arXiv subjects

Michael Allen

Publications and source records attributed to Michael Allen.

11 recordsLinked to original sources

arXiv2025

Atkin and Swinnerton-Dyer congruences for meromorphic modular forms

In the 1970's, Atkin and Swinnerton-Dyer conjectured that Fourier coefficients of holomorphic modular cusp forms on noncongruence subgroups of $\text{SL}_2(\mathbb{Z})$ satisfy certain $p$-adic recurrence relations which are analogous to Hecke's recurrence relations for congrunece subgroups. In 1985, this was proven in seminal work of Scholl and it was recently extended to weakly holomorphic modular forms by Kazalicki and Scholl. We show that Atkin and Swinnerton-Dyer type congruences extend to the setting of meromorphic modular forms and that the $p$-adic recurrence relations arise from Scholl's congruences in addition to a contribution of fibers of universal elliptic curves at the poles. Moreover, when the poles are located at CM points, we exploit the CM structure to reduce these $p$-adic recurrence relations to $2$-term relations and we give explicit examples. Using this framework, we partially prove conjectures that certain meromorphic modular forms are magnetic.

math.NT↗

arXiv2024

The Explicit Hypergeometric-Modularity Method II

In the first paper of this sequence, we provided an explicit hypergeometric modularity method by combining different techniques from the classical, $p$-adic, and finite field settings. In this article, we explore an application of this method from a motivic viewpoint through some known hypergeometric well-poised formulae of Whipple and McCarthy. We first use the method to derive a class of special weight three modular forms, labeled as $\mathbb{K}_2$-functions. Then using well-poised hypergeometric formulae we further construct a class of degree four Galois representations of the absolute Galois groups of the corresponding cyclotomic fields. These representations are then shown to be extendable to $G_{\mathbb{Q}}$ and the $L$-function of each extension coincides with the $L$-function of an automorphic form.

math.NT↗

arXiv2024

Holomorphic projection for sesquiharmonic Maass forms

We study the holomorphic projection of mixed mock modular forms involving sesquiharmonic Maass forms. As a special case, we numerically express the holomorphic projection of a function involving real quadratic class numbers multiplied by a certain theta function in terms of eta quotients. We also analyze certain shifted convolution $L$-series involving mock modular forms and bound certain shifted convolution sums.

math.NT↗

arXiv2024

The Explicit Hypergeometric-Modularity Method I

The theories of hypergeometric functions and modular forms are highly intertwined. For example, particular values of truncated hypergeometric functions and hypergeometric character sums are often congruent or equal to Fourier coefficients of modular forms. In this series of papers, we develop and explore an explicit "Hypergeometric-Modularity" method for associating a modular form to a given hypergeometric datum. In particular, for certain length three and four hypergeometric data we give an explicit method for finding a modular form $f$ such that the corresponding hypergeometric Galois representation has a subrepresentation isomorphic to the Deligne representation of $f$. Our method utilizes Ramanujan's theory of elliptic functions to alternative bases, commutative formal group laws, and supercongruences. As a byproduct, we give a collection of eta quotients with multiplicative coefficients constructed from hypergeometric functions. In the second paper, we discuss a number of applications, including explicit connections between hypergeometric values and periods of these explicit eta quotients as well as evaluation formulae for certain special $L$-values.

math.NT↗

arXiv2022

On Some Hypergeometric Supercongruence Conjectures of Long

In 2003, Rodriguez Villegas conjectured 14 supercongruences between hypergeometric functions arising as periods of certain families of rigid Calabi-Yau threefolds and the Fourier coefficients of weight 4 modular forms. Uniform proofs of these supercongruences were given in 2019 by Long, Tu, Yui, and Zudilin. Using p-adic techniques of Dwork, they reduce the original supercongruences to related congruences which involve only the hypergeometric series. We generalize their techniques to consider six further supercongruences recently conjectured by Long. In particular we prove an analogous version of Long, Tu, Yui, and Zudilin's reduced congruences for each of these six cases. We also conjecture a generalization of Dwork's work which has been observed computationally and which would, together with a proof of modularity for Galois representations associated to our hypergeometric data, yield a full proof of Long's conjectures.

math.NT↗

arXiv2021

Developing an OpenAI Gym-compatible framework and simulation environment for testing Deep Reinforcement Learning agents solving the Ambulance Location Problem

Background and motivation: Deep Reinforcement Learning (Deep RL) is a rapidly developing field. Historically most application has been made to games (such as chess, Atari games, and go). Deep RL is now reaching the stage where it may offer value in real world problems, including optimisation of healthcare systems. One such problem is where to locate ambulances between calls in order to minimise time from emergency call to ambulance on-scene. This is known as the Ambulance Location problem. Aim: To develop an OpenAI Gym-compatible framework and simulation environment for testing Deep RL agents. Methods: A custom ambulance dispatch simulation environment was developed using OpenAI Gym and SimPy. Deep RL agents were built using PyTorch. The environment is a simplification of the real world, but allows control over the number of clusters of incident locations, number of possible dispatch locations, number of hospitals, and creating incidents that occur at different locations throughout each day. Results: A range of Deep RL agents based on Deep Q networks were tested in this custom environment. All reduced time to respond to emergency calls compared with random allocation to dispatch points. Bagging Noisy Duelling Deep Q networks gave the most consistence performance. All methods had a tendency to lose performance if trained for too long, and so agents were saved at their optimal performance (and tested on independent simulation runs). Conclusions: Deep RL agents, developed using simulated environments, have the potential to offer a novel approach to optimise the Ambulance Location problem. Creating open simulation environments should allow more rapid progress in this field.

