuclear astrophysics in regards to science, diversity, education, and the interdisciplinarity and breadth of the field. Clearly nuclear astrophysics is a dynamic field with a bright future that is entering a new era of discovery opportunities. △ Less

Submitted 16 May, 2022; originally announced May 2022.

Comments: 96 pages. Submitted to Journal of Physics G

Report number: LA-UR-22-23997

Direct measurement of the 13C(α,n)16O cross section into the s-process Gamow peak

Authors: G. F. Ciani, L. Csedreki, D. Rapagnani, M. Aliotta, J. Balibrea-Correa, F. Barile, D. Bemmerer, A. Best, A. Boeltzig, C. Broggini, C. G. Bruno, A. Caciolli, F. Cavanna, T. Chillery, P. Corvisiero, S. Cristallo, T. Davinson, R. Depalo, A. DiLeva, Z. Elekes, F. Ferraro, E. Fiore, A. Formicola, Zs. Fulop, G. Gervino , et al. (23 additional authors not shown)

Abstract: One of the main neutron sources for the astrophysical s-process is the reaction 13C(α,n)16O, taking place in thermally pulsing Asymptotic Giant Branch stars at temperatures around 90 MK. To model the nucleosynthesis during this process the reaction cross section needs to be known in the 150-230keV energy window (Gamow peak). At these sub-Coulomb energies cross section direct measurements are sever… ▽ More One of the main neutron sources for the astrophysical s-process is the reaction 13C(α,n)16O, taking place in thermally pulsing Asymptotic Giant Branch stars at temperatures around 90 MK. To model the nucleosynthesis during this process the reaction cross section needs to be known in the 150-230keV energy window (Gamow peak). At these sub-Coulomb energies cross section direct measurements are severely affected by the low event rate, making us rely on input from indirect methods and extrapolations from higher-energy direct data. This leads to an uncertainty in the cross section at the relevant energies too high to reliably constrain the nuclear physics input to s-process calculations. We present the results from a new deep-underground measurement of 13C(α,n)16O, covering the energy range 230-300keV, with drastically reduced uncertainties over previous measurements and for the first time providing data directly inside the s-process Gamow peak. Selected stellar models have been computed to estimate the impact of our revised reaction rate. For stars of nearly solar composition, we find sizeable variations of some isotopes, whose production is influenced by the activation of close-by branching points that are sensitive to the neutron density, in particular the two radioactive nuclei 60Fe and 205Pb, as well as 152Gd △ Less

Submitted 1 October, 2021; originally announced October 2021.

Comments: 7 pages, 4 figures, accepted on PRL on 17th August 2021

Refined Selmer equations for the thrice-punctured line in depth two

Authors: Alex J. Best, L. Alexander Betts, Theresa Kumpitsch, Martin Lüdtke, Angus W. McAndrew, Lie Qian, Elie Studnia, Yujie Xu

Abstract: In [Kim05], Kim gave a new proof of Siegel's Theorem that there are only finitely many $S$-integral points on $\mathbb P^1_{\mathbb Z}\setminus\{0,1,\infty\}$. One advantage of Kim's method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of $S$ increases. In this paper, we implement a refinement of Kim's method to expl… ▽ More In [Kim05], Kim gave a new proof of Siegel's Theorem that there are only finitely many $S$-integral points on $\mathbb P^1_{\mathbb Z}\setminus\{0,1,\infty\}$. One advantage of Kim's method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of $S$ increases. In this paper, we implement a refinement of Kim's method to explicitly compute various examples where $S$ has size $2$ which has been introduced in [BD19]. In so doing, we exhibit new examples of a natural generalisation of a conjecture of Kim. △ Less

Submitted 28 April, 2023; v1 submitted 18 June, 2021; originally announced June 2021.

