arXiv subjects

Gareth Jones

Publications and source records attributed to Gareth Jones.

At least 37 records · Page 2Linked to original sources

arXiv2020

On algebraic values of Weierstrass $σ$-functions

Suppose that $Ω$ is a lattice in the complex plane and let $σ$ be the corresponding Weierstrass $σ$-function. Assume that the point $τ$ associated to $Ω$ in the standard fundamental domain has imaginary part at most 1.9. Assuming that $Ω$ has algebraic invariants $g_2,g_3$ we show that a bound of the form $c d^m (\log H)^n$ holds for the number of algebraic points of height at most $H$ and degree at most $d$ lying on the graph of $σ$. To prove this we apply results by Masser and Besson. What is perhaps surprising is that we are able to establish such a bound for the whole graph, rather than some restriction. We prove a similar result when, instead of $g_2,g_3$, the lattice points are algebraic. For this we naturally exclude those $(z,σ(z))$ for which $z\inΩ$.

math.NT↗

arXiv2020

Powers are easy to avoid

Suppose that $\widetilde{\mathbb R}$ is an o-minimal expansion of the real field in which restricted power functions are definable. We show that if $\widehat{\mathbb R}$ is both a reduct (in the sense of definability) of the expansion $\widetilde{\mathbb R}^{\mathbb R}$ of $\widetilde{\mathbb R}$ by all real power functions and an expansion (again in the sense of definability) of $\widetilde{\mathbb R}$, then, provided that $\widetilde{\mathbb R}$ and $\widehat{\mathbb R}$ have the same field of exponents, they define the same sets. This can be viewed as a polynomially bounded version of an old conjecture of van den Dries and Miller.

math.LO↗

arXiv2020

Integer Valued Definable Functions in $\mathbb{R}_{an,\exp}$

We give two variations on a result of Wilkie's on unary functions defianble in $\mathbb{R}_{an,\exp}$ that take integer values at positive integers. Provided that the functions grows slower than the function $2^x$, Wilkie showed that is must be eventually equal to a polynomial. We show the same conclusion under a stronger growth condition but only assuming that the function takes values sufficiently close to a integers at positive integers. In a different variation we show that it suffices to assume that the function takes integer values on a sufficiently dense subset of the positive integers(for instance primes), again under a stronger growth bound than that in Wilkie's result.

math.NT↗

arXiv2019

A Manin-Mumford theorem for the maximal compact subgroup of a universal vectorial extension of a product of elliptic curves

We study the intersection of an algebraic variety with the maximal compact subgroup of a universal vectorial extension of a product of elliptic curves. For this intersection we show a Manin-Mumford type statement. This answers some questions posed by Corvaja-Masser-Zannier which arose in connection with their investigation of the intersection of a curve with real analytic subgroups of various algebraic groups. They prove finiteness in the situation of a single elliptic curve. Using Khovanskii's zero-estimates combined with a stratification result of Gabrielov-Vorobjov and recent work of the authors we obtain effective bounds for this intersection that only depend on the degree of the algebraic variety, and the dimension of the group. This seems new even if restricted to the classical Manin-Mumford statement.

math.NT↗

arXiv2019

Rational values of transcendental functions and arithmetic dynamics

We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work with p-adic methods to obtain a lower bound of the form $cD^{n/4 - \varepsilon}$ on the degree of the splitting field of $P^{\circ n}(z)=P^{\circ n}(α)$, where $P$ is a polynomial of degree $D\geq 2$ over a number field, $P^{\circ n}$ is its $n$-th iterate and $c$ depends effectively on $P, α$ and $\varepsilon$. Our $c$ is positive for each algebraic $α$ for which the set $\{P^{\circ n}(α):n\in\mathbb{N}\}$ is infinite.

math.NT↗

arXiv2018

The Interstellar Medium Properties of Heavily Reddened Quasars & Companions at z~2.5 with ALMA & JVLA

We study the interstellar medium (ISM) properties of three heavily reddened quasars at $z\sim2.5$ as well as three millimetre-bright companion galaxies near these quasars. New JVLA and ALMA observations constrain the CO(1-0), CO(7-6) and [CI]$^3$P$_2-^3$P$_1$ line emission as well as the far infrared to radio continuum. The gas excitation and physical properties of the ISM are constrained by comparing our observations to photo-dissociation region (PDR) models. The ISM in our high-redshift quasars is composed of very high-density, high-temperature gas which is already highly enriched in elements like carbon. One of our quasar hosts is shown to be a close-separation ($<$2 arcsec) major merger with different line emission properties in the millimeter-bright galaxy and quasar components. Low angular resolution observations of high-redshift quasars used to assess quasar excitation properties should therefore be interpreted with caution as they could potentially be averaging over multiple components with different ISM conditions. Our quasars and their companion galaxies show a range of CO excitation properties spanning the full extent from starburst-like to quasar-like spectral line energy distributions. We compare gas masses based on CO, CI and dust emission, and find that these can disagree when standard assumptions are made regarding the values of $α_{\rm{CO}}$, the gas-to-dust ratio and the atomic carbon abundances. We conclude that the ISM properties of our quasars and their companion galaxies are diverse and likely vary spatially across the full extent of these complex, merging systems.

