{"title":"Beyond Dominated Convergence: Newer methods of Integration","authors":"P. Muldowney","doi":"10.33232/bims.0079.53.74","DOIUrl":"https://doi.org/10.33232/bims.0079.53.74","url":null,"abstract":"Lebesgue's dominated convergence theorem is a crucial pillar of modern analysis, but there are certain areas of the subject where this theorem is deficient. Deeper criteria for convergence of integrals are described in this article.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"41 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132359728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bends in the Plane with Variable Curvature","authors":"R. Sheehan, F. Peters","doi":"10.33232/bims.0078.61.80","DOIUrl":"https://doi.org/10.33232/bims.0078.61.80","url":null,"abstract":"Explicit formulae for planar variable curvature bends are constructed using Euler’s method of natural equations. The bend paths are expressed in terms of special functions. It is shown that the length of the different bend types varies linearly with increasing radius and that the curvature of variable curvature bends can be expressed as a multiple of the curvature of a circle.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128549628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mathematical models of seasonally migrating populations","authors":"J. Donohue","doi":"10.33232/bims.0076.27.28","DOIUrl":"https://doi.org/10.33232/bims.0076.27.28","url":null,"abstract":"This is an abstract of the PhD thesis Mathematical models of seasonally migrating populations written by J. Donohue under the supervision of Dr. P. T. Piiroinen at the School of Mathematics, Statistics, and Applied Mathematics, National University of Ireland, Galway and submitted in September 2015. The phenomenon of seasonal migration has attracted a wealth of attention from biologists. However, the dynamics of migratory populations have been little considered. In this thesis, we use differential equations to model the variation in abundance of seasonally migrating populations. Our contribution to the field begins with a representation of seasonal breeding. We use piecewise-smooth differential equations to model the variation in the size of a population that has a short interval each year during which successful reproduction is possible. We first consider a one-species model which illustrates the dynamics of a population of specialist feeders over the course of a single breeding season and use it to examine how reproductive success depends on the population’s distribution of breeding dates. We then introduce time-dependent switches to extend the model to a broader class of species. This allows us to consider the effect of climate change on populations that annually travel long distances. We then shift focus to consider interactions between migrants and species at higher levels in the food web. Predatory pressure influences almost all populations to some extent. Here, however, interactions may occur for just a brief period each year before the populations involved become spatially separated. The range of a migrating population may overlap with that of a population of predators for a","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129506997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The chain rule for $mathcal{F}$-differentiation","authors":"T. Chaobankoh, J. Feinstein, S. Morley","doi":"10.33232/bims.0077.19.34","DOIUrl":"https://doi.org/10.33232/bims.0077.19.34","url":null,"abstract":"Let $X$ be a perfect, compact subset of the complex plane, and let $D^{(1)}(X)$ denote the (complex) algebra of continuously complex-differentiable functions on $X$. Then $D^{(1)}(X)$ is a normed algebra of functions but, in some cases, fails to be a Banach function algebra. Bland and the second author investigated the completion of the algebra $D^{(1)}(X)$, for certain sets $X$ and collections $mathcal{F}$ of paths in $X$, by considering $mathcal{F}$-differentiable functions on $X$. \u0000In this paper, we investigate composition, the chain rule, and the quotient rule for this notion of differentiability. We give an example where the chain rule fails, and give a number of sufficient conditions for the chain rule to hold. Where the chain rule holds, we observe that the Fa'a di Bruno formula for higher derivatives is valid, and this allows us to give some results on homomorphisms between certain algebras of $mathcal{F}$-differentiable functions.