Comments: 35 pages, comments welcome

MSC Class: Primary 14G05; Secondary 11G55; 11Y50

Journal ref: Math. Comp. 93 (2024), 1497-1527

Polynomial ergodic averages for certain countable ring actions

Authors: Andrew Best, Andreu Ferré Moragues

Abstract: A recent result of Frantzikinakis establishes sufficient conditions for joint ergodicity in the setting of $\mathbb{Z}$-actions. We generalize this result for actions of second-countable locally compact abelian groups. We obtain two applications of this result. First, we show that, given an ergodic action $(T_n)_{n \in F}$ of a countable field $F$ with characteristic zero on a probability space… ▽ More A recent result of Frantzikinakis establishes sufficient conditions for joint ergodicity in the setting of $\mathbb{Z}$-actions. We generalize this result for actions of second-countable locally compact abelian groups. We obtain two applications of this result. First, we show that, given an ergodic action $(T_n)_{n \in F}$ of a countable field $F$ with characteristic zero on a probability space $(X,\mathcal{B},μ)$ and a family $\{p_1,\dots,p_k\}$ of independent polynomials, we have \[ \lim_{N \to \infty} \frac{1}{|Φ_N|}\sum_{n \in Φ_N} T_{p_1(n)}f_1\cdots T_{p_k(n)}f_k\ = \ \prod_{j=1}^k \int_X f_i \ dμ,\] where $f_i \in L^{\infty}(μ)$, $(Φ_N)$ is a Fø lner sequence of $(F,+)$, and the convergence takes place in $L^2(μ)$. This yields corollaries in combinatorics and topological dynamics. Second, we prove that a similar result holds for totally ergodic actions of suitable rings. △ Less

Submitted 13 June, 2022; v1 submitted 9 May, 2021; originally announced May 2021.

Comments: 35 pages. One definition corrected from journal version, all claimed results from journal version preserved

MSC Class: 37A44 (Primary) 28D15; 05D10; 37B05 (Secondary)

Multiple recurrence and large intersections for abelian group actions

Authors: Ethan Ackelsberg, Vitaly Bergelson, Andrew Best

Abstract: The purpose of this paper is to study the phenomenon of large intersections in the framework of multiple recurrence for measure-preserving actions of countable abelian groups. Among other things, we show: (1) If $G$ is a countable abelian group and $\varphi, ψ: G \to G$ are homomorphisms such that $\varphi(G)$, $ψ(G)$, and $(ψ- \varphi)(G)$ have finite index in $G$, then for every ergodic measure-… ▽ More The purpose of this paper is to study the phenomenon of large intersections in the framework of multiple recurrence for measure-preserving actions of countable abelian groups. Among other things, we show: (1) If $G$ is a countable abelian group and $\varphi, ψ: G \to G$ are homomorphisms such that $\varphi(G)$, $ψ(G)$, and $(ψ- \varphi)(G)$ have finite index in $G$, then for every ergodic measure-preserving system $(X, \mathcal{B}, μ, (T_g)_{g \in G})$, every set $A \in \mathcal{B}$, and every $\varepsilon > 0$, the set $\{g \in G : μ(A \cap T_{\varphi(g)}^{-1}A \cap T_{ψ(g)}^{-1}A) > μ(A)^3 - \varepsilon\}$ is syndetic. (2) If $G$ is a countable abelian group and $r,s \in \mathbb{Z}$ are integers such that $rG$, $sG$, and $(r \pm s)G$ have finite index in $G$, then for every ergodic measure-preserving system $(X, \mathcal{B}, μ, (T_g)_{g \in G})$, every set $A \in \mathcal{B}$, and every $\varepsilon > 0$, the set $\{g \in G : μ(A \cap T_{rg}^{-1}A \cap T_{sg}^{-1}A \cap T_{(r+s)g}^{-1}A) > μ(A)^4 - \varepsilon\}$ is syndetic. In particular, these extend and generalize results of Bergelson, Host, and Kra concerning $\mathbb{Z}$-actions and of Bergelson, Tao, and Ziegler concerning $\mathbb{F}_p^{\infty}$-actions. Using an ergodic version of the Furstenberg correspondence principle, we obtain new combinatorial applications. We also discuss numerous examples shedding light on the necessity of the various hypotheses above. Our results lead to a number of interesting questions and conjectures, formulated in the introduction and at the end of the paper. △ Less

Submitted 30 September, 2021; v1 submitted 7 January, 2021; originally announced January 2021.