astro-ph.GA↗

arXiv2018

Pfaffian definitions of Weierstrass elliptic functions

We give explicit definitions of the Weierstrass elliptic functions $\wp$ and $ζ$ in terms of pfaffian functions, with complexity independent of the lattice involved. We also give such a definition for a modification of the Weierstrass function $σ$. As immediate applications, we give an explicit uniform zero estimate for $\wp$ and answer a question of Corvaja, Masser and Zannier on additive extensions of elliptic curves.

math.NT↗

arXiv2017

On local definability of holomorphic functions

Given a collection A of holomorphic functions, we consider how to describe all the holomorphic functions locally definable from A. The notion of local definability of holomorphic functions was introduced by Wilkie, who gave a complete description of all functions locally definable from A in the neighbourhood of a generic point. We prove that this description is no longer complete in the neighbourhood of non-generic points. More precisely, we produce three examples of holomorphic functions which suggest that at least three new operations need to be added to Wilkie's description in order to capture local definability in its entirety. The constructions illustrate the interaction between resolution of singularities and definability in the o-minimal setting.

math.LO↗

arXiv2016

Critical current scaling in long diffusive graphene-based Josephson junctions

We present transport measurements on long diffusive graphene-based Josephson junctions. Several junctions are made on a single-domain crystal of CVD graphene and feature the same contact width of ~9$μ$m but vary in length from 400 to 1000 nm. As the carrier density is tuned with the gate voltage, the critical current in the these junctions spans a range from a few nA up to more than $5μ$A, while the Thouless energy, ETh, covers almost two orders of magnitude. Over much of this range, the product of the critical current and the normal resistance IcRn is found to scale linearly with ETh, as expected from theory. However, the ratio IcRn /ETh is found to be 0.1-0.2: much smaller than the predicted ~10 for long diffusive SNS junctions.

cond-mat.mes-hall↗

arXiv2014

Local interdefinability of Weierstrass elliptic functions

We explain which Weierstrass elliptic functions are locally definable from other elliptic functions and exponentiation in the context of o-minimal structures. The proofs make use of the predimension method from model theory to exploit functional transcendence theorems in a systematic way.

math.LO↗

arXiv2013

Gravitational-Wave Detection using Multivariate Analysis

Searches for gravitational-wave bursts (transient signals, typically of unknown waveform) require identification of weak signals in background detector noise. The sensitivity of such searches is often critically limited by non-Gaussian noise fluctuations which are difficult to distinguish from real signals, posing a key problem for transient gravitational-wave astronomy. Current noise rejection tests are based on the analysis of a relatively small number of measured properties of the candidate signal, typically correlations between detectors. Multivariate analysis (MVA) techniques probe the full space of measured properties of events in an attempt to maximise the power to accurately classify events as signal or background. This is done by taking samples of known background events and (simulated) signal events to train the MVA classifier, which can then be applied to classify events of unknown type. We apply the boosted decision tree (BDT) MVA technique to the problem of detecting gravitational-wave bursts associated with gamma-ray bursts. We find that BDTs are able to increase the sensitive distance reach of the search by as much as 50%, corresponding to a factor of ~3 increase in sensitive volume. This improvement is robust against trigger sky position, large sky localisation error, poor data quality, and the simulated signal waveforms that are used. Critically, we find that the BDT analysis is able to detect signals that have different morphologies to those used in the classifier training and that this improvement extends to false alarm probabilities beyond the 3σ significance level. These findings indicate that MVA techniques may be used for the robust detection of gravitational-wave bursts with a priori unknown waveform.

gr-qc↗

arXiv2012

Performance of an externally triggered gravitational-wave burst search

We present the performance of searches for gravitational wave bursts associated with external astrophysical triggers as a function of the search sky region. We discuss both the case of Gaussian noise and real noise of gravitational wave detectors for arbitrary detector networks. We demonstrate the ability to reach Gaussian limited sensitivity in real non-Gaussian data, and show the conditions required to attain it. We find that a single sky position search is ~20% more sensitive than an all-sky search of the same data.

gr-qc↗

arXiv2012

Multimessenger Science Reach and Analysis Method for Common Sources of Gravitational Waves and High-energy Neutrinos

We present the baseline multimessenger analysis method for the joint observations of gravitational waves (GW) and high-energy neutrinos (HEN), together with a detailed analysis of the expected science reach of the joint search. The analysis method combines data from GW and HEN detectors, and uses the blue-luminosity-weighted distribution of galaxies. We derive expected GW+HEN source rate upper limits for a wide range of source parameters covering several emission models. Using published sensitivities of externally triggered searches, we derive joint upper limit estimates both for the ongoing analysis with the initial LIGO-Virgo GW detectors with the partial IceCube detector (22 strings) HEN detector and for projected results to advanced LIGO-Virgo detectors with the completed IceCube (86 strings). We discuss the constraints these upper limits impose on some existing GW+HEN emission models.

astro-ph.HE↗

arXiv2011

2-Groups that factorise as products of cyclic groups, and regular embeddings of complete bipartite graphs