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"279 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122295764","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Theoretical and numerical analysis of rigid-body impacts with friction","authors":"S. Burns","doi":"10.33232/bims.0076.25.26","DOIUrl":"https://doi.org/10.33232/bims.0076.25.26","url":null,"abstract":"This is an abstract of the PhD thesis Theoretical and Numerical Analysis of Rigid-body Impacts with Friction written by S. Burns under the supervision of Petri Piiroinen at the School of Mathematics, Statistics, and Applied Mathematics, National University of Ireland, Galway and submitted in September 2015. This thesis gives a flavour of the area of rigid body impacts with friction, an area which has far reaching applications in engineering, sports science and every day life. The focus of this work will be on the two main streams of this field, theoretical and numerical. This thesis will present an overview of the general subject of rigidbody impact, including discussion and analysis of the validity of ones choice of impact law and the numerical techniques required for the simulation of rigid-body impacting systems. Two impact laws will be introduced in Chapter 3 and a direct comparison will be made in order to examine the varying dynamics that can be achieved using both a basic and a complex impact law and to explore some of the problems that can occur with a more basic formulation. It will be demonstrated that for certain regions in parameter space the two formulations are equivalent, however, for many other regions the two formulations can vary greatly. A hybrid event-driven numerical scheme is one in which smooth dynamics are described by differential equations, which can be solved numerically using standard techniques, and non smooth events which are described by maps. In Chapter 5, a hybrid event-driven numerical scheme for the implementation of the Energetic Impact Law","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122987808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Topics in cocyclic development of pairwise combinatorial designs","authors":"Ronan Egan","doi":"10.33232/bims.0076.29.30","DOIUrl":"https://doi.org/10.33232/bims.0076.29.30","url":null,"abstract":"This is an abstract of the PhD thesis Topics in Cocyclic Development of Pairwise Combinatorial Designs written by Ronan Egan under the supervision of Dane Flannery at the School of Mathematics, Statistics, and Applied Mathematics, National University of Ireland, Galway and submitted in July 2015. This thesis is a compilation of results dealing with cocyclic development of pairwise combinatorial designs. Motivated by a classification of the indexing and extension groups of the Paley Hadamard matrices due to de Launey and Stafford, we investigate cocyclic development of the so-called generalized Sylvester (or Drake) Hadamard matrices. We describe the automorphism groups and derive strict conditions on possible indexing groups, addressing research problems of de Launey and Flannery in doing so. The shift action, discovered by Horadam, is a certain action of any finite group on the set of its 2-cocycles with trivial coefficients, which preserves both cohomological equivalence and orthogonality. We answer questions posed by Horadam about the shift action, in particular regarding its fixed points. One of our main innovations is the concept of linear shift representation. We give an algorithm for calculating the matrix group representation of a shift action, which enables us to compute with the action in a natural setting. We prove detailed results on reducibility, and discuss the outcomes of some computational experiments, including searches for orthogonal cocycles. Using the algorithms developed for shift representations, and other methods, we classify up to equivalence all cocyclic BH(n, p)s where p is an odd prime (necessarily dividing n) and np ≤ 100. This was","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132243054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Incremental elastic surface waves and static wrinkles","authors":"A. Gower","doi":"10.33232/bims.0076.31.32","DOIUrl":"https://doi.org/10.33232/bims.0076.31.32","url":null,"abstract":"This is an abstract of the PhD thesis Incremental elastic surface waves and static wrinkles written by Artur L. Gower under the supervision of Prof. Michel Destrades at the School of Mathematics, Statistics, and Applied Mathematics, National University of Ireland, Galway and submitted in September 2015. This article-based thesis comprises a collection of four articles, each of which constitutes a chapter written and formatted in manuscript form. The general aim underlying these articles is to understand and predict how incremental elastic surface waves propagate or static wrinkles form on a deformed elastic substrate. The formation of these small-amplitude disturbances can be the end goal, such as in sending signals or creating functional coatings, or they can be used to measure and characterise the underlying elastic substrate. This thesis focuses on using surface waves or static wrinkles to characterise soft solids, such as biological tissues. For the complete thesis see [1]. Here we summarize the main conclusion of the thesis. Chapter 1 predicts a new phenomenon: oblique wrinkles, which should appear in a large range of materials. Yet oblique wrinkles have not been seen experimentally so far on soft solids. Another issue raised was why are the predicted critical strains greater than the experimentally observed critical strains? We showed that this is likely due to a skin effect caused by dehydration. In Chapter 