{"title":"Advances in Computing the Non-abelian Tensor Square of Polycyclic Groups","authors":"R. F. Morse","doi":"10.33232/bims.0056.115.123","DOIUrl":"https://doi.org/10.33232/bims.0056.115.123","url":null,"abstract":"The nonabelian tensor square G › G of the group G is the group generated by the symbols g › h, where g;h 2 G, subject to the relations gg 0 › h = ( g g 0 › g h)(g › h) and g › hh 0 = (g › h)( h g › h h 0 ) for all g;g;h;h 0 2 G, where g g 0 = gg 0 g i1 is conjugation on the left. Following the work of C. Miller [18], R. K. Dennis in [10] introduced the nonabelian tensor square which is a specialization of the more general nonabelian tensor product independently introduced by R. Brown and J.-L. Loday [6]. By computing the nonabelian tensor square we mean finding a standard or simplified presentation for it. In the case of finite groups, the definition of the nonabelian tensor square gives a finite presentation that can be simplified using Tietze transformations. This simplified presentation can then be examined to determine the nonabelian tensor square. This was the approach taken in [3], in which the nonabelian tensor square was computed for each nonabelian group of order up to 30. Creating a presentation from the definition of the nonabelian tensor square, simplifying it using Tietze transformations and computing a structure description from the simplified presentation can be implemented in few lines of GAP [16]. However, this strategy does not scale well to finite groups G having order greater than 100 since the initial presentation has jGj 2 generators and 2jGj 3 relations.","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":"129249536","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":"Weyl type Theorems and the Approximate Point Spectrum","authors":"M. Lahrouz, M. Zohry","doi":"10.33232/bims.0055.41.52","DOIUrl":"https://doi.org/10.33232/bims.0055.41.52","url":null,"abstract":"It is shown that, if an operator T on a complex Banach space or its adjoint T ⁄ has the single-valued extension property, then the generalized a-Browder's theorem holds for f(T) for every complex-valued analytic function f on a neigh- borhood of the spectrum of T. We also study the generalized a-Weyl's theorem in connection with the single-valued exten- sion property. Finally, we examine the stability of the gen- eralized a-Weyl's theorem under commutative perturbations by finite rank operators.","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":"129367915","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 simple proofs of Lima's two-term dilogarithm identity","authors":"S. Stewart","doi":"10.33232/bims.0089.43.49","DOIUrl":"https://doi.org/10.33232/bims.0089.43.49","url":null,"abstract":". Recently, Lima found a remarkable two-term dilogarithm identity whose proof was based on a hyperbolic form of a proof for the Basel problem given by Beukers, Kolk, and Calabi. A number of simple proofs for this identity that make use of known functional relations for the dilogarithm function are given and an application of Lima’s identity to another two-term dilogarithm evaluation is presented.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"30 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":"129462935","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":"Paths in a graph","authors":"J. Siemons","doi":"10.33232/bims.0007.55.59","DOIUrl":"https://doi.org/10.33232/bims.0007.55.59","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":"128695885","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":"Recent Trends on Order Bounded Disjointness Preserving Operators","authors":"K. Boulabiar","doi":"10.33232/bims.0062.43.69","DOIUrl":"https://doi.org/10.33232/bims.0062.43.69","url":null,"abstract":"Disjointness preserving operators have been introduced in some form or other in the forty’s. Indeed, linear multiplicative operations in the work [55] by Vulikh are disjointness preserving operators in disguise. However, only during the last decades have they made a formal entry into the development of vector lattices. A first systematic study of disjointness preserving operators goes back to the pioneering note [4] by Abramovich, Veksler, and Koldunov published in the end of the seventies. From then on, the interest in disjointness preserving operators has steadily grown and a series of works devoted to the subject appeared in the literature. In this regard, spectral properties of order bounded disjointness preserving operators were considered in great details in [8, 30]. On the other hand, invertible disjointness preserving operators occupied a prominent role in a vast literature, such as [7, 20, 33, 35] and mainly the remarkable memoir [3] by Abramovich and Kitover. One of the external reasons for the continuing interest in disjointness preserving operators is the fact that precisely the order bounded disjointness preserving operators allow multiplicative representations as weighted composition operators and, more generally, polar decompositions [2, 19, 28, 39]. They thus found applications in the theory of singular and integral equation, dynamical system, and differential equations with delayed time [38, 46, 51]. The present survey on order bounded disjointness preserving operators has two main objectives. First, convince the young researchers in vector lattices that disjointness preserving operators constitute an honorable research activity. Secondly, inform the experts about","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"84 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":"122487181","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":"Rational Maps and Images of Rational Points of Curves over Finite Fields","authors":"R. Guralnick","doi":"10.33232/bims.0050.71.96","DOIUrl":"https://doi.org/10.33232/bims.0050.71.96","url":null,"abstract":"We give a survey and some new results about covers of curves related to images of rational points. In par- ticular, we discuss exceptional covers and exceptional poly- nomials and pairs of covers which have the same image on rational points. Our approach uses group theoretic transla- tions of these problems. A variant of this problem is to study extensions of a number field with the same degree one primes. Dedicated to the memory of my good friend, colleague and collaborator Dennis Estes.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"168-169 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":"121282960","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}
Ameer Athavale, Abhijit Ranjekar, V. M. Sholapurkar
{"title":"On a Class of Alternatingly Hyperexpansive Subnormal Weighted Shifts","authors":"Ameer Athavale, Abhijit Ranjekar, V. M. Sholapurkar","doi":"10.33232/bims.0053.35.52","DOIUrl":"https://doi.org/10.33232/bims.0053.35.52","url":null,"abstract":"If T is a weighted shift operator on a Hilbert space with the associated weight sequence ffingn‚0 of pos- itive weights, then meaningful insights into the nature of T can be gained by examining the sequence fµn(T)gn‚0 where µ0(T) = 1 and µn(T) = ƒ ni1 k=0 fik 2 (n ‚ 1). We characterize those subnormal weighted shifts whose associated µn(T) are interpolated by members of a special subclass of the class of absolutely monotone functions on the non-negative real line. The special subclass has such pleasant properties as being closed under dierentiation and integration. We also attempt to highlight the operator theoretic significance of such characterizations.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"62 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":"126757182","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":"Banach Algebras in Which Every Left Ideal is Countably Generated","authors":"N. Boudi","doi":"10.33232/bims.0048.17.24","DOIUrl":"https://doi.org/10.33232/bims.0048.17.24","url":null,"abstract":"An algebra which is associative or alternative is Noetherian if it satisfies the ascending chain condition on left ideals or equivalently, every left ideal is finitely generated. A well-known result of Sinclair and Tullo [9] states that an associative Noetherian Banach algebra is finite dimensional. This result was extended in [2] to the alternative case. In this paper, we are concerned with associative and alternative Banach algebras in which every left ideal is countably generated. Several papers have appeared dealing with countably generated ideals in some Banach algebras (like in [4], [6]). It should be pointed out that sometimes we have some surprising facts, see for example [7] and [8]. It is clear that we can treat directly the alternative case, since every associative algebra is alternative. But our purpose here is to present the methods rather than the results. And the proof of the alternative case is rather more complicated.","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"164 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":"127374039","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":"Describing Ideals of Endomorphism Rings","authors":"B. Goldsmith, S. Pabst","doi":"10.33232/bims.0039.14.25","DOIUrl":"https://doi.org/10.33232/bims.0039.14.25","url":null,"abstract":"","PeriodicalId":103198,"journal":{"name":"Irish Mathematical Society Bulletin","volume":"12 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":"130406460","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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