We classify those 2-groups G which factorise as a product of two disjoint cyclic subgroups A and B, transposed by an automorphism of order 2. The case where G is metacyclic having been dealt with elsewhere, we show that for each e>2 there are exactly three such non-metacyclic groups G with $|A|=|B|=2^e$, and for e=2 there is one. These groups appear in a classification by Berkovich and Janko of 2-groups with one non-metacyclic maximal subgroup; we enumerate these groups, give simpler presentations for them, and determine their automorphism groups.

math.GR↗

arXiv2011

Bounding the Time Delay between High-energy Neutrinos and Gravitational-wave Transients from Gamma-ray Bursts

We derive a conservative coincidence time window for joint searches of gravita-tional-wave (GW) transients and high-energy neutrinos (HENs, with energies above 100GeV), emitted by gamma-ray bursts (GRBs). The last are among the most interesting astrophysical sources for coincident detections with current and near-future detectors. We take into account a broad range of emission mechanisms. We take the upper limit of GRB durations as the 95% quantile of the T90's of GRBs observed by BATSE, obtaining a GRB duration upper limit of ~150s. Using published results on high-energy (>100MeV) photon light curves for 8 GRBs detected by Fermi LAT, we verify that most high-energy photons are expected to be observed within the first ~150s of the GRB. Taking into account the breakout-time of the relativistic jet produced by the central engine, we allow GW and HEN emission to begin up to 100s before the onset of observable gamma photon production. Using published precursor time differences, we calculate a time upper bound for precursor activity, obtaining that 95% of precursors occur within ~250s prior to the onset of the GRB. Taking the above different processes into account, we arrive at a time window of tHEN - tGW ~ [-500s,+500s]. Considering the above processes, an upper bound can also be determined for the expected time window of GW and/or HEN signals coincident with a detected GRB, tGW - tGRB ~ tHEN - tGRB ~ [-350s,+150s].

astro-ph.HE↗

arXiv2010

X-Pipeline: An analysis package for autonomous gravitational-wave burst searches

Autonomous gravitational-wave searches -- fully automated analyses of data that run without human intervention or assistance -- are desirable for a number of reasons. They are necessary for the rapid identification of gravitational-wave burst candidates, which in turn will allow for follow-up observations by other observatories and the maximum exploitation of their scientific potential. A fully automated analysis would also circumvent the traditional "by hand" setup and tuning of burst searches that is both labourious and time consuming. We demonstrate a fully automated search with X-Pipeline, a software package for the coherent analysis of data from networks of interferometers for detecting bursts associated with GRBs and other astrophysical triggers. We discuss the methods X-Pipeline uses for automated running, including background estimation, efficiency studies, unbiased optimal tuning of search thresholds, and prediction of upper limits. These are all done automatically via Monte Carlo with multiple independent data samples, and without requiring human intervention. As a demonstration of the power of this approach, we apply X-Pipeline to LIGO data to search for gravitational-wave emission associated with GRB 031108. We find that X-Pipeline is sensitive to signals approximately a factor of 2 weaker in amplitude than those detectable by the cross-correlation technique used in LIGO searches to date. We conclude with the prospects for running X-Pipeline as a fully autonomous, near real-time triggered burst search in the next LSC-Virgo Science Run.

gr-qc↗

arXiv2010

Searching for gravitational waves emitted by binaries with spinning components

In this thesis we consider the data analysis problem of detecting gravitational waves emitted by inspiraling binary systems. Detection of gravitational waves will open a new window on the Universe enabling direct detection of systems such as binary black holes for the first time. In the first Chapter we show how gravitational waves are derived from Einstein's General theory of Relativity and discuss the emission of gravitational waves from inspiraling binaries and how this radiation may be detected using laser interferometers. Around two thirds of stars inhabit binary systems. As they orbit each other they will emit both energy and angular momentum in the form of gravitational waves which will inevitably lead to their inspiral and eventual merger. To date, searches for gravitational waves emitted during the inspiral of binary systems have concentrated on systems with non-spinning components. In Chapter 2 we detail the first dedicated search for binaries consisting of spinning stellar mass compact objects. We analysed 788 hours of data collected during the third science run (S3) of the LIGO detectors, no detection of gravitational waves was made and we set an upper limit on the rate of coalescences of stellar mass binaries. The inspiral of stellar mass compact objects into super massive black holes will radiate gravitational waves at frequencies detectable by the planned space-based LISA mission. In Chapter 3 we describe the development and testing of a computationally cheap method to detect the loudest few extreme mass ratio inspiral events that LISA will be sensitive to.

gr-qc↗

arXiv2009

Beauville surfaces and finite groups

Extending results of Bauer, Catanese and Grunewald, and of Fuertes and González-Diez, we show that Beauville surfaces of unmixed type can be obtained from the groups L_2(q) and SL_2(q) for all prime powers q>5, and the Suzuki groups Sz(2^e) and the Ree groups R(3^e) for all odd e>1. We also show that L_2(q) and SL_2(q) admit strongly real Beauville structures, yielding real Beauville surfaces, if and only if q>5.

math.GR↗