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In this paper we construct an abelian category of mixed perverse sheaves attached to any realization of a Coxeter group, in terms of the associated Elias–Williamson diagrammatic category. This construction extends previous work of the first two authors, where we worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias–Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization.
Third-wave psychological interventions have gained relevance in mental health service provision but their application to people with psychosis is in its infancy and interventions targeting wellbeing in psychosis are scarce. This study tested the feasibility and preliminary effectiveness of positive psychotherapy adapted for people with psychosis (WELLFOCUS PPT) to improve wellbeing.
Methods.
WELLFOCUS PPT was tested as an 11-week group intervention in a convenience sample of people with psychosis in a single centre randomised controlled trial (ISRCTN04199273) involving 94 people with psychosis. Patients were individually randomised in blocks to receive either WELLFOCUS PPT in addition to treatment as usual (TAU), or TAU only. Assessments took place before randomisation and after the therapy. The primary outcome was wellbeing (Warwick-Edinburgh Mental Well-Being Scale, WEMWBS). Secondary outcomes included symptoms (Brief Psychiatric Rating Scale), depression (Short Depression-Happiness Scale), self-esteem, empowerment, hope, sense of coherence, savouring beliefs and functioning, as well as two alternative measures of wellbeing (the Positive Psychotherapy Inventory and Quality of Life). Intention-to-treat analysis was performed. This involved calculating crude changes and paired-sample t-tests for all variables, as well as ANCOVA and Complier Average Causal Effect (CACE) Analysis to estimate the main effect of group on all outcomes.
Results.
The intervention and trial procedures proved feasible and well accepted. Crude changes between baseline and follow-up showed a significant improvement in the intervention group for wellbeing according to all three concepts assessed (i.e., WEMWBS, Positive Psychotherapy Inventory and Quality of Life), as well as for symptoms, depression, hope, self-esteem and sense of coherence. No significant changes were observed in the control group. ANCOVA showed no main effect on wellbeing according to the primary outcome scale (WEMWBS) but significant effects on symptoms (p = 0.006, ES = 0.42), depression (p = 0.03, ES = 0.38) and wellbeing according to the Positive Psychotherapy Inventory (p = 0.02, ES = 0.30). Secondary analysis adapting for therapy group further improved the results for symptom reduction (p = 0.004, ES = 0.43) and depression (p = 0.03, ES = 0.41) but did not lead to any more outcomes falling below the p = 0.05 significance level. CACE analysis showed a non-significant positive association between the intervention and WEMWBS scores at follow-up (b = 0.21, z = 0.9, p = 0.4).
Conclusions.
This study provides initial evidence on the feasibility of WELLFOCUS PPT in people with psychosis, positively affecting symptoms and depression. However, more work is needed to optimise its effectiveness. Future research might evaluate positive psychotherapy as a treatment for comorbid depression in psychosis, and consider alternative measurements of wellbeing.
We describe the equivariant cohomology of cofibers of spherical perverse sheaves on the affine Grassmannian of a reductive algebraic group in terms of the geometry of the Langlands dual group. In fact we give two equivalent descriptions: one in terms of $\mathscr{D}$-modules of the basic affine space, and one in terms of intertwining operators for universal Verma modules. We also construct natural collections of isomorphisms parameterized by the Weyl group in these three contexts, and prove that they are compatible with our isomorphisms. As applications we reprove some results of the first author and of Braverman and Finkelberg.
We prove an analogue of Koszul duality for category $ \mathcal{O} $ of a reductive group $G$ in positive characteristic $\ell $ larger than $1$ plus the number of roots of $G$. However, there are no Koszul rings, and we do not prove an analogue of the Kazhdan–Lusztig conjectures in this context. The main technical result is the formality of the dg-algebra of extensions of parity sheaves on the flag variety if the characteristic of the coefficients is at least the number of roots of $G$ plus $2$.
We present an analysis of the properties of H i holes detected in 20 galaxies that are part of “The H i Nearby Galaxy Survey”. We detected more than 1000 holes in total in the sampled galaxies. The holes are found throughout the disks of the galaxies, out to the edge of the H i disk. We find that shear limits the age of holes in spirals. Shear is less important in dwarf galaxies which explains why H i holes in dwarfs are rounder, on average than in spirals. Shear is particularly strong in the inner part of spiral galaxies, limiting the lifespan of holes there and explaining why we find that holes outside R25 are larger and older. We proceed to derive the surface and volume porosity and find that this correlates with the type of the host galaxy: later Hubble types tend to be more porous. The size distribution of the holes in our sample follows a power law with a slope of aν ~ −2.9. Assuming that the holes are the result of massive star formation, we derive values for the supernova rate (SNR) and star formation rate (SFR) which scales with the SFR derived based on other tracers. If we extrapolate the observed number of holes to include those that fall below our resolution limit, down to holes created by a single supernova, we find that our results are compatible with the hypothesis that H i holes result from star formation.
In this paper we construct, for F1 and F2 subbundles of a vector bundle E, a ‘Koszul duality’ equivalence between derived categories of 𝔾m-equivariant coherent(dg-)sheaves on the derived intersection , and the corresponding derived intersection . We also propose applications to Hecke algebras.
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