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The efficient and modular diversification of molecular scaffolds, particularly for the synthesis of diverse molecular libraries, remains a significant challenge in drug optimization campaigns. The late-stage introduction of alkyl fragments is especially desirable due to the high sp³-character and structural versatility of these motifs. Given their prevalence in molecular frameworks, C(sp²)−H bonds serve as attractive targets for diversification, though this process often requires difficult pre-functionalization or lengthy de novo syntheses. Traditionally, direct alkylations of arenes are achieved by employing Friedel–Crafts reaction conditions using strong Brønsted or Lewis acids. However, these methods suffer from poor functional group tolerance and low selectivity, limiting their broad implementation in late-stage functionalization and drug optimization campaigns. Herein, we report the application of a novel strategy for the selective coupling of differently hybridized radical species, which we term dynamic orbital selection. This mechanistic paradigm overcomes common limitations of Friedel-Crafts alkylations via the in situ formation of two distinct radical species, which are subsequently differentiated by a copper-based catalyst based on their respective binding properties. As a result, we demonstrate herein a general and highly modular reaction for the direct alkylation of native arene C−H bonds using abundant and benign alcohols and carboxylic acids as the alkylating agents. Ultimately, this solution overcomes the synthetic challenges associated with the introduction of complex alkyl scaffolds into highly sophisticated drug scaffolds in a late-stage fashion, thereby granting access to vast new chemical space. Based on the generality of the underlying coupling mechanism, dynamic orbital selection is expected to be a broadly applicable coupling platform for further challenging transformations involving two distinct radical species. 

A realization is a triple, (A,b,c), consisting of a d−tuple, A=(A1,⋯,Ad), d∈N, of bounded linear operators on a separable, complex Hilbert space, H, and vectors b,c∈H. Any such realization defines an analytic non-commutative (NC) function in an open neighbourhood of the origin, 0:=(0,⋯,0), of the NC universe of d−tuples of square matrices of any fixed size. For example, a univariate realization, i.e., where A is a single bounded linear operator, defines a holomorphic function of a single complex variable, z, in an open neighbourhood of the origin via the realization formula b∗(I−zA)−1c . It is well known that an NC function has a finite-dimensional realization if and only if it is a non-commutative rational function that is defined at 0 . Such finite realizations contain valuable information about the NC rational functions they generate. By extending to infinite-dimensional realizations, we construct, study and characterize more general classes of analytic NC functions. In particular, we show that an NC function is (uniformly) entire if and only if it has a jointly compact and quasinilpotent realization. Restricting our results to one variable shows that a formal Taylor series extends globally to an entire or meromorphic function in the complex plane, C, if and only if it has a realization whose component operator is compact and quasinilpotent, or compact, respectively. This motivates our definition of the field of global (uniformly) meromorphic NC functions as the field of fractions generated by NC rational expressions in the ring of NC functions with jointly compact realizations. This definition recovers the field of meromorphic functions in C when restricted to one variable. 

We extend results on complex analytic measures on the complex unit circle to a non-commutative multivariate setting. Identifying continuous linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz C ∗ - algebra, the free disk operator system, with non-commutative (NC) analogues of complex measures, we refine a previously developed Lebesgue decompo- sition for positive NC measures to establish an NC version of the Frigyes and Marcel Riesz Theorem for “analytic” measures, i.e. complex measures with vanishing positive moments. The proof relies on novel results on the order properties of positive NC measures that we develop and extend from classical measure theory. 

Alcohols represent a functional group class with unparalleled abundance and structural diversity. In an era of chemical synthesis that prioritizes reducing time to target and maximizing exploration of chemical space, harnessing these building blocks for carbon-carbon bond-forming reactions is a key goal in organic chemistry. In particular, leveraging a single activation mode to form a new C(sp³)–C(sp³) bond from two alcohol subunits would enable access to an extraordinary level of structural diversity. In this work, we report a nickel radical sorting–mediated cross-alcohol coupling wherein two alcohol fragments are deoxygenated and coupled in one reaction vessel, open to air. 

We extend the methodology in Yang et al. [SIAM J. Appl. Dyn. Syst. 22, 269–310 (2023)] to learn autonomous continuous-time dynamical systems from invariant measures. The highlight of our approach is to reformulate the inverse problem of learning ODEs or SDEs from data as a PDE-constrained optimization problem. This shift in perspective allows us to learn from slowly sampled inference trajectories and perform uncertainty quantification for the forecasted dynamics. Our approach also yields a forward model with better stability than direct trajectory simulation in certain situations. We present numerical results for the Van der Pol oscillator and the Lorenz-63 system, together with real-world applications to Hall-effect thruster dynamics and temperature prediction, to demonstrate the effectiveness of the proposed approach. 

Abstract We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over $$\mathbb {C} ^d$$ is defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov–Clark measure theory for noncommutative and contractive rational multipliers. Noncommutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz algebra, the unital $C^*$ -algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz–Toeplitz and Cuntz algebras, and the emerging field of noncommutative rational functions.