Daily temperature variations induce phase transitions and lattice strains in halide perovskites, challenging their stability in solar cells. We stabilized the perovskite black phase and improved solar cell performance using the ordered dipolar structure of β-poly(1,1-difluoroethylene) to control perovskite film crystallization and energy alignment. We demonstrated p-i-n perovskite solar cells with a record power conversion efficiency of 24.6% over 18 square millimeters and 23.1% over 1 square centimeter, which retained 96 and 88% of the efficiency after 1000 hours of 1-sun maximum power point tracking at 25° and 75°C, respectively. Devices under rapid thermal cycling between −60° and +80°C showed no sign of fatigue, demonstrating the impact of the ordered dipolar structure on the operational stability of perovskite solar cells.

We revisit localisation and patching method in the setting of Chevalley groups. Introducing certain subgroups of relative elementary Chevalley groups, we develop relative versions of the conjugation calculus and the commutator calculus in Chevalley groups G(Φ, R), rk(Φ) ≥ 2, which are both more general, and substantially easier than the ones available in the literature. For classical groups such relative commutator calculus has been recently developed by the authors in [34,33]. As an application we prove the mixed commutator formula,for two ideals a, b R. This answers a problem posed in a paper by Alexei Stepanov and the second author. O Life, you put thousand traps in my wayDare to try, is what you clearly say Omar Khayam IntroductionOne of the most powerful ideas in the study of groups of points of reductive groups over rings is localisation. It allows to reduce many important problems over arbitrary commutative rings, to similar problems for semi-local rings. Localisation comes in a number of versions. The two most familiar ones are localisation and patching, proposed by Daniel Quillen [55] and Andrei Suslin [65], and localisationcompletion, proposed by Anthony Bak [8].Originally, the above papers addressed the case of the general linear group GL(n, R). Soon thereafter, Suslin himself, Vyacheslav Kopeiko, Marat Tulenbaev, Giovanni Taddei, Leonid Vaserstein, Li Fuan, Eiichi Abe, You Hong, and others proposed workingThe work of the second author was supported by RFFI projects

This note revisits localisation and patching method in the setting of generalised unitary groups. Introducing certain subgroups of relative elementary unitary groups, we develop relative versions of the conjugation calculus and the commutator calculus in unitary groups, which are both more general, and substantially easier than the ones available in the literature. For the general linear group such relative commutator calculus has been recently developed by the first and the third authors. As an application we prove the mixed commutator formula, [EU(2n, I, Γ), GU(2n, J, ∆)] = [EU(2n, I, Γ), EU(2n, J, ∆)], for two form ideals (I, Γ) and (J, ∆) of a form ring (A, Λ). This answers two problems posed in a paper by Alexei Stepanov and the second author. 1 • Absolute standard unitary commutator formulae, Bak-Vavilov [9], Theorem 1.1 and Vaserstein-Hong You [50].• Relative unitary commutator formula at the stable level, under some additional stability assumptions, Habdank [18,19].• Relative commutator formula for the general linear group GL(n, R), 53,26]. This case is obtained, as one sets in our Theorem, A = R⊕R 0 .Observe, that in the above generality (relative, without stability conditions) our results are new already for the following familiar cases.• The case of symplectic groups Sp(2l, R), when the involution is trivial, and Λ = R.• The case of split orthogonal groups SO(2l, R), when the involution is trivial and Λ = 0.• The case of classical unitary groups SU(2l, R), when Λ = Λ max . See [20] §5.2B for further discussion on the generalised unitary groups.Actually, in § §8,9 we give another proof of Theorem 1, imitating that of [53]. Namely, we show, that Theorem 1 can be deduced from the absolute standard commutator formula by careful calculation of levels of the above commutator groups, and some group-theoretic arguments.Nevertheless, we believe that our localisation proof, based on the relative conjugation calculus and commutator calculus, we develop in § §5,6 of the present paper, and especially the calculations themselves, are of independent value, and will be used in many further applications.The paper is organised as follows. In § §2-4 we recall basic notation, and some background facts, used in the sequel. The next two sections constitute the technical core of the paper. Namely, in §5, and in §6 we develop relative unitary conjugation calculus, and relative unitary

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Let A be a quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. Let Ii, i = 0, ..., m, be two-sided ideals of A, GLn(A, Ii) the principal congruence subgroup of level Ii in GLn(A) and En(A, Ii) be the relative elementary subgroup of level Ii. We prove a multiple commutator formulaEn (A, I0), GLn(A, I1), GLn(A, I2), . . . , GLn(A, Im) = En(A, I0), En(A, I1), En(A, I2), . . . , En(A, Im) , which is a broad generalization of the standard commutator formulas. This result contains all the published results of commutator formulas over commutative rings and answers a problem posed by A. Stepanov and N. Vavilov (cf. Problem 4 in [24]).