Comments: 91 pages. Journal version

MSC Class: 37A15 (Primary) 37A30 (Secondary)

A Comparison of Proton Stopping Power Measured with Proton CT and x-ray CT in Fresh Post-Mortem Porcine Structures

Authors: Don F. DeJongh, Ethan A. DeJongh, Victor Rykalin, Greg DeFillippo, Mark Pankuch, Andrew W. Best, George Coutrakon, Kirk L. Duffin, Nicholas T. Karonis, Caesar E. Ordoñez, Christina Sarosiek, Reinhard W. Schulte, John R. Winans, Alec M. Block, Courtney L. Hentz, James S. Welsh

Abstract: Purpose: Currently, calculations of proton range in proton therapy patients are based on a conversion of CT Hounsfield Units of patient tissues into proton relative stopping power. Uncertainties in this conversion necessitate larger proximal and distal planned target volume margins. Proton CT can potentially reduce these uncertainties by directly measuring proton stopping power. We aim to demonstr… ▽ More Purpose: Currently, calculations of proton range in proton therapy patients are based on a conversion of CT Hounsfield Units of patient tissues into proton relative stopping power. Uncertainties in this conversion necessitate larger proximal and distal planned target volume margins. Proton CT can potentially reduce these uncertainties by directly measuring proton stopping power. We aim to demonstrate proton CT imaging with complex porcine samples, to analyze in detail three-dimensional regions of interest, and to compare proton stopping powers directly measured by proton CT to those determined from x-ray CT scans. Methods: We have used a prototype proton imaging system with single proton tracking to acquire proton radiography and proton CT images of a sample of porcine pectoral girdle and ribs, and a pig's head. We also acquired close in time x-ray CT scans of the same samples, and compared proton stopping power measurements from the two modalities. In the case of the pig's head, we obtained x-ray CT scans from two different scanners, and compared results from high-dose and low-dose settings. Results: Comparing our reconstructed proton CT images with images derived from x-ray CT scans, we find agreement within 1% to 2% for soft tissues, and discrepancies of up to 6% for compact bone. We also observed large discrepancies, up to 40%, for cavitated regions with mixed content of air, soft tissue, and bone, such as sinus cavities or tympanic bullae. Conclusions: Our images and findings from a clinically realistic proton CT scanner demonstrate the potential for proton CT to be used for low-dose treatment planning with reduced margins. △ Less

Submitted 29 October, 2021; v1 submitted 11 December, 2020; originally announced December 2020.

Comments: Accepted for publication in Medical Physics

The Furstenberg-Sárközy Theorem and Asymptotic Total Ergodicity Phenomena in Modular Rings

Authors: Vitaly Bergelson, Andrew Best

Abstract: The Furstenberg-Sárközy theorem asserts that the difference set $E-E$ of a subset $E \subset \mathbb{N}$ with positive upper density intersects the image set of any polynomial $P \in \mathbb{Z}[n]$ for which $P(0)=0$. Furstenberg's approach relies on a correspondence principle and a polynomial version of the Poincaré recurrence theorem, which is derived from the ergodic-theoretic result that for a… ▽ More The Furstenberg-Sárközy theorem asserts that the difference set $E-E$ of a subset $E \subset \mathbb{N}$ with positive upper density intersects the image set of any polynomial $P \in \mathbb{Z}[n]$ for which $P(0)=0$. Furstenberg's approach relies on a correspondence principle and a polynomial version of the Poincaré recurrence theorem, which is derived from the ergodic-theoretic result that for any measure-preserving system $(X,\mathcal{B},μ,T)$ and set $A \in \mathcal{B}$ with $μ(A) > 0$, one has $c(A):= \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N μ(A \cap T^{-P(n)}A) > 0.$ The limit $c(A)$ will have its optimal value of $μ(A)^2$ when $T$ is totally ergodic. Motivated by the possibility of new combinatorial applications, we define the notion of asymptotic total ergodicity in the setting of modular rings $\mathbb{Z}/N\mathbb{Z}$. We show that a sequence of modular rings $\mathbb{Z}/N_m\mathbb{Z}$, $m \in \mathbb{N},$ is asymptotically totally ergodic if and only if $\mathrm{lpf}(N_m)$, the least prime factor of $N_m$, grows to infinity. From this fact, we derive some combinatorial consequences, for example the following. Fix $δ\in (0,1]$ and a (not necessarily intersective) polynomial $Q \in \mathbb{Q}[n]$ such that $Q(\mathbb{Z}) \subseteq \mathbb{Z}$, and write $S = \{ Q(n) : n \in \mathbb{Z}/N\mathbb{Z}\}$. For any integer $N > 1$ with $\mathrm{lpf}(N)$ sufficiently large, if $A$ and $B$ are subsets of $\mathbb{Z}/N\mathbb{Z}$ such that $|A||B| \geq δN^2$, then $\mathbb{Z}/N\mathbb{Z} = A + B + S$. △ Less

