Irish Mathematical Society Bulletin最新文献

{"title":"Convergence from Below Suffices","authors":"J. Feinstein","doi":"10.33232/bims.0059.65.70","DOIUrl":"https://doi.org/10.33232/bims.0059.65.70","url":null,"abstract":"An elementary application of Fatou's lemma gives a strengthened version of the monotone convergence theorem. We call this the convergence from below theorem. We make the case that this result should be better known, and deserves a place in any introductory course on measure and integration.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133055563","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 Contribution to Metric Diophantine Approximation: the Lebesgue and Hausdorff Theories","authors":"F. Adiceam","doi":"10.33232/bims.0077.5.6","DOIUrl":"https://doi.org/10.33232/bims.0077.5.6","url":null,"abstract":"This thesis is concerned with the theory of Diophantine approximation from the point of \u0000view of measure theory. After the prolegomena which conclude with a number of conjectures set \u0000to understand better the distribution of rational points on algebraic planar curves, Chapter 1 \u0000provides an extension of the celebrated Theorem of Duffin and Schaeffer. This enables one to \u0000set a generalized version of the Duffin–Schaeffer conjecture. Chapter 2 deals with the topic of \u0000simultaneous approximation on manifolds, more precisely on polynomial curves. The aim is \u0000to develop a theory of approximation in the so far unstudied case when such curves are not \u0000defined by integer polynomials. A new concept of so–called “liminf sets” is then introduced in \u0000Chapters 3 and 4 in the framework of simultaneous approximation of independent quantities. \u0000In short, in this type of problem, one prescribes the set of integers which the denominators of \u0000all the possible rational approximants of a given vector have to belong to. Finally, a reasonably \u0000complete theory of the approximation of an irrational by rational fractions whose numerators \u0000and denominators lie in prescribed arithmetic progressions is developed in chapter 5. This \u0000provides the first example of a Khintchine type result in the context of so–called uniform \u0000approximation.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130233241","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":"Some Residually Solvable One-Relator Groups","authors":"Delaram Kahrobaei, Andrew F. Douglas, Katalin Bencs'ath","doi":"10.33232/bims.0065.23.31","DOIUrl":"https://doi.org/10.33232/bims.0065.23.31","url":null,"abstract":"This communication records some observations made in the course of studying one-relator groups from the point of view of residual solvability. As a contribution to clas- sication eorts we single out some relator types that render the corresponding one-relator groups residually solvable. a collection of facts and examples gathered while attempting to char- acterize the residually solvable one-relator groups in terms (of the form) of the (single) dening relator. In what follows we prove suf- ciency results for certain cases when the relator is a commutator, and then raise some questions. The class of one-relator groups shows a varied pattern of behavior with respect to residual properties. We begin with reviewing some of the literature that motivated our interest in the topic. G. Baum- slag in (3) showed that positive one-relator groups, which is to say that the relator has only positive exponents, are residually solvable. In the same paper he provided a specic example to demonstrate that not all one-relator groups are residually solvable. A free-by- cyclic group is necessarily residually solvable. As well are the free- by-solvable Baumslag{Solitar groups Bm;n (the groups with presen- tation ha;b;a 1 b m a = b n i for pairs of non-zero integers m;n), by a result of Peter Kropholler (15) who showed that in these groups the second derived subgroup is free. The Baumslag{Solitar groups","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124820313","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":"Ranges of bimodule projections and conditional expectations","authors":"R. Pluta","doi":"10.33232/bims.0069.9","DOIUrl":"https://doi.org/10.33232/bims.0069.9","url":null,"abstract":"This is an abstract of the PhD thesis Ranges of Bimodule Projections and Conditional Expectations written by R. Pluta under the supervision of Prof. Richard M. Timoney at the School of Mathematics, Trinity College Dublin and submitted in September 2011. The algebraic theory of corner rings introduced by Lam [1] (as an abstraction of the properties of Peirce corners eRe of a ring R associated with an idempotent e ∈ R) is investigated in the context of C∗-algebras and operator algebras. The main result is as follows.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131867487","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":"Automorphisms of pairwise combinatorial designs","authors":"Padraig Ó Catháin","doi":"10.33232/bims.0068.24","DOIUrl":"https://doi.org/10.33232/bims.0068.24","url":null,"abstract":"iv","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114557955","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 Two-Child Paradox: Dichotomy and Ambiguity","authors":"P. Lynch","doi":"10.33232/bims.0067.67.73","DOIUrl":"https://doi.org/10.33232/bims.0067.67.73","url":null,"abstract":"Given that one of the children in a two-child fam- ily is a boy, what are the chances that the other is also a boy. The intuitive answer is 50 : 50. More careful investigation leads us to a 1-in-3 chance. We investigate circumstances under which these answers are correct. The imposition of further conditions yields some very surprising results.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129130311","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":"$K$-Theory of Azumaya algebras","authors":"J. Millar","doi":"10.33232/bims.0066.27.28","DOIUrl":"https://doi.org/10.33232/bims.0066.27.28","url":null,"abstract":"For an Azumaya algebra $A$ which is free over its centre $R$, we prove that the $K$-theory of $A$ is isomorphic to $K$-theory of $R$ up to its rank torsion. We observe that a graded central simple algebra, graded by an abelian group, is a graded Azumaya algebra and it is free over its centre. So the above result, from the non-graded setting, covers graded central simple algebras. For a graded central simple algebra $A$, we can also consider graded projective modules. Let $Pgr (R)$ be the category of graded finitely generated projective $R$-modules and $K_i, igeq 0$, be the Quillen $K$-groups. Then $K_i^{gr} (R)$ is defined to be $K_i( Pgr (R))$. We give some examples to show that the graded $K$-theory of $A$ does not necessarily coincide with its usual $K$-theory. For a graded Azumaya algebra $A$, free over its centre $R$ and subject to some conditions, we show that $K_i^{gr} (A)$ is ``very close'' to $K_i^{gr}(R)$. Further, we consider additive commutators in the setting of graded division algebras. For a graded division algebra $D$ with a totally ordered abelian grade group, we show how the submodule generated by the additive commutators in $QD$ relates to that of $D$, where $QD$ is the quotient division ring.