cs.LG↗

arXiv2020

Integrating Deep Reinforcement Learning Networks with Health System Simulations

Background and motivation: Combining Deep Reinforcement Learning (Deep RL) and Health Systems Simulations has significant potential, for both research into improving Deep RL performance and safety, and in operational practice. While individual toolkits exist for Deep RL and Health Systems Simulations, no framework to integrate the two has been established. Aim: Provide a framework for integrating Deep RL Networks with Health System Simulations, and to ensure this framework is compatible with Deep RL agents that have been developed and tested using OpenAI Gym. Methods: We developed our framework based on the OpenAI Gym framework, and demonstrate its use on a simple hospital bed capacity model. We built the Deep RL agents using PyTorch, and the Hospital Simulatation using SimPy. Results: We demonstrate example models using a Double Deep Q Network or a Duelling Double Deep Q Network as the Deep RL agent. Conclusion: SimPy may be used to create Health System Simulations that are compatible with agents developed and tested on OpenAI Gym environments. GitHub repository of code: https://github.com/MichaelAllen1966/learninghospital

cs.LG↗

arXiv2019

Existence of $\eta$-Quotients For Squarefree Levels

In this paper, we investigate which modular form spaces can contain $\eta$-quotients, functions of the form $f(\tau) = \prod_{\delta \mid N} \eta(\delta\tau)^{r_\delta}$. For $k\geq 2$ even and $N$ coprime to 6, we give necessary conditions for the space $M_k(\Gamma_1(N))$ to contain $\eta$-quotients. We then show that these conditions are sufficient as well for a large family of squarefree levels $N$ coprime to 6.

math.NT↗

arXiv2019

Eta-quotients of Prime or Semiprime Level and Elliptic Curves

From the Modularity Theorem proven by Wiles, Taylor, et al, we know that all elliptic curves are modular. It has been shown by Martin and Ono exactly which are represented by eta-quotients, and some examples of elliptic curves represented by modular forms that are linear combinations of eta-quotients have been given by Pathakjee, RosnBrick, and Yoong. In this paper, we first show that eta-quotients which are modular for any congruence subgroup of level $N$ coprime to $6$ can be viewed as modular for $\Gamma_0(N)$. We then categorize when even weight eta-quotients can exist in $M_k (\Gamma_1(p))$ and $M_k (\Gamma_1(pq))$, for distinct primes $p,q$. We conclude by providing some new examples of elliptic curves whose corresponding modular forms can be written as a linear combination of eta-quotients, and describe an algorithmic method for finding additional examples.

math.NT↗

arXiv2017

Proof of Concept of Wireless TERS Monitoring

Temporary earth retaining structures (TERS) help prevent collapse during construction excavation. To ensure that these structures are operating within design specifications, load forces on supports must be monitored. Current monitoring approaches are expensive, sparse, off-line, and thus difficult to integrate into predictive models. This work aims to show that wirelessly connected battery powered sensors are feasible, practical, and have similar accuracy to existing sensor systems. We present the design and validation of ReStructure, an end-to-end prototype wireless sensor network for collection, communication, and aggregation of strain data. ReStructure was validated through a six months deployment on a real-life excavation site with all but one node producing valid and accurate strain measurements at higher frequency than existing ones. These results and the lessons learnt provide the basis for future widespread wireless TERS monitoring that increase measurement density and integrate closely with predictive models to provide timely alerts of damage or potential failure.

cs.NI↗

arXiv2004

Submillimeter Observations of Low Metallicity Galaxy NGC 4214

Results of submillimeter (450 micron and 850 micron) observations of a nearby dwarf irregular galaxy NGC 4214 with SCUBA on JCMT are presented. We aimed at examining the far-infrared to submillimeter spectral energy distribution (SED) and properties of dust thermal emission in a low metallicity environment by choosing NGC 4214 of which gas metallicity (log O/H + 12) is 8.34. We found that the SED is quite similar to those of IRAS bright galaxies sample (IBGS) which are local bright star-forming galaxies with metallicity comparable to the solar abundance. A dust temperature and an emissivity index for NGC 4214 obtained by a fitting to the single temperature greybody model are T_d = 35 \pm 0.8 K and beta = 1.4 \pm 0.1, respectively, which are typical values for IBGS. Compiling the previous studies on similar nearby dwarf irregular galaxies, we found that NGC 1569 shows similar results to those of NGC 4214, while NGC 4449 and IC 10 SE show different SEDs and low emissivity indices. There seems to be a variety of SEDs among metal poor dwarf irregular galaxies. We examined dependence on intensity of interstellar radiation field as well as two-temperature model, but the origin of the difference is not clear. Some mechanism(s) other than metallicity and interstellar radiation field must be responsible for controlling dust emission property.

astro-ph↗