Tilting mutation of weakly symmetric algebras and stable equivalence

We consider tilting mutations of a weakly symmetric algebra at a subset of simple modules, as recently introduced by T. Aihara. These mutations are defined as the endomorphism rings of certain tilting complexes of length 1. Starting from a weakly symmetric algebra A, presented by a quiver with relations, we give a detailed description of the quiver and relations of the algebra obtained by mutating at a single loopless vertex of the quiver of A. In this form the mutation procedure appears similar to, although significantly more complicated than, the mutation procedure of Derksen, Weyman and Zelevinsky for quivers with potentials. By definition, weakly symmetric algebras connected by a sequence of tilting mutations are derived equivalent, and hence stably equivalent. The second aim of this article is to study these stable equivalences via a result of Okuyama describing the images of the simple modules. As an application we answer a question of Asashiba on the derived Picard groups of a class of self-injective algebras of finite representation type. We conclude by introducing a mutation procedure for maximal systems of orthogonal bricks in a triangulated category, which is motivated by the effect that a tilting mutation has on the set of simple modules in the stable category.Comment: Description and proof of mutated algebra made more rigorous (Prop. 3.1 and 4.2). Okuyama's Lemma incorporated: Theorem 4.1 is now Corollary 5.1, and proof is omitted. To appear in Algebras and Representation Theor

The homotopy theory of dg-categories and derived Morita theory

The main purpose of this work is the study of the homotopy theory of dg-categories up to quasi-equivalences. Our main result provides a natural description of the mapping spaces between two dg-categories $C$ and $D$ in terms of the nerve of a certain category of $(C,D)$-bimodules. We also prove that the homotopy category $Ho(dg-Cat)$ is cartesian closed (i.e. possesses internal Hom's relative to the tensor product). We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dg-categories of modules over two dg-categories $C$ and $D$ as the dg-category of $(C,D)$-bi-modules. Finally, we give three applications of our results. The first one expresses Hochschild cohomology as endomorphisms of the identity functor, as well as higher homotopy groups of the \emph{classifying space of dg-categories} (i.e. the nerve of the category of dg-categories and quasi-equivalences between them). The second application is the existence of a good theory of localization for dg-categories, defined in terms of a natural universal property. Our last application states that the dg-category of (continuous) morphisms between the dg-categories of quasi-coherent (resp. perfect) complexes on two schemes (resp. smooth and proper schemes) is quasi-equivalent to the dg-category of quasi-coherent complexes (resp. perfect) on their product.Comment: 50 pages. Few mistakes corrected, and some references added. Thm. 8.15 is new. Minor corrections. Final version, to appear in Inventione

Gastric Outlet Obstruction among Adult Patient at Two Rwandan Referral Hospitals: Etiology. H. pylori Infection and Outcomes

No Abstract

Seasonal dependence of peroxy radical concentrations at a Northern hemisphere marine boundary layer site during summer and winter: evidence for radical activity in winter

Peroxy radicals (HO2+Σ RO2) were measured at the Weybourne Atmospheric Observatory (52° N, 1° E), Norfolk using a PEroxy Radical Chemical Amplifier (PERCA) during the winter and summer of 2002. The peroxy radical diurnal cycles showed a marked difference between the winter and summer campaigns with maximum concentrations of 12 pptv at midday in the summer and maximum concentrations as high as 30 pptv (10 min averages) in winter at night. The corresponding nighttime peroxy radical concentrations were not as high in summer (3 pptv). The peroxy radical concentration shows a distinct anti-correlation with increasing NOx during the daylight hours. At night, peroxy radicals increase with increasing NOx indicative of the role of NO3 chemistry. The average diurnal cycles for net ozone production, N(O3) show a large variability in ozone production, P(O3), and a large ozone loss, L(O3) in summer relative to winter. For a daylight average, net ozone production in summer was higher than winter (1.51±0.5 ppbv h−1 and 1.11±0.47 ppbv h−1, respectively). The variability in NO concentration has a much larger effect on N(O3) than the peroxy radical concentrations. Photostationary state (PSS) calculations show an NO2 lifetime of 5 min in summer and 21 minutes in the winter, implying that steady-state NO-NO2 ratios are not always attained during the winter months. The results show an active peroxy radical chemistry at night and that significant oxidant levels are sustained in winter. The net effect of this with respect to production of ozone in winter is unclear owing to the breakdown in the photostationary state

H3+ in Diffuse Interstellar Clouds: a Tracer for the Cosmic-Ray Ionization Rate

Using high resolution infrared spectroscopy we have surveyed twenty sightlines for H3+ absorption. H3+ is detected in eight diffuse cloud sightlines with column densities varying from 0.6x10^14 cm^-2 to 3.9x10^14 cm^-2. This brings to fourteen the total number of diffuse cloud sightlines where H3+ has been detected. These detections are mostly along sightlines concentrated in the Galactic plane, but well dispersed in Galactic longitude. The results imply that abundant H3+ is common in the diffuse interstellar medium. Because of the simple chemistry associated with H3+ production and destruction, these column density measurements can be used in concert with various other data to infer the primary cosmic-ray ionization rate, zeta_p. Values range from 0.5x10^-16 s^-1 to 3x10^-16 s^-1 with an average of 2x10^-16 s^-1. Where H3+ is not detected the upper limits on the ionization rate are consistent with this range. The average value of zeta_p is about an order of magnitude larger than both the canonical rate and rates previously reported by other groups using measurements of OH and HD. The discrepancy is most likely due to inaccurate measurements of rate constants and the omission of effects which were unknown when those studies were performed. We believe that the observed column density of H3+ is the most direct tracer for the cosmic-ray ionization rate due to its simple chemistry. Recent models of diffuse cloud chemistry require cosmic-ray ionization rates on the order of 10^-16 s^-1 to reproduce observed abundances of various atomic and molecular species, in rough accord with our observational findings.Comment: Accepted to ApJ, 35 pages, 5 figures, 5 table

