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We examine the claimed observations of a gravitational external field effect (EFE) reported by Chae et al. We show that observations suggestive of the EFE can be interpreted without violating Einstein’s equivalence principle, namely from known correlations between the morphology, the environment, and dynamics of galaxies. While Chae et al.’s analysis provides a valuable attempt at a clear test of modified Newtonian dynamics, an evidently important topic, a re-analysis of the observational data does not permit us to confidently assess the presence of an EFE or to distinguish this interpretation from that proposed in this article. 

The kernel $$\mathcal{K}^{\operatorname{st}}$$ of a descent statistic $$\operatorname{st}$$, introduced by Grinberg, is a subspace of the algebra $$\operatorname{QSym}$$ of quasisymmetric functions defined in terms of $$\operatorname{st}$$-equivalent compositions, and is an ideal of $$\operatorname{QSym}$$ if and only if $$\operatorname{st}$$ is shuffle-compatible. This paper continues the study of kernels of descent statistics, with emphasis on the peak set $$\operatorname{Pk}$$ and the peak number $$\operatorname{pk}$$. The kernel $$\mathcal{K}^{\operatorname{Pk}}$$ in particular is precisely the kernel of the canonical projection from $$\operatorname{QSym}$$ to Stembridge's algebra of peak quasisymmetric functions, and is the orthogonal complement of Nyman's peak algebra. We prove necessary and sufficient conditions for obtaining spanning sets and linear bases for the kernel $$\mathcal{K}^{\operatorname{st}}$$ of any descent statistic $$\operatorname{st}$$ in terms of fundamental quasisymmetric functions, and give characterizations of $$\mathcal{K}^{\operatorname{Pk}}$$ and $$\mathcal{K}^{\operatorname{pk}}$$ in terms of the fundamental basis and the monomial basis of $$\operatorname{QSym}$$. Our results imply that the peak set and peak number statistics are $$M$$-binomial, confirming a conjecture of Grinberg. 

Abstract One of the most important problems vexing the ΛCDM cosmological model is the Hubble tension. It arises from the fact that measurements of the present value of the Hubble parameter performed with low-redshift quantities, e.g. the Type IA supernova, tend to yield larger values than measurements from quantities originating at high-redshift, e.g. fits of cosmic microwave background radiation. It is becoming likely that the discrepancy, currently standing at 5σ, is not due to systematic errors in the measurements. Here we explore whether the self-interaction of gravitational fields in General Relativity, which are traditionally neglected when studying the evolution of the Universe, can contribute to explaining the tension. We find that with field self-interaction accounted for, both low- and high-redshift data aresimultaneouslywell-fitted, thereby showing that gravitational self-interaction yield consistentH₀values when inferred from SnIA and cosmic microwave background observations. Crucially, this is achieved without introducing additional parameters. 

This Perspective evaluates recent progress in modeling nature–society systems to inform sustainable development. We argue that recent work has begun to address longstanding and often-cited challenges in bringing modeling to bear on problems of sustainable development. For each of four stages of modeling practice—defining purpose, selecting components, analyzing interactions, and assessing interventions—we highlight examples of dynamical modeling methods and advances in their application that have improved understanding and begun to inform action. Because many of these methods and associated advances have focused on particular sectors and places, their potential to inform key open questions in the field of sustainability science is often underappreciated. We discuss how application of such methods helps researchers interested in harnessing insights into specific sectors and locations to address human well-being, focus on sustainability-relevant timescales, and attend to power differentials among actors. In parallel, application of these modeling methods is helping to advance theory of nature–society systems by enhancing the uptake and utility of frameworks, clarifying key concepts through more rigorous definitions, and informing development of archetypes that can assist hypothesis development and testing. We conclude by suggesting ways to further leverage emerging modeling methods in the context of sustainability science. 

We derive conditions for a nonholonomic system subject to nonlinear constraints (obeying Chetaev's rule) to preserve a smooth volume form. When applied to affine constraints, these conditions dictate that a basic invariant density exists if and only if a certain 1-form is exact and a certain function vanishes (this function automatically vanishes for linear constraints). Moreover, this result can be extended to geodesic flows for arbitrary metric connections and the sufficient condition manifests as integrability of the torsion. As a consequence, volume-preservation of a nonholonomic system is closely related to the torsion of the nonholonomic connection. Examples of nonlinear/affine/linear constraints are considered.