Mathematical Sciences- Journal Articles

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Abel's limit theorem, its converse, and multiplication formulae for Γ(x)
(Irish Mathematical Society, 2022) Holland, Finbarr
Abel's well-known limit theorem for power series, and its corrected converse due to J. E. Littlewood, form the basis for a general identity that is presented here, which is shown to be equivalent to Gauss's multiplication theorem for the Gamma function.
Absence of evidence ≠ evidence of absence: statistical analysis of inclusions in multiferroic thin films
(Nature Publishing Group, Macmillan Publishers Limited, 2014-07-16) Schmidt, Michael; Amann, Andreas; Keeney, Lynette; Pemble, Martyn E.; Holmes, Justin D.; Petkov, Nikolay; Whatmore, Roger W.; Science Foundation Ireland
Assertions that a new material may offer particularly advantageous properties should always be subjected to careful critical evaluation, especially when those properties can be affected by the presence of inclusions at trace level. This is particularly important for claims relating to new multiferroic compounds, which can easily be confounded by unobserved second phase magnetic inclusions. We demonstrate an original methodology for the detection, localization and quantification of second phase inclusions in thin Aurivillius type films. Additionally, we develop a dedicated statistical model and demonstrate its application to the analysis of Bi6Ti2.8Fe1.52Mn0.68O18 (B6TFMO) thin films, that makes it possible to put a high, defined confidence level (e.g. 99.5%) to the statement of ‘new single phase multiferroic materials’. While our methodology has been specifically developed for magnetic inclusions, it can easily be adapted to any other material system that can be affected by low level inclusions.
Accelerating dynamics of collective attention
(Springer Nature, 2019-04-15) Lorenz-Spreen, Philipp; Mønsted, Bjarke Mørch; Hövel, Philipp; Lehmann, Sune; Deutsche Forschungsgemeinschaft
With news pushed to smart phones in real time and social media reactions spreading across the globe in seconds, the public discussion can appear accelerated and temporally fragmented. In longitudinal datasets across various domains, covering multiple decades, we find increasing gradients and shortened periods in the trajectories of how cultural items receive collective attention. Is this the inevitable conclusion of the way information is disseminated and consumed? Our findings support this hypothesis. Using a simple mathematical model of topics competing for finite collective attention, we are able to explain the empirical data remarkably well. Our modeling suggests that the accelerating ups and downs of popular content are driven by increasing production and consumption of content, resulting in a more rapid exhaustion of limited attention resources. In the interplay with competition for novelty, this causes growing turnover rates and individual topics receiving shorter intervals of collective attention.
Affine connections on complex manifolds with algebraic dimension zero
(Independent University of Moscow and Higher School of Economics, Moscow, 2016) McKay, Benjamin; Dumitrescu, Sorin
We prove that any compact complex manifold with finite fundamental group and algebraic dimension zero admits no holomorphic affine connection or holomorphic conformal structure.
An algebraic construction of quantum flows with unbounded generators
(Institut Henri Poincaré, 2015-02) Belton, Alexander C. R.; Wills, Stephen J.; Lancaster University
It is shown how to construct ∗-homomorphic quantum stochastic Feller cocycles for certain unbounded generators, and so obtain dilations of strongly continuous quantum dynamical semigroups on C∗ algebras; this generalises the construction of a classical Feller process and semigroup from a given generator. Our construction is possible provided the generator satisfies an invariance property for some dense subalgebra A0 of the C∗ algebra A and obeys the necessary structure relations; the iterates of the generator, when applied to a generating set for A0, must satisfy a growth condition. Furthermore, it is assumed that either the subalgebra A0 is generated by isometries and A is universal, or A0 contains its square roots. These conditions are verified in four cases: classical random walks on discrete groups, Rebolledo’s symmetric quantum exclusion process and flows on the non-commutative torus and the universal rotation algebra. Des cocycles de Feller stochastiques quantiques ∗-homomorphes sont construits pour certains générateurs non bornés, et ainsi nous obtenons des dilatations pour des semigroupes dynamiques quantiques fortement continus sur des C∗ algèbres. Ceci généralise la construction d’un processus de Feller classique et de son semigroupe à partir d’un générateur donné. Notre construction est possible à condition que le générateur satisfasse une propriété d’invariance pour une sous-algèbre dense A0 de la C∗ algèbre A et obéisse aux relations de structure nécessaires; les itérations du générateur, lorsqu’elles sont appliquées à une famille génératrice de A0, doivent satisfaire à une condition de croissance. De plus, il est supposé que soit la sous-algèbre A0 est engendrée par les isométries et A est universelle, ou bien A0 contient ses racines carrées. Ces conditions sont vérifiées dans quatre cas: marches aléatoires classiques sur les groupes discrets, le processus d’exclusion quantique symétrique introduit par Rebolledo et des flux sur le tore non commutatif et l’algèbre de rotation universelle.
