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On Tannaka Duality
Martín Szyld
Mathematics , 2011,
Abstract: The purpose of this work is twofold: to expose the existing similarities between the generalizations of the Tannaka and Galois theories, and on the other hand, to develop in detail our own treatment of part of the content of Joyal and Street [1] paper, generalizing from vector spaces to an abstract tensor category. We also develop in detail the proof of the Tannaka equivalence of categories in the case of vector spaces. Saavedra Rivano [2], Deligne and Milne [3] generalize classical Tannaka theory to the context of K-linear tensor (or monoidal) categories. They obtain a lifting-equivalence into a category of \group representations" for a ?nite-dimensional vector space valued monoidal functor. This lifting theorem is similar to the one of Grothendieck Galois theory [4] for a ?nite sets valued functor. On the other hand, Joyal and Street [1] work on the algebraic side of the duality between algebra and geometry, and also obtain a lifting-equivalence, but now to the category of ?nite-dimensional comodules over a Hopf algebra. In this work we follow the ideas of Joyal and Street and prove their lifting-equivalence, and then we show how it corresponds, by dualizing, to the ones of Saavedra Rivano [2] and Deligne and Milne [3]. Text is in spanish, but a brief english introduction is provided.
Tannaka Theory over Sup-Lattices
Martín Szyld
Mathematics , 2015,
Abstract: The main result of this thesis is the construction of a tannakian context over the category of sup-lattices, associated with an arbitrary Grothendieck topos, and the attainment of new results in tannakian representation theory from it. Although many results were obtained and published historically linking Galois and Tannaka theory (see our introduction), these are different and less general since they assume the existence of Galois closures and work on Galois topos rather than on arbitrary topos. Instead we, when talking about Galois theory, mean the extension to arbitrary topos of Joyal and Tierney, critical to get the results of this thesis. The tannakian context associated with a Grothendieck topos is obtained through the process of taking relations of its localic cover. Then, through an investigation and exhaustive comparison of the constructions of the Galois and Tannaka theories, we prove the equivalence of their fundamental recognition theorems (see section 8). Since the (bi)categories of relations of a Grothendieck topos were characterized by Carboni and Walters, a new recognition-type tannakian theorem (theorem 8.12) is obtained, essentially different from those known so far.
A Tannakian Context for Galois
Eduardo J. Dubuc,Martin Szyld
Mathematics , 2011,
Abstract: Strong similarities have been long observed between the Galois (Categories Galoisiennes) and the Tannaka (Categories Tannakiennes) theories of representation of groups. In this paper we construct an explicit (neutral) Tannakian context for the Galois theory of atomic topoi, and prove the equivalence between its fundamental theorems. Since the theorem is known for the Galois context, this yields, in particular, a proof of the fundamental (recognition) theorem for a new Tannakian context. This example is different from the additive cases or their generalization, where the theorem is known to hold, and where the unit of the tensor product is always an object of finite presentation, which is not the case in our context.
Tannaka Theory for Topos
Eduardo J. Dubuc,Martin Szyld
Mathematics , 2015,
Abstract: We consider locales $B$ as algebras in the tensor category $s\ell$ of sup-lattices. We show the equivalence between the Joyal-Tierney descent theorem for open localic surjections $sh(B) \stackrel{q}{\longrightarrow} \mathcal{E}$ in Galois theory [An extension of the Galois Theory of Grothendieck, AMS Memoirs 151] and a Tannakian recognition theorem over $s\ell$ for the $s\ell$-functor $Rel(E) \stackrel{Rel(q^*)}{\longrightarrow} Rel(sh(B)) \cong (B$-$Mod)_0$ into the $s\ell$-category of discrete $B$-modules. Thus, a new Tannaka recognition theorem is obtained, essentially different from those known so far. This equivalence follows from two independent results. We develop an explicit construction of the localic groupoid $G$ associated by Joyal-Tierney to $q$, and do an exhaustive comparison with the Deligne Tannakian construction of the Hopf algebroid $L$ associated to $Rel(q^*)$, and show they are isomorphic, that is, $L \cong \mathcal{O}(G)$. On the other hand, we show that the $s\ell$-category of relations of the classifying topos of any localic groupoid $G$, is equivalent to the $s\ell$-category of $L$-comodules with discrete subjacent $B$-module, where $L = \mathcal{O}(G)$. We are forced to work over an arbitrary base topos because, contrary to the neutral case developed over Sets in [A Tannakian Context for Galois Theory, Advances in Mathematics 234], here change of base techniques are unavoidable.
Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. II. Interior eigenvalues
Daniel B. Szyld,Eugene Vecharynski,Fei Xue
Mathematics , 2015,
Abstract: We consider the solution of large-scale nonlinear algebraic Hermitian eigenproblems of the form $T(\lambda)v=0$ that admit a variational characterization of eigenvalues. These problems arise in a variety of applications and are generalizations of linear Hermitian eigenproblems $Av\!=\!\lambda Bv$. In this paper, we propose a Preconditioned Locally Minimal Residual (PLMR) method for efficiently computing interior eigenvalues of problems of this type. We discuss the development of search subspaces, preconditioning, and eigenpair extraction procedure based on the refined Rayleigh-Ritz projection. Extension to the block methods is presented, and a moving-window style soft deflation is described. Numerical experiments demonstrate that PLMR methods provide a rapid and robust convergence towards interior eigenvalues. The approach is also shown to be efficient and reliable for computing a large number of extreme eigenvalues, dramatically outperforming standard preconditioned conjugate gradient methods.
Asynchronous iterative computations with Web information retrieval structures: The PageRank case
Giorgos Kollias,Efstratios Gallopoulos,Daniel B. Szyld
Computer Science , 2006,
Abstract: There are several ideas being used today for Web information retrieval, and specifically in Web search engines. The PageRank algorithm is one of those that introduce a content-neutral ranking function over Web pages. This ranking is applied to the set of pages returned by the Google search engine in response to posting a search query. PageRank is based in part on two simple common sense concepts: (i)A page is important if many important pages include links to it. (ii)A page containing many links has reduced impact on the importance of the pages it links to. In this paper we focus on asynchronous iterative schemes to compute PageRank over large sets of Web pages. The elimination of the synchronizing phases is expected to be advantageous on heterogeneous platforms. The motivation for a possible move to such large scale distributed platforms lies in the size of matrices representing Web structure. In orders of magnitude: $10^{10}$ pages with $10^{11}$ nonzero elements and $10^{12}$ bytes just to store a small percentage of the Web (the already crawled); distributed memory machines are necessary for such computations. The present research is part of our general objective, to explore the potential of asynchronous computational models as an underlying framework for very large scale computations over the Grid. The area of ``internet algorithmics'' appears to offer many occasions for computations of unprecedent dimensionality that would be good candidates for this framework.
Exposición ocupacional al virus de hepatitis C Occupational exposure to hepatitis C virus
Eduardo Warley,Javier Desse,Edgardo Szyld,Fernando Silva Nieto
Medicina (Buenos Aires) , 2006,
Abstract: La transmisión ocupacional de virus de hepatitis C (VHC) es un área de creciente preocupación dada la falta de profilaxis y la poca información de su prevalencia en el medio hospitalario. Sobre 128 exposiciones ocupacionales ocurridas en el Hospital Diego Paroissien entre 1999 y 2003 hubieron 8 casos de exposición a VHC (6.3%) y un caso de seroconversión posterior a la exposición (0.8%). No existiendo en la actualidad terapia preventiva para VHC resulta de gran interés la posibilidad de tratamiento de la infección aguda. La mayor parte de los autores coincide en recomendar el tratamiento del episodio agudo de hepatitis por VHC basado en la evidencia actual, aunque aún no está bien definida la mejor estrategia diagnóstica y terapéutica. El acatamiento de las Normas de Precauciones Universales sigue siendo en la actualidad la más importante medida preventiva para evitar la infección ocupacional por VHC en el personal de salud y la de mejor equilibrio costo-beneficio. Occupational exposure to Hepatitis C virus is an area of growing concern due to a lack of prophylaxis and limited knowledge regarding prevalence in hospital environment. Between 1999 and 2003, a total of 128 occupational exposures to this virus were registered in Diego Paroissien Hospital, eight of which led to hepatitis (6.3%) and one case to serum conversion (0.8%). Currently there is no preventive therapy and great interest is focused on acute infection therapy the effectiveness of which is still controversial. This study reinforces the fact that adherence to the Universal Precautions is still the most important preventive measure for health care workers, and the most cost beneficial.
Krylov Subspace Recycling for Sequences of Shifted Linear Systems
Kirk M. Soodhalter,Daniel B. Szyld,Fei Xue
Computer Science , 2013, DOI: 10.1016/j.apnum.2014.02.006
Abstract: We study the use of Krylov subspace recycling for the solution of a sequence of slowly-changing families of linear systems, where each family consists of shifted linear systems that differ in the coefficient matrix only by multiples of the identity. Our aim is to explore the simultaneous solution of each family of shifted systems within the framework of subspace recycling, using one augmented subspace to extract candidate solutions for all the shifted systems. The ideal method would use the same augmented subspace for all systems and have fixed storage requirements, independent of the number of shifted systems per family. We show that a method satisfying both requirements cannot exist in this framework. As an alternative, we introduce two schemes. One constructs a separate deflation space for each shifted system but solves each family of shifted systems simultaneously. The other builds only one recycled subspace and constructs approximate corrections to the solutions of the shifted systems at each cycle of the iterative linear solver while only minimizing the base system residual. At convergence of the base system solution, we apply the method recursively to the remaining unconverged systems. We present numerical examples involving systems arising in lattice quantum chromodynamics.
