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Superconductivity emerges in both La$_3$Ni$_2$O$_7$ and La$_4$Ni$_3$O$_{10}$ under high pressure by suppressing their density-wave transitions, but critical temperature (Tc) differs significantly between these two compounds. To gain deeper insights into the distinct superconducting states, it is essential to unravel the nature of the density-wave states at ambient pressure, a topic that remains largely unexplored. Here, using scanning tunneling microscopy/spectroscopy (STM/STS), we report the direct visualization of an incommensurate unidirectional charge density wave (CDW) in La$_4$Ni$_3$O$_{10}$ in real space. The density of states (DOS) is strongly depleted near $E_F$, indicating the opening of a CDW gap of $2{\Delta} \approx 71$ meV, which is unfavorable for the formation of superconductivity at ambient pressure. We propose that the CDW arises from Fermi surface nesting, and is likely a subsidiary phase of a spin density wave. Compared to La$_3$Ni$_2$O$_7$, the weaker electronic correlation in La$_4$Ni$_3$O$_{10}$ is likely one reason for the lower $T_c$., Comment: 11 pages, 4 figures

We study an online joint assortment-inventory optimization problem, in which we assume that the choice behavior of each customer follows the Multinomial Logit (MNL) choice model, and the attraction parameters are unknown a priori. The retailer makes periodic assortment and inventory decisions to dynamically learn from the customer choice observations about the attraction parameters while maximizing the expected total profit over time. In this paper, we propose a novel algorithm that can effectively balance exploration and exploitation in the online decision-making of assortment and inventory. Our algorithm builds on a new estimator for the MNL attraction parameters, an innovative approach to incentivize exploration by adaptively tuning certain known and unknown parameters, and an optimization oracle to static single-cycle assortment-inventory planning problems with given parameters. We establish a regret upper bound for our algorithm and a lower bound for the online joint assortment-inventory optimization problem, suggesting that our algorithm achieves nearly optimal regret rate, provided that the static optimization oracle is exact. Then we incorporate more practical approximate static optimization oracles into our algorithm, and bound from above the impact of static optimization errors on the regret of our algorithm. We perform numerical studies to demonstrate the effectiveness of our proposed algorithm. At last, we extend our study by incorporating inventory carryover and the learning of customer arrival distribution.

Kagome metals show rich competing quantum phases due to geometry frustration, flat bands, many-body effects, and non-trivial topology. Recently, a novel charge density wave (CDW) was discovered deep inside the antiferromagnetic phase of FeGe, attracting intense attention due to close relation with magnetism. Here, via a scanning tunneling microscope (STM), we find the 2*2 CDW in FeGe is very fragile and can be readily disrupted into the initial 1*1 phase; Small sqrt3*sqrt3 CDW puddles are found to coexist with the 2*2 CDW in as-grown samples, and can also be induced in the intermediate process of CDW disruption, which will eventually transform into the initial 1*1 phase. Moreover, an exotic intermediate CDW state and standalone CDW nuclei appear unexpectedly during the disruption process. Our first-principle calculations find equal softening of a flat optical phonon mode in a large momentum region around the CDW wave vector, corresponding to numerous competing CDWs with close energies. This might lead to strong instability of the CDW ground state, responsible for STM observations. Our findings provide more novel experimental aspects to understand the CDW in FeGe and suggest FeGe-like Kagome metals are ideal platforms for studying the physics of competing CDW instabilities., Comment: 16 pages, 7 figures

Tensor Robust Principal Component Analysis (TRPCA), which aims to recover a low-rank tensor corrupted by sparse noise, has attracted much attention in many real applications. This paper develops a new Global Weighted TRPCA method (GWTRPCA), which is the first approach simultaneously considers the significance of intra-frontal slice and inter-frontal slice singular values in the Fourier domain. Exploiting this global information, GWTRPCA penalizes the larger singular values less and assigns smaller weights to them. Hence, our method can recover the low-tubal-rank components more exactly. Moreover, we propose an effective adaptive weight learning strategy by a Modified Cauchy Estimator (MCE) since the weight setting plays a crucial role in the success of GWTRPCA. To implement the GWTRPCA method, we devise an optimization algorithm using an Alternating Direction Method of Multipliers (ADMM) method. Experiments on real-world datasets validate the effectiveness of our proposed method., Comment: 8 pages