Isaac Newton Institute for Mathematical Scienceshttps://www.repository.cam.ac.uk/handle/1810/2275632024-11-04T17:58:04Z2024-11-04T17:58:04Z41Code supporting "GIT stability of divisors in products of projective spaces"Karagiorgis, IoannisOrtscheidt, TheresaPapazachariou, Theodoroshttps://www.repository.cam.ac.uk/handle/1810/3621332023-12-09T01:40:58Zdc.title: Code supporting "GIT stability of divisors in products of projective spaces"
dc.contributor.author: Karagiorgis, Ioannis; Ortscheidt, Theresa; Papazachariou, Theodoros
dc.description: This software package is a complement to the article "GIT stability of divisors in products of projective spaces". The software package implements a series of algorithms for the study of Geometric Invariant Theory quotients of divisors of arbitrary bidegrees k1,...,kl in products of projective spaces of dimensions m1-1,...ml-1. Given the m_i and bidegres k1,..., kl, the code finds all relevant one-parameter subgroups which determine all the GIT quotients for these divisors. In addition, it finds all maximal orbits of not stable and strictly semistable divisors, as well as minimal closed orbits of strictly semistable pairs in terms of families of pairs defined by monomials with non-zero coefficients. It outputs the maximal orbits of not stable and strictly semistable divisors as a list of polynomials defined by monomials with non-zero coefficients. The user can decide to run the code locally, via the Jupyter file, where the output is printed in jupyter, or locally via the run file GIT_divisors_in_products_of_projective_spaces.sage, which prints the output in a txt output file. Further details can be found at the article "GIT stability of divisors in products of projective spaces". The sotware package is implemented as a Sage 9.2 package. The program can be run directly from the Sage Shell (Windows) or from the terminal (Mac and Linux). The user inserts the dimensions m1,...ml and bidegrees k1,...,kl, with arbitrary degree of generality.
An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problemsJoyce, DParnell, WJAssier, RCAbrahams, IDhttps://www.repository.cam.ac.uk/handle/1810/2667022024-01-05T15:17:33Z2017-05-01T00:00:00Zdc.title: An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problems
dc.contributor.author: Joyce, D; Parnell, WJ; Assier, RC; Abrahams, ID
dc.description.abstract: In Parnell & Abrahams (2008 Proc. R. Soc. A 464, 1461–1482. (doi:10.1098/rspa.2007.0254)), a homogenization scheme was developed that gave rise to explicit forms for the effective antiplane shear moduli of a periodic unidirectional fibre-reinforced medium where fibres have non-circular cross section. The explicit expressions are rational functions in the volume fraction. In that scheme, a (non-dilute) approximation was invoked to determine leading-order expressions. Agreement with existing methods was shown to be good except at very high volume fractions. Here, the theory is extended in order to determine higher-order terms in the expansion. Explicit expressions for effective properties can be derived for fibres with non-circular cross section, without recourse to numerical methods. Terms appearing in the expressions are identified as being associated with the lattice geometry of the periodic fibre distribution, fibre cross-sectional shape and host/fibre material properties. Results are derived in the context of antiplane elasticity but the analogy with the potential problem illustrates the broad applicability of the method to, e.g. thermal, electrostatic and magnetostatic problems. The efficacy of the scheme is illustrated by comparison with the well-established method of asymptotic homogenization where for fibres of general cross section, the associated cell problem must be solved by some computational scheme.
2017-05-01T00:00:00ZDelay-independent asymptotic stability in monotone systemsDevane, ELestas, Ihttps://www.repository.cam.ac.uk/handle/1810/2525122024-06-11T22:46:03Z2015-01-01T00:00:00Zdc.title: Delay-independent asymptotic stability in monotone systems
dc.contributor.author: Devane, E; Lestas, I
dc.description.abstract: Monotone systems comprise an important class of dynamical systems that are of interest both for their wide applicability and because of their interesting mathematical properties. It is known that under the property of quasimono-tonicity time-delayed systems become monotone, and some remarkable properties have been reported for such systems. These include, for example, the fact that for linear systems global asymptotic stability of the undelayed system implies global asymptotic stability for the delayed system under arbitrary bounded delays. Nevertheless, extensions to nonlinear systems have thus far relied on various restrictive conditions, such as homogeneity and subhomogeneity, and it has been conjectured that these can be relaxed. Our aim in this paper is to show that this is feasible for a general class of nonlinear monotone systems, by deriving asymptotic stability results in which simple properties of the undelayed system lead to delay-independent stability. In particular, one of our results is to show that if the undelayed system has a convergent trajectory that is unbounded in all components as t → -∞ then the system is globally asymptotically stable for arbitrary time-varying delays. This follows from a more general result derived in the paper where delay-independent regions of attraction are quantified from the asymptotic behavior of individual trajectories of the undelayed system. This result recovers various known delay-independent stability results, and several examples are included in the paper to illustrate the significance of the proposed stability conditions.
2015-01-01T00:00:00ZBayesPeak: Bayesian analysis of ChIP-seq data.Spyrou, ChristianaStark, RoryLynch, Andy GTavaré, Simonhttps://www.repository.cam.ac.uk/handle/1810/2440852024-01-05T00:22:33Z2009-09-21T00:00:00Zdc.title: BayesPeak: Bayesian analysis of ChIP-seq data.
dc.contributor.author: Spyrou, Christiana; Stark, Rory; Lynch, Andy G; Tavaré, Simon
dc.description.abstract: BACKGROUND: High-throughput sequencing technology has become popular and widely used to study protein and DNA interactions. Chromatin immunoprecipitation, followed by sequencing of the resulting samples, produces large amounts of data that can be used to map genomic features such as transcription factor binding sites and histone modifications. METHODS: Our proposed statistical algorithm, BayesPeak, uses a fully Bayesian hidden Markov model to detect enriched locations in the genome. The structure accommodates the natural features of the Solexa/Illumina sequencing data and allows for overdispersion in the abundance of reads in different regions. Moreover, a control sample can be incorporated in the analysis to account for experimental and sequence biases. Markov chain Monte Carlo algorithms are applied to estimate the posterior distributions of the model parameters, and posterior probabilities are used to detect the sites of interest. CONCLUSION: We have presented a flexible approach for identifying peaks from ChIP-seq reads, suitable for use on both transcription factor binding and histone modification data. Our method estimates probabilities of enrichment that can be used in downstream analysis. The method is assessed using experimentally verified data and is shown to provide high-confidence calls with low false positive rates.
2009-09-21T00:00:00Z