An analysis of taxation supports for private pension provision in Ireland
(Economic and Social Review, 2018) Whelan, Shane; Hally, Maeve
The size and distribution of the taxation supports for private pension provision has been a contentious issue. Research produced or commissioned by representative groups of the pensions industry in Ireland maintains that the tax supports are merely tax deferment, and the effective tax relief is lower than the ‘headline’ relief on pension contributions. Research by the OECD, on the other hand, suggests that pension savings are essentially tax free to the majority of pension savers. This paper estimates the value of the favourable tax treatment to private pensions provision, expressed as a percentage of the original amount invested, and analyses how it varies with income level, gender, saving period, and other factors. The net effective tax relief on pension savings on each Euro invested in a private pension is estimated by comparing the increase in the present value of pension savings over the lifetime of the individual when compared to other savings. We report that the net effective relief is considerably higher than estimated by the widely cited industry research, and depends on the value of the pension fund at retirement. We identify three distinct groups of individuals in the current regime of incentivising pension savings: those on low incomes who are offered no incentive, the standard rate tax-payers where the net effective tax relief is about 25-30 per cent, and the higher rate tax-payers where the net effective relief is about 31-51 per cent. We argue that current regressive taxation supports for pension savings should be reformed, and reformed before the proposed imminent introduction of an auto-enrolment retirement saving scheme.
An analysis of the attitudes of dental patients attending general dental practice in Galway
(Irish Dental Association, 2013-08) Hayes, Martina; Burke, Francis M.; McKenna, Gerald; Madden, Jamie M.; Cronin, Michael
Aim: To describe the patterns of dental attendance and attitudes towards tooth loss of general dental practice patients in Galway. Objectives: 1. To determine the pattern of adult dental attendance in general practices in Galway; and, 2. To examine the oral health attitudes of these patients. Method: Questionnaires were distributed to 311 consecutive adult patients in the waiting rooms of ten general dental practices in Galway, which were randomly selected from the telephone directory. Results: A total of 254 of the 311 questionnaires distributed were fully completed, returned and included in the results, giving a response rate of 81.7%. A total of 59% of dentate participants attended their dentist for annual or biannual examinations compared to 23% of edentate patients. Some 10.5% of medical card holders and 0.5% of non-medical card holders were edentulous. Conclusions: The data from the survey indicated that medical card holders in Galway were more likely to be edentulous than nonmedical card holders. Edentate patients were less likely to be regular dental attenders than dentate patients.
An analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutions
(Elsevier B.V., 2019-01-15) Chen, Hu; Holland, Finbarr; Stynes, Martin; China Postdoctoral Science Foundation; National Natural Science Foundation of China
A convergence analysis is given for the Grünwald–Letnikov discretisation of a Riemann–Liouville fractional initial-value problem on a uniform mesh tm=mτ with m=0,1,…,M. For given smooth data, the unknown solution of the problem will usually have a weak singularity at the initial time t=0. Our analysis is the first to prove a convergence result for this method while assuming such non-smooth behaviour in the unknown solution. In part our study imitates previous analyses of the L1 discretisation of such problems, but the introduction of some additional ideas enables exact formulas for the stability multipliers in the Grünwald–Letnikov analysis to be obtained (the earlier L1 analyses yielded only estimates of their stability multipliers). Armed with this information, it is shown that the solution computed by the Grünwald–Letnikov scheme is O(τtmα−1) at each mesh point tm; hence the scheme is globally only O(τα) accurate, but it is O(τ) accurate for mesh points tm that are bounded away from t=0. Numerical results for a test example show that these theoretical results are sharp.
