Differentiable spaces that are subcartesian

We show that the differential structure of the orbit space of a proper action of a Lie group on a smooth manifold is weakly reflexive. This implies that the orbit space is a differentiable space in the sense of Smith, which ensures that the orbit space has an exterior algebra of differential forms, that satisfies Smith’s version of de Rham’s theorem. Because the orbit space is a locally closed subcartesian space, it has vector fields and their flows.     

Fabricating Dual-Atom Iron Catalysts for Efficient Oxygen Evolution Reaction: A Heteroatom Modulator Approach

Understanding the thermal aggregation behavior of metal atoms is important for the synthesis of supported metal clusters. Here, derived from a metal–organic framework encapsulating a trinuclear Fe^III₂Fe^II complex (denoted as Fe₃) within the channels, a well-defined nitrogen-doped carbon layer is fabricated as an ideal support for stabilizing the generated iron nanoclusters. Atomic replacement of Fe^II by other metal(II) ions (e.g., Zn^II/Co^II) via synthesizing isostructural trinuclear-complex precursors (Fe₂Zn/Fe₂Co), namely the “heteroatom modulator approach”, is inhibiting the aggregation of Fe atoms toward nanoclusters with formation of a stable iron dimer in an optimal metal–nitrogen moiety, clearly identified by direct transmission electron microscopy and X-ray absorption fine structure analysis. The supported iron dimer, serving as cooperative metal–metal site, acts as efficient oxygen evolution catalyst. Our findings offer an atomic insight to guide the future design of ultrasmall metal clusters bearing outstanding catalytic capabilities.     

Mass distribution and skewness for passive scalar transport in pipes with polygonal and smooth cross sections

We extend our previous results characterizing the loading properties of a diffusing passive scalar advected by a laminar shear flow in ducts and channels to more general cross‐sectional shapes, including regular polygons and smoothed corner ducts originating from deformations of ellipses. For the case of the triangle and localized, cross‐wise uniform initial distributions, short‐time skewness is calculated exactly to be positive, while long‐time asymptotics shows it to be negative. Monte Carlo simulations confirm these predictions, and document the timescale for sign change. The equilateral triangle appears to be the only regular polygon with this property—all others possess positive skewness at all times. Alternatively, closed‐form flow solutions can be constructed for smooth deformations of ellipses, and illustrate how both nonzero short‐time skewness and the possibility of multiple sign switching in time is unrelated to domain corners. Exact conditions relating the median and the skewness to the mean are developed which guarantee when the sign for the skewness implies front (more mass to the right of the mean) or back (more mass to the left of the mean) “loading” properties of the evolving tracer distribution along the pipe. Short‐ and long‐time asymptotics confirm this condition, and Monte Carlo simulations verify this at all times. The simulations are also used to examine the role of corners and boundaries on the distribution for short‐time evolution of point source , as opposed to cross‐wise uniform, initial data.     

Strategies for Improving the Catalytic Performance of 2D Covalent Organic Frameworks for Hydrogen Evolution and Oxygen Evolution Reactions

Hydrogen evolution reaction (HER) and oxygen evolution reaction (OER) have been deemed as clean and sustainable strategies to solve the energy crisis and environmental problems. Various catalysts have been developed to promote the process of HER and OER. Among them, two-dimensional covalent organic frameworks (2D COFs) have received great attention due to their diverse and designable structure. In this minireview, we mainly summarize the diverse linkages of 2D COFs and strategies for enhancing the catalytic performance of 2D COFs for HER and OER, such as introducing active building blocks, metal ions and tailored linkages. Furthermore, a brief outlook for the development directions of COFs in the field of HER and OER is provided, expecting to stimulate new opportunities in future research.     

Time dependent Schrödinger equation for black hole evaporation: No information loss

In 1976 S. Hawking claimed that “Because part of the information about the state of the system is lost down the hole, the final situation is represented by a density matrix rather than a pure quantum state”. ¹ This was the starting point of the popular “black hole (BH) information paradox”.     

Properties of bounded stochastic processes employed in biophysics

Abstract

Realistic stochastic modeling is increasingly requiring the use of bounded noises. In this work, properties and relationships of commonly employed bounded stochastic processes are investigated within a solid mathematical ground. Four families are object of investigation: the Sine-Wiener (SW), the Doering–Cai–Lin (DCL), the Tsallis–Stariolo–Borland (TSB), and the Kessler–Sørensen (KS) families. We address mathematical questions on existence and uniqueness of the processes defined through Stochastic Differential Equations, which often conceal non-obvious behavior, and we explore the behavior of the solutions near the boundaries of the state space. The expression of the time-dependent probability density of the Sine-Wiener noise is provided in closed form, and a close connection with the Doering–Cai–Lin noise is shown. Further relationships among the different families are explored, pathwise and in distribution. Finally, we illustrate an analogy between the Kessler–Sørensen family and Bessel processes, which allows to relate the respective local times at the boundaries.