{"title":"Corrigendum: Modular Metric Spaces","authors":"Hanan. M. Abobaker, R. Ryan","doi":"10.33232/bims.0089.41","DOIUrl":"https://doi.org/10.33232/bims.0089.41","url":null,"abstract":". We indicate a corrected version of the paper [1], which had some errors. We are grateful to V. V. Chistyakov for bringing these to our attention.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117039605","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 Number of Generators of a Finite Group","authors":"F. Menegazzo","doi":"10.33232/bims.0050.117.128","DOIUrl":"https://doi.org/10.33232/bims.0050.117.128","url":null,"abstract":"In this expository article, which is a slightly expanded version of the lecture given at the All Ireland Algebra Days (Belfast, 16–19 May, 2001), we first recall a technique recently developed by F. Dalla Volta and A. Lucchini to study generation properties of finite groups. We then discuss some problems in permutation groups, linear groups and profinite groups where this technique has proved useful. Finally, we comment on some results and problems related to probability","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131247298","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":"Irreducible characters of small degree of the unitriangular group","authors":"M. Marjoram","doi":"10.33232/bims.0042.21.31","DOIUrl":"https://doi.org/10.33232/bims.0042.21.31","url":null,"abstract":"","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132015223","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":"“Odd” Matrices and Eigenvalue Accuracy","authors":"David Judge","doi":"10.33232/bims.0069.25.32","DOIUrl":"https://doi.org/10.33232/bims.0069.25.32","url":null,"abstract":"A definition of even and odd matrices is given, and some of their elementary properties stated. The basic result is that if λ is an eigenvalue of an odd matrix, then so is −λ. Starting from this, there is a consideration of some ways of using odd matrices to test the order of accuracy of eigenvalue routines. 1. Definitions and some elementary properties Let us call a matrix W even if its elements are zero unless the sum of the indices is even – i.e. Wij = 0 unless i + j is even; and let us call a matrix B odd if its elements are zero unless the sum of the indices is odd – i.e. Bij = 0 unless i + j is odd. The non-zero elements of W and B (the letters W and B will always denote here an even and an odd matrix, respectively) can be visualised as lying on the white and black squares, respectively, of a chess board (which has a white square at the top left-hand corner). Obviously, any matrix A can be written as W+B; we term W and B the even and odd parts, respectively, of A. Under multiplication, even and odd matrices have the properties, similar to those of even and odd functions, that even × even and odd × odd are even, even × odd and odd × even are odd. From now on, we consider only square matrices. It is easily seen that, if it exists, the inverse of an even matrix is even, the inverse of an odd matrix is odd. It is also easily seen that in the PLU decomposition of a non-singular matrix A which is either even or odd, L and U are always even, while P is even or odd according as A is. 2010 Mathematics Subject Classification. 15A18,15A99,65F15.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134101285","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":"Old, Recent and New Results on Quasinormal Subgroups","authors":"S. Stonehewer","doi":"10.33232/bims.0056.125.133","DOIUrl":"https://doi.org/10.33232/bims.0056.125.133","url":null,"abstract":"Thus normal subgroups are always quasinormal, but not conversely. For, if p is a prime, then any cyclic group Cpn extended by any cyclic group Cpm has all subgroups quasinormal (provided, when p = 2 and n > 2, the cyclic subgroup of order 4 in C2n is central in the extension). The same is true if Cpn is replaced by any abelian p-group H of finite exponent, with Cpm acting on H as a group of power automorphisms (and elements of order 4 in H are again central in the extension if p = 2). These results can be found in sections 2.3 and 2.4 of [16]. One of the earliest results about quasinormal subgroups is due to Ore, who proved in 1938 that a quasinormal subgroup of a finite group G is always subnormal in G ([14]). When G is infinite, then A does not have to be subnormal, but it is always ascendant in G (see [17]). Clearly the extent to which a quasinormal subgroup A can differ from being normal is of interest and a measure of this was given by Ito and Szep in 1962 when they proved that, again with G finite, and denoting the core of A in G by AG, the quotient A/AG is always nilpotent","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134590137","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":"Infinite Powers: The Story of Calculus, by Steven H. Strogatz","authors":"P. O'Kane","doi":"10.33232/bims.0088.79.81","DOIUrl":"https://doi.org/10.33232/bims.0088.79.81","url":null,"abstract":"","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133836795","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":"Yang-Mills instantons on the Taub-NUT space and supersymmetric $N = 2$ gauge theories with impurities","authors":"Clare O'Hara","doi":"10.33232/bims.0066.30.31","DOIUrl":"https://doi.org/10.33232/bims.0066.30.31","url":null,"abstract":"","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122887434","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":"A comparison technique for integral equations","authors":"M. Meehan, D. O’Regan","doi":"10.33232/bims.0042.54.71","DOIUrl":"https://doi.org/10.33232/bims.0042.54.71","url":null,"abstract":"Using the results obtained for (1:1) in Se tion 2, a omparison te hnique is presented in Se tion 3 whi h rstly guarantees that the solution y 2 C[0; T ) of (1:1) (with T = 1), is su h that lim t!1 y(t) exists, and se ondly, allows us to read o what this limit is. The te hnique is illustrated with some examples. A result of Miller [9℄, whi h pertains to a spe ial ase of (1:1) is in luded and dis ussed for ompleteness.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128717432","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":"Higher order symmetry of graphs","authors":"Ronald Brown","doi":"10.33232/bims.0032.46.59","DOIUrl":"https://doi.org/10.33232/bims.0032.46.59","url":null,"abstract":"1 Symmetry in analogues of set theory This lecture gives background to and results of work of my student John Shrimpton [19, 20, 21]. It advertises the joining of two themes: groups and symmetry; and categorical methods and analogues of set theory. Groups are expected to be associated with symmetry. Klein's famous Erlanger Programme asserted that the study of a geometry was the study of the group of automorphisms of that geometry. The structure of group alone may not give all the expression one needs of the intuitive idea of symmetry. One often needs structured groups (for example topological, Lie, algebraic, order, ...). Here we consider groups with the additional structure of directed graph, which we abbreviate to graph. This type of structure appears in [18, 14]. We shall associate with a graph A a group AUT(A) which is also a graph. The vertices of AUT(A) are the automorphism of the graph A and the edges between automorphisms give an expression of \" adjacency \" of automorphisms. The vertices of this graph form a group, and so also do the edges. The automorphisms of A adjacent to the identity will be called the inner automorphisms of the graph A. One aspect of the problem is to describe these inner automorphisms in terms of the internal structure of the graph A. The second theme is that of regarding the usual category of sets and mappings as but one environment for doing mathematics, and one which may be replaced by others. We use the word \" environment \" here rather than \" foundation \" , because the former word implies a more relativistic approach. The other environment we choose here is the category of directed graphs and their morphisms. We define this category, and then use methods analogous to those of set theory within this category. * This paper is an account of a lecture \" Groups which are graphs (and vice versa!) \" given to the Fifth September Meeting of the Irish Mathematical Society at Waterford Regional Technical College, 1992. It was published as [4].","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129070333","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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