Analytical limitation for time-delayed feedback control in autonomous systems
(American Physical Society, 2012) Hooton, Edward W.; Amann, Andreas; Science Foundation Ireland
We prove an analytical limitation on the use of time-delayed feedback control for the stabilization of periodic orbits in autonomous systems. This limitation depends on the number of real Floquet multipliers larger than unity, and is therefore similar to the well-known odd number limitation of time-delayed feedback control. Recently, a two-dimensional example has been found, which explicitly demonstrates that the unmodified odd number limitation does not apply in the case of autonomous systems. We show that our limitation correctly predicts the stability boundaries in this case.
Assessment of a statistical AIF extraction method for dynamic PET studies with 15O water and 18F fluorodeoxyglucose in locally advanced breast cancer patients
(Society of Photo-optical Instrumentation Engineers (SPIE), 2018) O'Sullivan, Finbarr; O'Sullivan, Janet N.; Huang, Jian; Doot, Robert; Muzi, Mark; Schubert, Erin; Peterson, Lanell; Dunnwald, Lisa K.; Mankoff, David M.; Science Foundation Ireland; National Institutes of Health
Blood flow-metabolism mismatch from dynamic positron emission tomography (PET) studies with O-15-labeled water (H2O) and F-18-labeled fluorodeoxyglucose (FDG) has been shown to be a promising diagnostic for locally advanced breast cancer (LABCa) patients. The mismatch measurement involves kinetic analysis with the arterial blood time course (AIF) as an input function. We evaluate the use of a statistical method for AIF extraction (SAIF) in these studies. Fifty three LABCa patients had dynamic PET studies with H2O and FDG. For each PET study, two AIFs were recovered, an SAIF extraction and also a manual extraction based on a region of interest placed over the left ventricle (LV-ROI). Blood flow-metabolism mismatch was obtained with each AIF, and kinetic and prognostic reliability comparisons were made. Strong correlations were found between kinetic assessments produced by both AIFs. SAIF AIFs retained the full prognostic value, for pathologic response and overall survival, of LV-ROI AIFs. (c) The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.
Assessment of the prognostic value of radiomic features in 18F-FMISO PET imaging of hypoxia in postsurgery brain cancer patients: secondary analysis of imaging data from a single-center study and the multicenter ACRIN 6684 trial
(MDPI, 2020-03-01) Muzi, Mark; Wolsztynski, Eric; Fink, James R.; O'Sullivan, Janet N.; O'Sullivan, Finbarr; Krohn, Kenneth A.; Mankoff, David A.; National Institutes of Health; National Cancer Institute; Science Foundation Ireland
Hypoxia is associated with resistance to radiotherapy and chemotherapy in malignant gliomas, and it can be imaged by positron emission tomography with 18F-fluoromisonidazole (18F-FMISO). Previous results for patients with brain cancer imaged with 18F-FMISO at a single center before conventional chemoradiotherapy showed that tumor uptake via T/Bmax (tissue SUVmax/blood SUV) and hypoxic volume (HV) was associated with poor survival. However, in a multicenter clinical trial (ACRIN 6684), traditional uptake parameters were not found to be prognostically significant, but tumor SUVpeak did predict survival at 1 year. The present analysis considered both study cohorts to reconcile key differences and examine the potential utility of adding radiomic features as prognostic variables for outcome prediction on the combined cohort of 72 patients with brain cancer (30 University of Washington and 42 ACRIN 6684). We used both 18F-FMISO intensity metrics (T/Bmax, HV, SUV, SUVmax, SUVpeak) and assessed radiomic measures that determined first-order (histogram), second-order, and higher-order radiomic features of 18F-FMISO uptake distributions. A multivariate model was developed that included age, HV, and the intensity of 18F-FMISO uptake. HV and SUVpeak were both independent predictors of outcome for the combined data set (P < .001) and were also found significant in multivariate prognostic models (P < .002 and P < .001, respectively). Further model selection that included radiomic features showed the additional prognostic value for overall survival of specific higher order texture features, leading to an increase in relative risk prediction performance by a further 5%, when added to the multivariate clinical model.