Submitted 31 March, 2023; v1 submitted 29 November, 2020; originally announced November 2020.

Comments: 29 pages, update with referee's comments

MSC Class: 37A05 (Primary) 11B13 (Secondary)

Journal ref: Journal of Number Theory, Volume 243, 2023, Pages 615-645

Automatic selection of eye tracking variables in visual categorization in adults and infants

Authors: Samuel Rivera, Catherine A. Best, Hyungwook Yim, Dirk B. Walther, Vladimir M. Sloutsky, Aleix M. Martinez

Abstract: Visual categorization and learning of visual categories exhibit early onset, however the underlying mechanisms of early categorization are not well understood. The main limiting factor for examining these mechanisms is the limited duration of infant cooperation (10-15 minutes), which leaves little room for multiple test trials. With its tight link to visual attention, eye tracking is a promising m… ▽ More Visual categorization and learning of visual categories exhibit early onset, however the underlying mechanisms of early categorization are not well understood. The main limiting factor for examining these mechanisms is the limited duration of infant cooperation (10-15 minutes), which leaves little room for multiple test trials. With its tight link to visual attention, eye tracking is a promising method for getting access to the mechanisms of category learning. But how should researchers decide which aspects of the rich eye tracking data to focus on? To date, eye tracking variables are generally handpicked, which may lead to biases in the eye tracking data. Here, we propose an automated method for selecting eye tracking variables based on analyses of their usefulness to discriminate learners from non-learners of visual categories. We presented infants and adults with a category learning task and tracked their eye movements. We then extracted an over-complete set of eye tracking variables encompassing durations, probabilities, latencies, and the order of fixations and saccadic eye movements. We compared three statistical techniques for identifying those variables among this large set that are useful for discriminating learners form non-learners: ANOVA ranking, Bayes ranking, and L1 regularized logistic regression. We found remarkable agreement between these methods in identifying a small set of discriminant variables. Moreover, the same eye tracking variables allow us to classify category learners from non-learners among adults and 6- to 8-month-old infants with accuracies above 71%. △ Less

Submitted 26 November, 2020; v1 submitted 28 October, 2020; originally announced October 2020.

Sums of Powers in Large Finite Fields: A Mix of Methods

Authors: Vitaly Bergelson, Andrew Best, Alex Iosevich

Abstract: Can any element in a sufficiently large finite field be represented as a sum of two $d$th powers in the field? In this article, we recount some of the history of this problem, touching on cyclotomy, Fermat's last theorem, and diagonal equations. Then, we offer two proofs, one new and elementary, and the other more classical, based on Fourier analysis and an application of a nontrivial estimate fro… ▽ More Can any element in a sufficiently large finite field be represented as a sum of two $d$th powers in the field? In this article, we recount some of the history of this problem, touching on cyclotomy, Fermat's last theorem, and diagonal equations. Then, we offer two proofs, one new and elementary, and the other more classical, based on Fourier analysis and an application of a nontrivial estimate from the theory of finite fields. In context and juxtaposition, each will have its merits. △ Less

Submitted 16 December, 2020; v1 submitted 20 September, 2020; originally announced September 2020.

