Improved Complexity Analysis of the Sinkhorn and Greenkhorn Algorithms for Optimal Transport

The Sinkhorn algorithm is a widely used method for solving the optimal transport problem, and the Greenkhorn algorithm is one of its variants. While there are modified versions of these two algorithms whose computational complexities are $O({n^2\|C\|_\infty^2\log n}/{\varepsilon^2})$ to achieve an $\varepsilon$-accuracy, the best known complexities for the vanilla versions are $O({n^2\|C\|_\infty^3\log n}/{\varepsilon^3})$. In this paper we fill this gap and show that the complexities of the vanilla Sinkhorn and Greenkhorn algorithms are indeed $O({n^2\|C\|_\infty^2\log n}/{\varepsilon^2})$. The analysis relies on the equicontinuity of the dual variables of the entropic regularized optimal transport problem, which is of independent interest

High Yield Synthesis Of Helical Carbon Nanotubes Catalyzed By Porous Precursor With Terrace Morphology

Superfine and uniformly dispersed catalyst precursor was obtained through a novel precipitation/sol-gel/reduction technique, which had multi-scale multi-layered porous structures with large surface areas. Helical carbon nanotubes (HCNTs) with good purity were then synthesized through chemical vapor deposition (CVD) by acetylene decomposition over the as-obtained precursor. The yield of the HCNTs was nearly three times of the highest value ever reported. The high yield and good quality of the HCNTs was attributed to the peculiar structures of the precursor that could decompose into catalyst with large surface area and high activity. Furthermore, the structures and catalytic efficiencies of the catalyst precursors prepared through other two different approaches were also investigated to reveal the relationship between the structures of the precursor and their capability differences in catalyzing the HCNTs growth

Gas-Induced Formation of Cu Nanoparticle as Catalyst for High-Purity Straight and Helical Carbon Nanofibers

The facile preparation of high-purity carbon nanofibers (CNFs) remains challenging due to the high complexity and low controllability in reaction. A novel approach using gas-induced formation of Cu crystals to control the growth of CNFs is developed in this study. By adjusting the atmospheric composition, controllable preparation of Cu nanoparticles (NPs) with specific size and shape is achieved, and they are further used as a catalyst for the growth of straight or helical CNFs with good selectivity and high yield. The preparation of Cu NPs and the formation of CNFs are completed by a one-step process. The inducing effect of N₂, Ar, H₂, and C₂H₂ on the formation of Cu NPs is systematically investigated through a combined experimental and computational approach. The morphology of CNFs obtained under different conditions is rationalized in terms of Cu NP and CNF growth models. The results suggest that the shapes of CNFs, namely, straight or helical, depend closely on the size, shape, and facet activity of Cu NPs, while such a gas-inducing method offers a simple way to control the formation of Cu NPs

Lithium Deposition-Induced Fracture of Carbon Nanotubes and Its Implication to Solid-State Batteries

The increasing demand for safe and dense energy storage has shifted research focus from liquid electrolyte-based Li-ion batteries toward solid-state batteries (SSBs). However, the application of SSBs is impeded by uncontrollable Li dendrite growth and short circuiting, the mechanism of which remains elusive. Herein, we conceptualize a scheme to visualize Li deposition in the confined space inside carbon nanotubes (CNTs) to mimic Li deposition dynamics inside solid electrolyte (SE) cracks, where the high-strength CNT walls mimic the mechanically strong SEs. We observed that the deposited Li propagates as a creeping solid in the CNTs, presenting an effective pathway for stress relaxation. When the stress-relaxation pathway is blocked, the Li deposition-induced stress reaches the gigapascal level and causes CNT fracture. Mechanics analysis suggests that interfacial lithiophilicity critically governs Li deposition dynamics and stress relaxation. Our study offers critical strategies for suppressing Li dendritic growth and constructing high-energy-density, electrochemically and mechanically robust SSBs