The benefits of individual versus group work in a biology based laboratory setting
(Consortia Academia Publishing, 2017-04-04) Cronin, Michael; McCabe, Anthony
Laboratory science classes and group work are almost inextricably linked in student college life today. The merits of both are extolled by both educators and future employers as a fundamental part of a rounded qualified graduate. However, there is little evidence to the contrary, particularly in laboratory based programs. This study aims to determine the students’ perception of the benefits of different working scenarios in a biological laboratory setting. The study followed 32 students in their third year of their college program working in three different biology based laboratory classes. During each of these laboratory classes the students either worked alone, worked with a preferred partner or worked with an assigned partner. The students were asked to grade their experiences in a survey. The students were asked to reflect on the following areas: enjoyment of experience, understanding of activities, laboratory technique and time efficiency. Mixed modal analysis of variance and pair-wise analyses were used to determine significance between the differing work group scenarios. The laboratory group make-up had no impact on the student enjoyment of the laboratory class. There was also no significant impact on the students’ ability to perform the laboratory exercise in a time efficient manner. The analysis did however indicate that students who work alone or with their preferred partner have a greater understanding of the subject matter against those who worked with an assigned partner. The results also suggest that students who worked alone had significantly improved laboratory technique compared to student working in a group.
Boolean rings are definitely commutative!
(Irish Mathematical Society, 2015) MacHale, Desmond
A ring {R, +, .} is called Boolean if r2 = r for all r ∈ R. We present four proofs that a Boolean ring is commutative.
Border-collision bifurcations in a driven time-delay system
(AIP Publishing, 2020-02-07) Ryan, Pierce; Keane, Andrew; Amann, Andreas; Irish Research Council; McAfee
We show that a simple piecewise-linear system with time delay and periodic forcing gives rise to a rich bifurcation structure of torus bifurcations and Arnold tongues, as well as multistability across a significant portion of the parameter space. The simplicity of our model enables us to study the dynamical features analytically. Specifically, these features are explained in terms of border-collision bifurcations of an associated Poincaré map. Given that time delay and periodic forcing are common ingredients in mathematical models, this analysis provides widely applicable insight.
Bounds on the distribution of the number of gaps when circles and lines are covered by fragments: theory and practical application to genomic and metagenomic projects
(BioMed Central, 2007-03-02) Moriarty, John; Marchesi, Julian R.; Metcalfe, Anthony; Science Foundation Ireland; Irish Government
Background: The question of how a circle or line segment becomes covered when random arcs are marked off has arisen repeatedly in bioinformatics. The number of uncovered gaps is of particular interest. Approximate distributions for the number of gaps have been given in the literature, one motivation being ease of computation. Error bounds for these approximate distributions have not been given. Results: We give bounds on the probability distribution of the number of gaps when a circle is covered by fragments of fixed size. The absolute error in the approximation is typically on the order of 0.1% at 10× coverage depth. The method can be applied to coverage problems on the interval, including edge effects, and applications are given to metagenomic libraries and shotgun sequencing.
Capacitance and conductance for an MOS system in inversion, with oxide capacitance and minority carrier lifetime extractions
(Institute of Electrical and Electronics Engineers (IEEE), 2014-10) Monaghan, Scott; O'Connor, Éamon; Rios, Rafael; Ferdousi, Fahmida; Floyd, Liam; Ryan, Eimear; Cherkaoui, Karim; Povey, Ian M.; Kuhn, Kelin J.; Hurley, Paul K.; Science Foundation Ireland; Intel Corporation; Seventh Framework Programme
Experimental observations for the In 0.53 Ga 0.47 As metal-oxide-semiconductor (MOS) system in inversion indicate that the measured capacitance (C) and conductance (G or G m ), are uniquely related through two functions of the alternating current angular frequency (ω). The peak value of the first function (G/ω) is equal to the peak value of the second function (-dC/dlog e (ω) ≡ -ωdC/dω). Moreover, these peak values occur at the same angular frequency (ω m ), that is, the transition frequency. The experimental observations are confirmed by physics-based simulations, and applying the equivalent circuit model for the MOS system in inversion, the functional relationship is also demonstrated mathematically and shown to be generally true for any MOS system in inversion. The functional relationship permits the discrimination between high interface state densities and genuine surface inversion. The two function peak values are found to be equal to C ox 2 /(2(C ox + C D )) where C ox is the oxide capacitance per unit area and C D is the semiconductor depletion capacitance in inversion. The equal peak values of the functions, and their observed symmetry relation about ω m on a logarithmic ω plot, opens a new route to experimentally determining C ox . Finally, knowing ω m permits the extraction of the minority carrier generation lifetime in the bulk of the In 0.53 Ga 0.47 As layer.