Comments: 15 pages, to appear in The American Mathematical Monthly. Change from previous version: fixed a misspelled name in a footnote

MSC Class: 11D85 (Primary) 11T06 (Secondary)

Elliptic curves with good reduction outside of the first six primes

Authors: Alex J. Best, Benjamin Matschke

Abstract: We present a database of rational elliptic curves, up to Q-isomorphism, with good reduction outside {2,3,5,7,11,13}. We provide a heuristic involving the abc and BSD conjectures that the database is likely to be the complete set of such curves. Moreover, proving completeness likely needs only more computation time to conclude. We present data on the distribution of various quantities associated to… ▽ More We present a database of rational elliptic curves, up to Q-isomorphism, with good reduction outside {2,3,5,7,11,13}. We provide a heuristic involving the abc and BSD conjectures that the database is likely to be the complete set of such curves. Moreover, proving completeness likely needs only more computation time to conclude. We present data on the distribution of various quantities associated to curves in the set. We also discuss the connection to S-unit equations and the existence of rational elliptic curves with maximal conductor. △ Less

Submitted 20 July, 2020; originally announced July 2020.

Comments: 19 pages, 4 figures; the data is available at https://github.com/elliptic-curve-data/ec-data-S6

MSC Class: 11G05 (Primary) 11Y50 (Secondary)

A user's guide to the local arithmetic of hyperelliptic curves

Authors: Alex J. Best, L. Alexander Betts, Matthew Bisatt, Raymond van Bommel, Vladimir Dokchitser, Omri Faraggi, Sabrina Kunzweiler, Céline Maistret, Adam Morgan, Simone Muselli, Sarah Nowell

Abstract: A new approach has been recently developed to study the arithmetic of hyperelliptic curves $y^2=f(x)$ over local fields of odd residue characteristic via combinatorial data associated to the roots of $f$. Since its introduction, numerous papers have used this machinery of "cluster pictures" to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide… ▽ More A new approach has been recently developed to study the arithmetic of hyperelliptic curves $y^2=f(x)$ over local fields of odd residue characteristic via combinatorial data associated to the roots of $f$. Since its introduction, numerous papers have used this machinery of "cluster pictures" to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide is to summarise and centralise all of these results in a self-contained fashion, complemented by an abundance of examples. △ Less

Submitted 1 June, 2021; v1 submitted 3 July, 2020; originally announced July 2020.

Comments: Minor changes. To appear in the Bulletin of the London Mathematical Society

MSC Class: 11G20 (11G10; 14D10; 14G20; 14H45; 14Q05)

Re-evaluation of the $^{22}$Ne($α,γ$)$^{26}$Mg and $^{22}$Ne($α,n$)$^{25}$Mg reaction rates

Authors: Philip Adsley, Umberto Battino, Andreas Best, Antonio Caciolli, Alessandra Guglielmetti, Gianluca Imbriani, Heshani Jayatissa, Marco La Cognata, Livio Lamia, Eliana Masha, Cristian Massimi, Sara Palmerini, Ashley Tattersall, Raphael Hirschi

Abstract: The competing $^{22}$Ne($α,γ$)$^{26}$Mg and $^{22}$Ne($α,n$)$^{25}$Mg reactions control the production of neutrons for the weak $s$-process in massive and AGB stars. In both systems, the ratio between the corresponding reaction rates strongly impacts the total neutron budget and strongly influences the final nucleosynthesis. The $^{22}$Ne($α,γ$)$^{26}$Mg and $^{22}$Ne($α,n$)$^{25}$Mg reaction rate… ▽ More The competing $^{22}$Ne($α,γ$)$^{26}$Mg and $^{22}$Ne($α,n$)$^{25}$Mg reactions control the production of neutrons for the weak $s$-process in massive and AGB stars. In both systems, the ratio between the corresponding reaction rates strongly impacts the total neutron budget and strongly influences the final nucleosynthesis. The $^{22}$Ne($α,γ$)$^{26}$Mg and $^{22}$Ne($α,n$)$^{25}$Mg reaction rates was re-evaluated by using newly available information on $^{26}$Mg given by various recent experimental studies. Evaluations of The evaluated $^{22}$Ne($α,γ$)$^{26}$Mg reaction rate remains substantially similar to that of Longland {\it et al.} but, including recent results from Texas A\&M, the $^{22}$Ne($α,n$)$^{25}$Mg reaction rate is lower at a range of astrophysically important temperatures. Stellar models computed with NEWTON and MESA predict decreased production of the weak branch $s$-process due to the decreased efficiency of $^{22}$Ne as a neutron source. Using the new reaction rates in the MESA model results in $^{96}$Zr/$^{94}$Zr and $^{135}$Ba/$^{136}$Ba ratios in much better agreement with the measured ratios from presolar SiC grains. △ Less