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129230877","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":"Large fluctuations of stochastic differential equations with regime switching: applications to simulation and finance","authors":"T. Lynch","doi":"10.33232/bims.0066.25.26","DOIUrl":"https://doi.org/10.33232/bims.0066.25.26","url":null,"abstract":"This thesis deals with the asymptotic behaviour of various classes of stochastic differential equations (SDEs) and their discretisations. More specifically, it concerns the largest fluctuations of such equations by considering the rate of growth of the almost sure running maxima of the solutions. The first chapter gives a brief overview of the main ideas and motivations for this thesis. Chapter 2 examines a class of nonlinear finite–dimensional SDEs which have mean– reverting drift terms and bounded noise intensity or, by extension, unbounded noise intensity. Equations subject to Markovian switching are also studied, allowing the drift and diffusion coefficients to switch randomly according to a Markov jump process. The assumptions are motivated by the large fluctuations experienced by financial markets which are subjected to random regime shifts. We determine sharp upper and lower bounds on the rate of growth of the large fluctuations of the process by means of stochastic comparison methods and time change techniques. Chapter 3 applies similar techniques to a variant of the classical Geometric Brownian Motion (GBM) market model which is subject to random regime shifts. We prove that the model exhibits the same long–run growth properties and deviations from the trend rate of growth as conventional GBM. The fourth chapter examines the consistency of the asymptotic behaviour of a discretisation of the model detailed in Chapter 3. More specifically, it is shown that the discrete approximation to the stock price grows exponentially and that the large fluctuations from this exponential growth trend are governed by a Law of the Iterated Logarithm. The results about the asymptotic behaviour of discretised SDEs found in Chapter 4, rely on the use of an exponential martingale inequality (EMI). Chapter 5 considers a discrete version of the EMI driven by independent Gaussian sequences. Some extensions, applications and ramifications of the results are detailed. The final chapter uses the EMI developed in Chapter 5 to analyse the asymptotic behaviour of discretised SDEs. Two different methods of discretisation are considered: a standard Euler–Maruyama method and an implicit split–step variant of Euler–Maruyama. v","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129507798","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 Information-carrying capacity of certain quantum channels","authors":"C. Morgan","doi":"10.33232/bims.0066.28.30","DOIUrl":"https://doi.org/10.33232/bims.0066.28.30","url":null,"abstract":"In this thesis we analyse the type of states and ensembles which achieve the capacity for certain quantum channels carrying classical information. We first concentrate on the product-state capacity of a particular quantum channel, that is, the capacity which is achieved by encoding the output states from a source into codewords comprised of states taken from ensembles of non-entangled states and sending them over copies of the quantum channel. Using the \"single-letter\" formula proved independently by Holevo and by Schumacher and Westmoreland we obtain the product-state capacity of the qubit quantum amplitude-damping channel, which is determined by a transcendental equation in a single real variable and can be solved numerically. We demonstrate that the product-state capacity of this channel can be achieved using a minimal ensemble of non-orthogonal pure states. Next we consider the classical capacity of two quantum channels with memory, namely a periodic channel with quantum depolarising channel branches and a convex combination of quantum channels. We prove that the classical capacity for each of the classical memory channels mentioned above is, in fact, equal to the respective product-state capacities. For those channels this means that the classical capacity is achieved without the use of entangled input-states. Next we introduce the channel coding theorem for memoryless quantum channels, providing a known proof by Winter for the strong converse of the theorem. We then consider the strong converse to the channel coding theorem for a periodic quantum channel.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117068770","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 Theoretical Study of Spin Filtering and its Application to Polarizing Antiprotons","authors":"D. O’Brien","doi":"10.33232/bims.0062.26.27","DOIUrl":"https://doi.org/10.33232/bims.0062.26.27","url":null,"abstract":"There has been much recent research into possible methods of polarizing an antiproton beam, the most promising being spin filtering, the theoretical understanding of which is currently incomplete. The method of polarization buildup by spin filtering requires many of the beam particles to remain within the beam after repeated interaction with an internal target in a storage ring. Hence small scattering angles, where we show that electromagnetic effects dominate hadronic effects, are important. All spin-averaged and spin-dependent electromagnetic cross-sections and spin observables for elastic spin 1/2 - spin 1/2 scattering, for both point-like particles and non-point-like particles with internal structure defined by electromagnetic form factors, are derived to first order in QED. Particular attention is paid to spin transfer and depolarization cross-sections in antiproton-proton, antiproton-electron and positron-electron scattering, in the low | t | region of momentum transfer. A thorough mathematical treatment of spin filtering is then presented, identifying the key physical processes involved and highlighting the dynamical properties of the physical system. We present and solve sets of differential equations which describe the buildup of polarization by spin filtering in many different scenarios of interest. The advantages of using a lepton target are outlined, and finally a proposal to polarize antiprotons by spin filtering off an opposing polarized electron beam is investigated.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"145 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121616913","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}