The capacity of sets of divergence of Certain Taylor Series on the unit circle
(Springer, 2019-05-09) Twomey, J. Brian
A simple and direct proof is given of a generalization of a classical result on the convergence of ∑∞k=0ak e ikx outside sets of x of an appropriate capacity zero, where f(z)=∑∞k=0akzk is analytic in the unit disc U and ∑∞k=0kα|ak|2<∞ with α∈(0,1]. We also discuss some convergence consequences of our results for weighted Besov spaces, for the classes of analytic functions in U for which ∑∞k=1kγ|ak|p<∞, and for trigonometric series of the form ∑∞k=1(αkcoskx+βksinkx) with ∑∞k=1kγ(|αk|p+|βk|p)<∞ , where γ>0,p>1.
Characterizing complex networks using entropy-degree diagrams: unveiling changes in functional brain connectivity induced by Ayahuasca
(MDPI, 2019-01-30) Viol, Aline; Palhano-Fontes, Fernanda; Onias, Heloisa; de Araujo, Draulio B.; Hövel, Philipp; Viswanathan, Gandhi M.; Deutsche Forschungsgemeinschaft; Coordenação de Aperfeiçoamento de Pessoal de Nível Superior; Conselho Nacional de Desenvolvimento Científico e Tecnológico
With the aim of further advancing the understanding of the human brain’s functional connectivity, we propose a network metric which we term the geodesic entropy. This metric quantifies the Shannon entropy of the distance distribution to a specific node from all other nodes. It allows to characterize the influence exerted on a specific node considering statistics of the overall network structure. The measurement and characterization of this structural information has the potential to greatly improve our understanding of sustained activity and other emergent behaviors in networks. We apply this method to study how the psychedelic infusion Ayahuasca affects the functional connectivity of the human brain in resting state. We show that the geodesic entropy is able to differentiate functional networks of the human brain associated with two different states of consciousness in the awaking resting state: (i) the ordinary state and (ii) a state altered by ingestion of the Ayahuasca. The functional brain networks from subjects in the altered state have, on average, a larger geodesic entropy compared to the ordinary state. Finally, we discuss why the geodesic entropy may bring even further valuable insights into the study of the human brain and other empirical networks.
Chimera states in networks of type-I Morris-Lecar neurons
(American Physical Society, 2018-12) Calim, Ali; Hövel, Philipp; Ozer, Mahmut; Uzuntarla, Muhammet; Erasmus+; Deutsche Forschungsgemeinschaft
Chimeras are complex spatiotemporal patterns that emerge as coexistence of both coherent and incoherent groups of coupled dynamical systems. Here, we investigate the emergence of chimera states in nonlocal networks of type-I Morris-Lecar neurons coupled via chemical synapses. This constitutes a more realistic neuronal modeling framework than previous studies of chimera states, since the Morris-Lecar model provides biophysically more relevant control parameters to describe the activity in actual neural systems. We explore systematically the transitions of dynamic behavior and find that different types of synchrony appear depending on the excitability level and nonlocal network features. Furthermore, we map the transitions between incoherent states, traveling waves, chimeras, coherent states, and global amplitude death in the parameter space of interest. This work contributes to a better understanding of biological conditions giving rise to the emergence of chimera states in neural medium.
Choosing parameter values for a geometric CUSUM chart for detecting an upward shift in a proportion
(Taylor & Francis, 2018-04-04) Bourke, Patrick D.
This article considers the task of providing practical advice to quality engineers and practitioners on the choice of values for the two parameters of a geometric CUSUM control chart. This CUSUM chart was developed to enable the detection of sudden shifts from an acceptable level for a proportion (p) such as fraction nonconforming. In the first part of the article, tables are presented listing recommended parameter-choices for each of 18 in-control levels for p in the range (0.04 to 0.001) for detection of each of five sizes of upward shift. In the second part of the article, some empirical relationships among the parameter values are identified. These relationships are used with interpolation to design geometric CUSUM schemes for any in-control level (pa) of p within the range covered by the tables, and with extrapolation for levels of pa as low as 0.0001. Because of the equivalence between a geometric CUSUM and a Bernoulli CUSUM, the tables and the proposed methods of interpolation and extrapolation also provide assistance in the design of a Bernoulli CUSUM chart.

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