Submitted 21 March, 2021; v1 submitted 29 May, 2020; originally announced May 2020.

Comments: 23 pages, 15 figures, updated

Setup commissioning for an improved measurement of the D(p,gamma)3He cross section at Big Bang Nucleosynthesis energies

Authors: V. Mossa, K. Stöckel, F. Cavanna, F. Ferraro, M. Aliotta, F. Barile, D. Bemmerer, A. Best, A. Boeltzig, C. Broggini, C. G. Bruno, A. Caciolli, L. Csedreki, T. Chillery, G. F. Ciani, P. Corvisiero, T. Davinson, R. Depalo, A. Di Leva, Z. Elekes, E. M. Fiore, A. Formicola, Zs. Fülöp, G. Gervino, A. Guglielmetti , et al. (22 additional authors not shown)

Abstract: Among the reactions involved in the production and destruction of deuterium during Big Bang Nucleosynthesis, the deuterium-burning D(p,gamma)3He reaction has the largest uncertainty and limits the precision of theoretical estimates of primordial deuterium abundance. Here we report the results of a careful commissioning of the experimental setup used to measure the cross-section of the D(p,gamma)3H… ▽ More Among the reactions involved in the production and destruction of deuterium during Big Bang Nucleosynthesis, the deuterium-burning D(p,gamma)3He reaction has the largest uncertainty and limits the precision of theoretical estimates of primordial deuterium abundance. Here we report the results of a careful commissioning of the experimental setup used to measure the cross-section of the D(p,gamma)3He reaction at the Laboratory for Underground Nuclear Astrophysics of the Gran Sasso Laboratory (Italy). The commissioning was aimed at minimising all sources of systematic uncertainty in the measured cross sections. The overall systematic error achieved (< 3 %) will enable improved predictions of BBN deuterium abundance. △ Less

Submitted 29 April, 2020; originally announced May 2020.

Comments: 11 pages, 11 figures

Computing classical modular forms

Authors: Alex J. Best, Jonathan Bober, Andrew R. Booker, Edgar Costa, John Cremona, Maarten Derickx, Min Lee, David Lowry-Duda, David Roe, Andrew V. Sutherland, John Voight

Abstract: We discuss practical and some theoretical aspects of computing a database of classical modular forms in the L-functions and Modular Forms Database (LMFDB). We discuss practical and some theoretical aspects of computing a database of classical modular forms in the L-functions and Modular Forms Database (LMFDB). △ Less

Submitted 28 May, 2022; v1 submitted 11 February, 2020; originally announced February 2020.

Comments: 63 pages; minor edits, including a correction to Conjecture 8.5.1

Journal ref: Arithmetic Geometry, Number Theory, and Computation, Simons Symp. (2021), 131-213

SPA: Verbal Interactions between Agents and Avatars in Shared Virtual Environments using Propositional Planning