On the Wegener granulomatosis associated region on chromosome 6p21.3
Pawe? Szyld, Peter Jagiello, Elena Csernok, Wolfgang L Gross, Joerg T Epplen
BMC Medical Genetics , 2006, DOI: 10.1186/1471-2350-7-21
Abstract: 150–180 German WG patients and a corresponding cohort of healthy controls (n = 100–261) were used in a two-step study. A panel of 94 microsatellites was designed for the initial step using a DNA pooling approach. Markers with significantly differing allele frequencies between patient and control pools were individually genotyped. The RXRB gene was analysed for single strand conformation polymorphisms (SSCP) and restriction fragment length polymorphisms (RFLP). The splice-site polymorphism in the BTNL2 gene was also investigated by RFLP analysis.A previously investigated microsatellite (#1.0.3.7, Santa Cruz genome browser (UCSC) May 2004 Freeze localisation: chr6:31257596-34999883), which was used as a positive control, remained associated throughout the whole two-step approach. Yet, no additional evidence for association of other microsatellite markers was found in the entire investigated region. Analysis of the RXRB gene located in the WG associated region revealed associations of two variations (rs10548957 pallelic = 0.02 and rs6531 pallelic = 5.20 × 10-5, OR = 1.88). Several alleles of markers located between HLA-DPB1, SNP rs6531 and microsatellite 1.0.3.7 showed linkage disequilibrium with r2 values exceeding 0.10. Significant differences were not demonstrable for the sarcoidosis associated splice-site variation (rs2076530 pallelic = 0.80) in our WG cohort.Since a microsatellite flanking the RXRB gene and two intragenic polymorphisms are associated significantly with WG on chromosome 6p21.3, further investigations should be focussed on extensive fine-mapping in this region by densely mapping with additional markers such as SNPs. This strategy may reveal even deeper insights into the genetic contributions of the respective region for the pathogenesis of WG.Wegener granulomatosis (WG) is a granulomatous disorder belonging to the heterogeneous group of systemic vasculitides (SV). A common feature of SV is the inflammation of the endothelium [1,2]. SV are classified
Prácticas en la asistencia de ni?os nacidos con líquido amniótico meconial en la sala de partos: una encuesta nacional
Aguilar,Adriana M.; Satragno,Daniela S.; Vain,Néstor E.; Szyld,Edgardo G.; Prudent,Luis M.;
Archivos argentinos de pediatr?-a , 2010,
Abstract: based on the results of two large rcts, guidelines from nrp/aap, ilcor, acog, and from argentinean scientific societies for intrapartum and postpartum management of pregnancies with meconium stained amniotic fluid (msaf) have radically changed. however, there are no data as to current delivery room management of infants born through msaf. objectives. 1) to assess current delivery room approaches in infants born through msaf in argentina; 2) to evaluate the adherence to nrp/aap recommendations among different centers. design/methods. a 20-question anonymous web-mail based survey was sent to 121 institutions with ≥ 500 births a year. results. the response rate was 63% representing 182,540 deliveries/year (26% of the total number of births in the country). among the surveyed institutions, 24% were located in the city of buenos aires and 72% were public hospitals. although no longer recommended, the following procedures are frequently performed: 1) oro-nasopharingeal suctioning of meconiumstained infants before the delivery of the shoulders: sometimes, 50% of centers; always, 7 %. 2) routine tracheal suctioning in vigorous infants, 30%. 3) glottis exposure under laryngoscopy, 13 % for thin-consistency msaf and 43% for thick-consistency msaf. on the contrary, and although strongly recommended, only 69% of depressed infants have tracheal suctioning before ventilation. up to 7% of centers perform other non recommended procedures such as chest compression and pressure over the larynx, to avoid breathing, and gastric lavage. conclusions. in spite of not being currently recommended, intrapartum suction and postnatal intubation and suction of vigorous infants with msaf are commonly performed in argentina while in depressed newborns the latter procedure is not always carried out. our study confirms the frequent gaps observed between scientific evidence, guidelines and clinical practice.
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