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This paper focuses on the cohomology of operator algebras associated with the free semigroup generated by the set $\{z_{\alpha}\}_{\alpha\in\Lambda}$, with the left regular free semigroup algebra $\mathfrak{L}_{\Lambda}$ and the non-commutative disc algebra $\mathfrak{A}_{\Lambda}$ serving as two typical examples. We establish that all derivations of these algebras are automatically continuous. By introducing a novel computational approach, we demonstrate that the first Hochschild cohomology group of $\mathfrak{A}_{\Lambda}$ with coefficients in $\mathfrak{L}_{\Lambda}$ is zero. Utilizing the Ces\`aro operators and conditional expectations, we show that the first normal cohomology group of $\mathfrak{L}_{\Lambda}$ is trivial. Finally, we prove that the higher cohomology groups of the non-commutative disc algebras with coefficients in the complex field vanish when $|\Lambda|<\infty$. These methods extend to compute the cohomology groups of a specific class of operator algebras generated by the left regular representations of cancellative semigroups, which notably include Thompson's semigroup., Comment: 19 pages

Connected and Autonomous Vehicles (CAVs) technology facilitates the advancement of intelligent transportation. However, intelligent control techniques for mixed traffic flow at signalized intersections involving both CAVs and Human-Driven Vehicles (HDVs) require further investigation into the impact of backward-looking effect. This paper proposes the concept of 1+n+1 mixed platoon considering the backward-looking effect, consisting of one leading CAV, n following HDVs, and one trailing CAV. The leading and trailing CAVs collectively guide the movement of intermediate HDVs at intersections, forming an optimal control framework for platoon-based CAVs at signalized intersections. Initially, a linearized dynamic model for the 1+n+1 mixed platoon is established and compared with a benchmark model focusing solely on controlling the lead vehicle. Subsequently, constraints are formulated for the optimal control framework, aiming to enhance overall intersection traffic efficiency and fuel economy by directly controlling the leading and trailing CAVs in the platoon. Finally, extensive numerical simulations compare vehicle throughput and fuel consumption at signalized intersections under different mixed platoon control methods, validating that considering both front and backward-looking effects in the mixed platoon control method outperforms traditional methods focusing solely on the lead CAV.

This paper investigates Voiculescu's theorem on approximate equivalence in separable properly infinite factors. We establish the norm-denseness of the set of all reducible operators and prove Voiculescu's bicommutant theorem. Additionally, we extend these results to the multiplier algebras within separable type $\mathrm{III}$ factors.

In this paper, we prove the equidistribution property of high-frequency eigensections of a certain series of unitary flat bundles, using the mixture of semiclassical and geometric quantizations., Comment: 24 pages, comments are welcome!

Three types of Cantor sets are studied.For any integer $m\ge 4$, we show that every real number in $[0,k]$ is the sum of at most $k$ $m$-th powers of elements in the Cantor ternary set $C$ for some positive integer $k$, and the smallest such $k$ is $2^m$.Moreover, we generalize this result to middle-$\frac 1\alpha$ Cantor set for $1<\alpha<2+\sqrt{5}$ and $m$ sufficiently large.For the naturally embedded image $W$ of the Cantor dust $C\times C$ into the complex plane $\mathbb{C}$, we prove that for any integer $m\ge 3$, every element in the closed unit disk in $\mathbb C$ can be written as the sum of at most $2^{m+8}$ $m$-th powers of elements in $W$.At last, some similar results on $p$-adic Cantor sets are also obtained.

The finite model property of quasi-transitive modal logic $\mathsf{K}_2^3=\mathsf{K}\oplus \Box\Box p\rightarrow \Box\Box\Box p$ is established. This modal logic is conservatively extended to the tense logic $\mathsf{Kt}_2^3$. We present a Gentzen sequent calculus $\mathsf{G}$ for $\mathsf{Kt}_2^3$. The sequent calculus $\mathsf{G}$ has the finite algebra property by a finite syntactic construction. It follows that $\mathsf{Kt}_2^3$ and $\mathsf{K}_2^3$ have the finite model property., Comment: 11 pages