Authors: Andrew Best, Sahil Narang, Dinesh Manocha

Abstract: We present a novel approach for generating plausible verbal interactions between virtual human-like agents and user avatars in shared virtual environments. Sense-Plan-Ask, or SPA, extends prior work in propositional planning and natural language processing to enable agents to plan with uncertain information, and leverage question and answer dialogue with other agents and avatars to obtain the need… ▽ More We present a novel approach for generating plausible verbal interactions between virtual human-like agents and user avatars in shared virtual environments. Sense-Plan-Ask, or SPA, extends prior work in propositional planning and natural language processing to enable agents to plan with uncertain information, and leverage question and answer dialogue with other agents and avatars to obtain the needed information and complete their goals. The agents are additionally able to respond to questions from the avatars and other agents using natural-language enabling real-time multi-agent multi-avatar communication environments. Our algorithm can simulate tens of virtual agents at interactive rates interacting, moving, communicating, planning, and replanning. We find that our algorithm creates a small runtime cost and enables agents to complete their goals more effectively than agents without the ability to leverage natural-language communication. We demonstrate quantitative results on a set of simulated benchmarks and detail the results of a preliminary user-study conducted to evaluate the plausibility of the virtual interactions generated by SPA. Overall, we find that participants prefer SPA to prior techniques in 84\% of responses including significant benefits in terms of the plausibility of natural-language interactions and the positive impact of those interactions. △ Less

Submitted 8 February, 2020; originally announced February 2020.

A new approach to monitor 13C-targets degradation in situ for 13C(alpha,n)16O cross-section measurements at LUNA

Authors: G. F. Ciani, L. Csedreki, J. Balibrea-Correa, A. Best, M. Aliotta, F. Barile, D. Bemmerer, A. Boeltzig, C. Broggini, C. G. Bruno, A. Caciolli, F. Cavanna, T. Chillery, P. Colombetti, P. Corvisiero, T. Davinson, R. Depalo, A. Di Leva, L. Di Paolo, Z. Elekes, F. Ferraro, E. M. Fiore, A. Formicola, Zs. Fulop, G. Gervino , et al. (24 additional authors not shown)

Abstract: Direct measurements of reaction cross-sections at astrophysical energies often require the use of solid targets able to withstand high ion beam currents for extended periods of time. Thus, monitoring target thickness, isotopic composition, and target stoichiometry during data taking is critical to account for possible target modifications and to reduce uncertainties in the final cross-section resu… ▽ More Direct measurements of reaction cross-sections at astrophysical energies often require the use of solid targets able to withstand high ion beam currents for extended periods of time. Thus, monitoring target thickness, isotopic composition, and target stoichiometry during data taking is critical to account for possible target modifications and to reduce uncertainties in the final cross-section results. A common technique used for these purposes is the Nuclear Resonant Reaction Analysis (NRRA), which however requires that a narrow resonance be available inside the dynamic range of the accelerator used. In cases when this is not possible, as for example the 13C(alpha,n)16O reaction recently studied at low energies at the Laboratory for Underground Nuclear Astrophysics (LUNA) in Italy, alternative approaches must be found. Here, we present a new application of the shape analysis of primary gamma rays emitted by the 13C(p,g)14N radiative capture reaction. This approach was used to monitor 13C target degradation {\em in situ} during the 13C(alpha,n)16O data taking campaign. The results obtained are in agreement with evaluations subsequently performed at Atomki (Hungary) using the NRRA method. A preliminary application for the extraction of the 13C(alpha,n)16O reaction cross-section at one beam energy is also reported. △ Less

Submitted 3 March, 2020; v1 submitted 23 January, 2020; originally announced January 2020.

Comments: 10 pages, 6 figures, to be published in EPJ A

Journal ref: The European Physical Journal A volume 56, Article number: 75 (2020)

Automatic Testing With Reusable Adversarial Agents

Authors: Xin Qin, Nikos Aréchiga, Andrew Best, Jyotirmoy Deshmukh

Abstract: Autonomous systems such as self-driving cars and general-purpose robots are safety-critical systems that operate in highly uncertain and dynamic environments. We propose an interactive multi-agent framework where the system-under-design is modeled as an ego agent and its environment is modeled by a number of adversarial (ado) agents. For example, a self-driving car is an ego agent whose behavior i… ▽ More Autonomous systems such as self-driving cars and general-purpose robots are safety-critical systems that operate in highly uncertain and dynamic environments. We propose an interactive multi-agent framework where the system-under-design is modeled as an ego agent and its environment is modeled by a number of adversarial (ado) agents. For example, a self-driving car is an ego agent whose behavior is influenced by ado agents such as pedestrians, bicyclists, traffic lights, road geometry etc. Given a logical specification of the correct behavior of the ego agent, and a set of constraints that encode reasonable adversarial behavior, our framework reduces the adversarial testing problem to the problem of synthesizing controllers for (constrained) ado agents that cause the ego agent to violate its specifications. Specifically, we explore the use of tabular and deep reinforcement learning approaches for synthesizing adversarial agents. We show that ado agents trained in this fashion are better than traditional falsification or testing techniques because they can generalize to ego agents and environments that differ from the original ego agent. We demonstrate the efficacy of our technique on two real-world case studies from the domain of self-driving cars. △ Less

