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We prove that for any generating set $S$ of $\mathbb {Z}^n$, the continuous edge chromatic number of the Schreier graph of the Bernoulli shift action $G=F(S,2^{\mathbb{Z}^n})$ is $\chi'_c(G)=\chi'(G)+1$. In particular, for the standard generating set, the continuous edge chromatic number of $F(2^{\mathbb {Z}^n})$ is $2n+1$.

Although the deep learning recognition model has been widely used in the condition monitoring of rotating machinery. However, it is still a challenge to understand the correspondence between the structure and function of the model and the diagnosis process. Therefore, this paper discusses embedding distributed attention modules into dense connections instead of traditional dense cascading operations. It not only decouples the influence of space and channel on fault feature adaptive recalibration feature weights, but also forms a fusion attention function. The proposed dense fusion focuses on the visualization of the network diagnosis process, which increases the interpretability of model diagnosis. How to continuously and effectively integrate different functions to enhance the ability to extract fault features and the ability to resist noise is answered. Centrifugal fan fault data is used to verify this network. Experimental results show that the network has stronger diagnostic performance than other advanced fault diagnostic models., Comment: The research direction between authors needs to be adjusted, so the preprint is requested to be revoked

Mid-infrared semiconductor lasers in photonic integrated circuits are of considerable interest for a variety of industrial, environmental, and medical applications. However, photonic integration technologies in the mid-infrared lag far behind the near-infrared range. Here we present the monolithic integration of mid-infrared quantum cascade lasers with low-loss passive waveguides via butt-coupling. The passive waveguide losses are experimentally evaluated to be only 1.2 +- 0.3 dB/cm, with negligible butt-coupling losses. We demonstrate continuous-wave lasing at room temperature of these active-to-passive waveguide coupled devices. Moreover, we report a frequency comb operation paving the way toward on-chip, mid-infrared, dual-comb sensors.

Semiconductor lasers with ultra-low thresholds and minimal footprints are a topic of active research. Such devices require a combination of high quality factor laser cavities with small active region volumes, which drives the quest for novel cavity geometries exploiting nano-optic concepts. For high-reflectivity coated ridge lasers, where light is tightly confined in the waveguide, a low threshold can only be achieved by strongly reducing the diffraction losses arising at the laser facet. We show here that, somewhat counter-intuitively, opening a carefully designed aperture in a metallic facet coating can simultaneously enhance both its transmission and modal reflectivity by correcting the phase front at the subwavelength scale. Numerical simulations and experimental results demonstrate a reduction of optical mirror loss by up to 40% while the transmission is increased by four orders of magnitude. Applying this approach to both facets of a short cavity quantum cascade laser, we achieve laser operation at room temperature with an electrical dissipation of only 143 mW. Such light sources are especially suitable for portable and battery-operated chemical agent sensing applications operating in the mid-infrared wavelength range, where multiple greenhouse and pollutant gases have their fundamental absorption lines. Our work suggests possibilities for further applications including frequency comb dispersion engineering, and can be implemented in a broad range of optoelectronic systems., Comment: 8 pages, 3 figures

We present quantum cascade laser (QCL) frequency comb devices with engineered waveguides for managing the dispersion. The QCL waveguide consists of multiple sections with different waveguide widths. The narrow and wide sections of the waveguide are designed in a way to compensate the group velocity dispersion (GVD) of each other and thereby produce a flat and slightly negative GVD for the QCL. The QCL exhibits continuous comb operation over a large part of the dynamic range of the laser. Strong and narrow-linewidth intermode beatnotes are achieved in more than 300 mA wide operation current range. The comb device features also considerably high output power (>380 mW) and wide optical bandwidth (>55 cm-1)