Background Against a long-term trend of increasing demand, the COVID-19 pandemic has led to a global rise in common mental disorders. Now more than ever, there is an urgent need for scalable, evidence-based interventions to support mental well-being. Objective The aim of this proof-of-principle study was to evaluate the efficacy of a mobile-based app in adults with self-reported symptoms of anxiety and stress in a randomized control trial that took place during the first wave of the COVID-19 pandemic in the United Kingdom. Methods Adults with mild to severe anxiety and moderate to high levels of perceived stress were randomized to either the intervention or control arm. Participants in the intervention arm were given access to the Foundations app for the duration of the 4-week study. All participants were required to self-report a range of validated measures of mental well-being (10-item Connor-Davidson Resilience scale [CD-RISC-10], 7-item Generalized Anxiety Disorder scale [GAD-7], Office of National Statistics Four Subjective Well-being Questions [ONS-4], World Health Organization-5 Well-Being Index [WHO-5]) and sleep (Minimal Insomnia Scale [MISS]) at baseline and at weeks 2 and 4. The self-reported measures of perceived stress (10-item Perceived Stress Score [PSS-10]) were obtained weekly. Results A total of 136 participants completed the study and were included in the final analysis. The intervention group (n=62) showed significant improvements compared to the control group (n=74) on measures of anxiety, with a mean GAD-7 score change from baseline of –1.35 (SD 4.43) and –0.23 (SD 3.24), respectively (t134=1.71, P=.04); resilience, with a mean change in CD-RISC score of 1.79 (SD 4.08) and –0.31 (SD 3.16), respectively (t134=–3.37, P<.001); sleep, with a mean MISS score change of –1.16 (SD 2.67) and –0.26 (SD 2.29), respectively (t134=2.13, P=.01); and mental well-being, with a mean WHO-5 score change of 1.53 (SD 5.30) and –0.23 (SD 4.20), respectively (t134=–2.16, P=.02), within 2 weeks of using Foundations, with further improvements emerging at week 4. Perceived stress was also reduced within the intervention group, although the difference did not reach statistical significance relative to the control group, with a PSS score change from baseline to week 2 of –2.94 (SD 6.84) and –2.05 (SD 5.34), respectively (t134= 0.84, P=.20). Conclusions This study provides a proof of principle that the digital mental health app Foundations can improve measures of mental well-being, anxiety, resilience, and sleep within 2 weeks of use, with greater effects after 4 weeks. Foundations therefore offers potential as a scalable, cost-effective, and accessible solution to enhance mental well-being, even during times of crisis such as the COVID-19 pandemic. Trial Registration OSF Registries osf.io/f6djb; https://osf.io/vm3xq

Abstract. In this paper we introduce certain "combinatorial sheaves" on posets, which we call weighted sheaves, and we relate their cohomology, computed by a weighted complex, with the twisted cohomology of Artin groups. It turns out that to each Artin group one can associate a weighted sheaf, where the poset is given by the simplicial complex of all finite parabolic subgroups, and the cohomology of the Artin group with coefficients in a module of Laurent polynomials (interesting for geometrical reasons) is computed by the associated weighted complex. We connect this theory with the so called "Discrete Morse Theory": a weighted matching on the weighted complex gives rise to a Morse complex computing the cohomology. We give a natural filtration of the weighted complex, which is compatible with the weighted matching, so obtaining a converging spectral sequence. We use such machinery to compute the twisted conomology for all exceptional type affine Artin groups.

Abstract. The C 1 -harmonic capacity κ c plays a central role in problems of approximation by harmonic functions in the C 1 -norm in R n+1 . In this paper we prove the comparability between the capacity κ c and its positive version κ c + . As a corollary, we deduce the semiadditivity of κ c . This capacity can be considered as a generalization in R n+1 of the continuous analytic capacity α in C. Moreover, we also show that the so-called inner boundary conjecture fails for dimensions n > 1, unlike in the case n = 1.

In the first part we review some topological and algebraic aspects in the theory of Artin and Coxeter groups, both in the finite and infinite case (but still, finitely generated). In the following parts, among other things, we compute the Schwartz genus of the covering associated to the orbit space for all affine Artin groups. We also give a partial computation of the cohomology of the braid group with non-abelian coefficients coming from geometric representations. We introduce an interesting class of "sheaves over posets", which we call "weighted sheaves over posets", and use them for explicit computations.

In this paper we prove that, given s ≥ 0, and a Borel non zero measure µ in R m , if for µ-almost every x ∈ R m the limit lim ε→0 |x−y|>ε x − y |x − y| s+1 dµ(y) exists and 0 < lim sup r →0 µ(B(x, r))/r s < ∞, then s in an integer. In particular, if E ⊂ R m is a set with positive and bounded s-dimensional Hausdorff measure H s and for H s-almost every x ∈ E the limit lim ε→0 |x−y|>ε x − y |x − y| s+1 dH s |E (y) exists, then s is an integer.