Submitted 5 July, 2021; v1 submitted 29 October, 2019; originally announced October 2019.

Two recent p-adic approaches towards the (effective) Mordell conjecture

Authors: Jennifer S. Balakrishnan, Alex J. Best, Francesca Bianchi, Brian Lawrence, J. Steffen Müller, Nicholas Triantafillou, Jan Vonk

Abstract: We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The latter method, which is usually called the method of Chabauty--Kim or non-abelian Chabauty in the literature, has the advantage that in some cases it has been turned into an effective method to determine the set of rational points on a curve, and we… ▽ More We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The latter method, which is usually called the method of Chabauty--Kim or non-abelian Chabauty in the literature, has the advantage that in some cases it has been turned into an effective method to determine the set of rational points on a curve, and we illustrate this by presenting three new examples of modular curves where this set can be determined. △ Less

Submitted 19 January, 2020; v1 submitted 28 October, 2019; originally announced October 2019.

Comments: Corrected Theorem 6.3 and its proof. Added short discussion of non-proper hyperbolic curves

Direct capture cross section and the $E_p$ = 71 and 105 keV resonances in the $^{22}$Ne($p,γ$)$^{23}$Na reaction

Authors: F. Ferraro, M. P. Takács, D. Piatti, F. Cavanna, R. Depalo, M. Aliotta, D. Bemmerer, A. Best, A. Boeltzig, C. Broggini, C. G. Bruno, A. Caciolli, T. Chillery, G. F. Ciani, P. Corvisiero, T. Davinson, G. D'Erasmo, A. DiLeva, Z. Elekes, E. M. Fiore, A. Formicola, Zs. Fülöp, G. Gervino, A. Guglielmetti, C. Gustavino , et al. (19 additional authors not shown)

Abstract: The $^{22}$Ne($p,γ$)$^{23}$Na reaction, part of the neon-sodium cycle of hydrogen burning, may explain the observed anticorrelation between sodium and oxygen abundances in globular cluster stars. Its rate is controlled by a number of low-energy resonances and a slowly varying non-resonant component. Three new resonances at $E_p$ = 156.2, 189.5, and 259.7 keV have recently been observed and confirm… ▽ More The $^{22}$Ne($p,γ$)$^{23}$Na reaction, part of the neon-sodium cycle of hydrogen burning, may explain the observed anticorrelation between sodium and oxygen abundances in globular cluster stars. Its rate is controlled by a number of low-energy resonances and a slowly varying non-resonant component. Three new resonances at $E_p$ = 156.2, 189.5, and 259.7 keV have recently been observed and confirmed. However, significant uncertainty on the reaction rate remains due to the non-resonant process and to two suggested resonances at $E_p$ = 71 and 105 keV. Here, new $^{22}$Ne($p,γ$)$^{23}$Na data with high statistics and low background are reported. Stringent upper limits of 6$\times$10$^{-11}$ and 7$\times$10$^{-11}$\,eV (90\% confidence level), respectively, are placed on the two suggested resonances. In addition, the off-resonant S-factor has been measured at unprecedented low energy, constraining the contributions from a subthreshold resonance and the direct capture process. As a result, at a temperature of 0.1 GK the error bar of the $^{22}$Ne($p,γ$)$^{23}$Na rate is now reduced by three orders of magnitude. △ Less

Submitted 3 October, 2018; originally announced October 2018.

Comments: Submitted to Phys. Rev. Lett

Journal ref: Phys. Rev. Lett. 121, 172701 (2018)