2 the effects of a stiffer skin on an elastic substrate on surface wrinkles was initially studied, and therein we also studied the","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116547988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Preconditioning techniques for singularly perturbed differential equations","authors":"Anh Nhan","doi":"10.33232/bims.0076.35.36","DOIUrl":"https://doi.org/10.33232/bims.0076.35.36","url":null,"abstract":"This is an abstract of the PhD thesis Preconditioning techniques for singularly perturbed differential equations written by Thái Anh Nhan, under the supervision of Dr Niall Madden, at the School of Mathematics, Statistics, and Applied Mathematics, National University of Ireland, Galway and submitted in July 2015. This dissertation is concerned with the numerical solution of linear systems arising from finite difference and finite element discretizations of singularly perturbed reaction-diffusion problems. Such linear systems present several difficulties that make computing accurate solutions efficiently a nontrivial challenge for both direct and iterative solvers. The poor performance of direct solvers, such as Cholesky factorization, is due to the presence of subnormal floating point numbers in the factors. This thesis provides a careful analysis of this phenomenon by giving a concrete formula for the magnitude of the fill-in entries in the Cholesky factors in terms of the perturbation parameter, ε, and the discretization parameter, N . It shows that, away from the main diagonal, the magnitude of fill-in entries decreases exponentially. Furthermore, with our analysis, the location of corresponding fill-in entries associated with some given magnitude can also be determined. This can be used to predict the number and location of subnormals in the factors. Since direct solvers scale badly with ε, one must use iterative solvers. However, the application of finite difference and finite element discretizations on layer-adapted meshes results in ill-conditioned","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125413047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Torsion and ground state maxima: close but not the same","authors":"B. Benson, R. Laugesen, M. Minion, B. Siudeja","doi":"10.33232/bims.0078.81.88","DOIUrl":"https://doi.org/10.33232/bims.0078.81.88","url":null,"abstract":"Could the location of the maximum point for a positive solution of a semilinear Poisson equation on a convex domain be independent of the form of the nonlinearity? Cima and Derrick found certain evidence for this surprising conjecture. \u0000We construct counterexamples on the half-disk, by working with the torsion function and first Dirichlet eigenfunction. On an isosceles right triangle the conjecture fails again. Yet the conjecture has merit, since the maxima of the torsion function and eigenfunction are unexpectedly close together. It is an open problem to quantify this closeness in terms of the domain and the nonlinearity.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"99 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123761342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Completions of partial matrices","authors":"James R McTigue","doi":"10.33232/bims.0076.33.34","DOIUrl":"https://doi.org/10.33232/bims.0076.33.34","url":null,"abstract":"This is an abstract of the PhD thesis Completions of Partial Matrices written by J. McTigue under the supervision of Rachel Quinlan at the School of Mathematics, Statistics, and Applied Mathematics, National University of Ireland, Galway and submitted in March 2015. A partial matrix over a field F is a matrix whose entries are either elements of the field or independent indeterminates. A completion of a partial matrix is any matrix that results from assigning a field element to each indeterminate. The set of completions of an m× n partial matrix forms an affine subspace of Mm×n(F). This thesis investigates partial matrices whose sets of completions satisfy particular rank properties specifically partial matrices whose completions all have ranks that are bounded below and partial matrices whose completions all have the same rank. The maximum possible number of indeterminates in such partial matrices is determined, and the partial matrices that attain these bounds are fully characterized for all fields. These characterizations utilize a duality between properties of affine spaces of matrices that are related by the trace bilinear form. Precise conditions (based on field order, rank and size) are provided to determine if a partial matrix whose completions all have rank r must possess an r × r partial submatrix whose completions are all nonsingular. Finally a characterization of maximal nonsingular partial matrices is provided a maximal nonsingular partial matrix is a square partial matrix each of whose completions has full rank, with the property","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134313630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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