On Morita and derived equivalences for cohomological Mackey algebras

By results of the second author, a source algebra equivalence between two p-blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories ofcohomological Mackey functors. The main result of this paper proves a partial converse: an equivalence (resp. Rickard equivalence) between the categories of cohomological Mackey functors of two blocks of finite groups induces a permeable Morita (resp. derived) equivalence between the two block algebras

Torsion pairs and simple-minded systems in triangulated categories

Let T be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu, we say that a family S of pairwise orthogonal objects in T with trivial endomorphism rings is a simple-minded system if its closure under extensions is all of T. We construct torsion pairs in T associated to any subset X of a simple-minded system S, and use these to define left and right mutations of S relative to X. When T has a Serre functor \nu, and S and X are invariant under \nu[1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where T is the stable module category of a self-injective algebra \Lambda. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in the derived category of \Lambda. It follows that the mutation of the set of simple \Lambda-modules relative to X yields the images of the simple \Gamma-modules under a stable equivalence between \Gamma\ and \Lambda, where \Gamma\ is the tilting mutation of \Lambda\ relative to X.Comment: Minor corrections. To appear in Applied Categorical Structures. The final publication is available at springerlink.com: http://link.springer.com/article/10.1007%2Fs10485-014-9365-

Estimation of rate coefficients and branching ratios for gas-phase reactions of OH with aromatic organic compounds for use in automated mechanism construction

Reaction with the hydroxyl (OH) radical is the dominant removal process for volatile organic compounds (VOCs) in the atmosphere. Rate coefficients for the reactions of OH with VOCs are therefore essential parameters for chemical mechanisms used in chemistry transport models, and are required more generally for impact assessments involving estimation of atmospheric lifetimes or oxidation rates for VOCs. A structure–activity relationship (SAR) method is presented for the reactions of OH with aromatic organic compounds, with the reactions of aliphatic organic compounds considered in the preceding companion paper. The SAR is optimized using a preferred set of data including reactions of OH with 67 monocyclic aromatic hydrocarbons and oxygenated organic compounds. In each case, the rate coefficient is defined in terms of a summation of partial rate coefficients for H abstraction or OH addition at each relevant site in the given organic compound, so that the attack distribution is defined. The SAR can therefore guide the representation of the OH reactions in the next generation of explicit detailed chemical mechanisms. Rules governing the representation of the reactions of the product radicals under tropospheric conditions are also summarized, specifically the rapid reaction sequences initiated by their reactions with O2

Topological Orthoalgebras

We define topological orthoalgebras (TOAs) and study their properties. While every topological orthomodular lattice is a TOA, the lattice of projections of a Hilbert space is an example of a lattice-ordered TOA that is not a toplogical lattice. On the other hand, we show that every compact Boolean TOA is a topological Boolean algebra. We also show that a compact TOA in which 0 is an isolated point is atomic and of finite height. We identify and study a particularly tractable class of TOAs, which we call {\em stably ordered}: those in which the upper-set generated by an open set is open. This includes all topological OMLs, and also the projection lattices of Hilbert spaces. Finally, we obtain a topological version of the Foulis-Randall representation theory for stably ordered TOAsComment: 16 pp, LaTex. Minor changes and corrections in sections 1; more substantial corrections in section

Derived equivalence classification of the cluster-tilted algebras of Dynkin type E

We obtain a complete derived equivalence classification of the cluster-tilted algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E6, E7 and E8 which turn out to fall into 6, 14, 15 derived equivalence classes, respectively. This classification can be achieved computationally and we outline an algorithm which has been implemented to carry out this task. We also make the classification explicit by giving standard forms for each derived equivalence class as well as complete lists of the algebras contained in each class; as these lists are quite long they are provided as supplementary material to this paper. From a structural point of view the remarkable outcome of our classification is that two cluster-tilted algebras of Dynkin type E are derived equivalent if and only if their Cartan matrices represent equivalent bilinear forms over the integers which in turn happens if and only if the two algebras are connected by a sequence of "good" mutations. This is reminiscent of the derived equivalence classification of cluster-tilted algebras of Dynkin type A, but quite different from the situation in Dynkin type D where a far-reaching classification has been obtained using similar methods as in the present paper but some very subtle questions are still open.Comment: 19 pages. v4: completely rewritten version, to appear in Algebr. Represent. Theory. v3: Main theorem strengthened by including "good" mutations (cf. also arXiv:1001.4765). Minor editorial changes. v2: Third author added. Major revision. All questions left open in the earlier version by the first two authors are now settled in v2 and the derived equivalence classification is completed. arXiv admin note: some text overlap with arXiv:1012.466