arXiv subjects
Publications and source records attributed to Ruijun Wang.
Friedman and Stanley asked whether an invariant Borel class of countable structures must be Borel complete whenever every countable level of their hierarchy reduces to it. We construct a counterexample in one fixed countable language. The structural counterexample and its principal supporting results have been formalized in Lean~4 and it is available on github.
We prove that the family of coset relations of $F_σ$ real linear subspaces of Banach space $\ell_1$ has no common upper bound among Borel equivalence relations in the sense of Borel reducibility. These relations are indexed by a well-founded tree. Consequently, on a Polish space, a universal $Σ^0_α$ equivalence relation exists exactly when $α=1$, and a universal $Π^0_α$ equivalence relation exists exactly when $α=1$ or $2$. This strengthens the earlier nonexistence theorem for a universal $F_σ$ equivalence relation. This also answers a question of Kanovei and Lyubetsky about universal equivalence relations in Borel classes.
We determine all Borel-reducibility comparisons within the family of benchmark equivalence relations in Gao's monograph, enlarged by the universal treeable relation $E_{\infty T}$ and its countable power $E_{\infty T}^ω$. We prove $E_\infty^ω\leq_B c_0$, $E_\infty\nleq_B\ell_p$ for $2<p<\infty$, $\ell_p\nleq_B{\equiv_m}$ for $2<p<\infty$, $c_0\nleq_B{\equiv_m}$, $c_0\nleq_B\ell_\infty^+$, and $\equiv_m\nleq_B\ell_\infty^+$. Together with the classical result, these give $\ell_p\leq_B{\equiv_m}$ exactly when $p\leq2$. The same methods show that $E_{\infty T}$ is incomparable with every $\ell_p$, $1\leq p<\infty$. The comparisons answer questions recorded in Gao's monograph and Kanovei's monograph. We also prove that unitary equivalence of all bounded linear operators is not Borel reducible to $\equiv_m$, answering a question of Ding and Gao. Equivalently, unitary equivalence of arbitrary representations of $F_\infty$ is strictly more complicated than that of arbitrary representations of $\mathbb Z$, answering a question of Thomas.
Rosendal proved that any $K_σ$ equivalence relation is Borel reducible to $\ell_\infty$. We prove that for every $F_σ$ equivalence relation $F$ on a Polish space, there is an $F_σ$ real linear subspace $H$ of $\ell_2$ such that the coset relation $x\mathrel{E_H}y$ if and only if $x-y\in H$ has no Baire-measurable reduction to $F$. In particular, there is no $F_σ$ equivalence relation universal under Borel reducibility for all $F_σ$ equivalence relations. This answers a question of Rosendal.
We prove that there is a Borel quasi-kernel in any locally countable Borel directed graph with finite Borel chromatic number. We prove that the Borel chromatic number of a Borel directed graph with bounded out-degree $n$ is either infinite or less than or equal to $\frac{(n+1)(n+2)}{2}$. This is an alternative proof of Palamourdas' theorem.
For every fixed integer $d\geq2$, we prove that the finite graphs receiving a continuous homomorphism from the standard Schreier graph $F(2^{\mathbb Z^d})$ form a $Σ^0_1$-complete set. This extends a theorem of Gao, Jackson, Krohne and Seward when $d=2$. For the positive part of the reduction, we show that if a graph $H$ satisfies a certain property then there is a continuous graph homomorphism from $F(2^{\mathbb Z^d})$ to $H$, in particular, the complete graph $K_4$ satisfies this property. This extends a theorem of Gao and Jackson. We also prove that $\big (\mathcal H_d\big )$ is a strictly decreasing sequence.
We prove that every bounded-to-one action of a finitely generated commutative monoid has finite asymptotic dimension, without a freeness assumption. If the group completion has torsion-free rank $r$, we give an explicit upper bound $(3^{r+1}-3)/2$. This answers a question of Shinko, Weilacher and Yu, and extends a theorem of theirs.
We show that a Borel graph generated by finitely many bounded-to-1 commuting functions is hyperfinite. This recovers a theorem in the forthcoming paper by Naryshkin, Shinko, Weilacher and Yu.
Let $\vec F(2^{\mathbb Z^2})$ be the directed Schreier graph on the free part of the Bernoulli shift $\mathbb Z^2\curvearrowright 2^{\mathbb Z^2}$, with arcs in the two coordinate directions. We prove that the continuous oriented chromatic number of it is 7, that is, there is a tournament on 7 vertices receiving a continuous graph homomorphism from $\vec F(2^{\mathbb Z^2})$ and there is no continuous graph homomorphism from $\vec F(2^{\mathbb Z^2})$ to any tournament on 6 vertices. And we prove that the Borel and measurable oriented chromatic number of directed Schreier graph $\vec F(2^{\mathbb Z^n})$, $n>1$ is 5.
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 $χ'_c(G)=χ'(G)+1$. 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.
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.
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)