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Abstract We introduce a distributional Jacobian determinant $detD{V}_{\beta }\left(Dv\right)$\det DV_{\beta}(Dv)in dimension two for the nonlinear complex gradient $V_{\beta }\left(Dv\right)={\left|Dv\right|}^{\beta }(v_{x_1},-v_{x_2})$V_{\beta}(Dv)=\lvert Dv\rvert^{\beta}(v_{x_{1}},-v_{x_{2}})for any $\beta >-1$\beta>-1, whenever $v\in W_{\mathrm{loc}}^{1,2}$v\in W^{1\smash{,}2}_{\mathrm{loc}}and $\beta {\left|Dv\right|}^{1+\beta }\in W_{\mathrm{loc}}^{1,2}$\beta\lvert Dv\rvert^{1+\beta}\in W^{1\smash{,}2}_{\mathrm{loc}}.This is new when $\beta \ne 0$\beta\neq 0.Given any planar ∞-harmonic function 𝑢, we show that such distributional Jacobian determinant $detD{V}_{\beta }\left(Du\right)$\det DV_{\beta}(Du)is a nonnegative Radon measure with some quantitative local lower and upper bounds.We also give the following two applications. Applying this result with $\beta =0$\beta=0, we develop an approach to build up a Liouville theorem, which improves that of Savin.Precisely, if 𝑢 is an ∞-harmonic function in the whole ${\mathbb{R}}^2$\mathbb{R}^{2}with $\underset{R\to \mathrm{\infty }}{lim\; inf}\underset{c\in \mathbb{R}}{inf}\frac{1}{R}{⨍}_{B(0,R)}\left|u\left(x\right)-c\right|dx<\mathrm{\infty },$\liminf_{R\to\infty}\inf_{c\in\mathbb{R}}\frac{1}{R}\barint_{B(0,R)}\lvert u(x)-c\rvert\,dx<\infty,then $u=b+a\cdot x$u=b+a\cdot xfor some $b\in \mathbb{R}$b\in\mathbb{R}and $a\in {\mathbb{R}}^2$a\in\mathbb{R}^{2}.Denoting by $u_p$u_{p}the 𝑝-harmonic function having the same nonconstant boundary condition as 𝑢, we show that $detD{V}_{\beta }\left(D{u}_p\right)\to detD{V}_{\beta }\left(Du\right)$\det DV_{\beta}(Du_{p})\to\det DV_{\beta}(Du)as $p\to \mathrm{\infty }$p\to\inftyin the weak-⋆ sense in the space of Radon measure.Recall that $V_{\beta }\left(D{u}_p\right)$V_{\beta}(Du_{p})is always quasiregular mappings, but $V_{\beta }\left(Du\right)$V_{\beta}(Du)is not in general. 

Neuromorphic vision sensors or event cameras have made the visual perception of extremely low reaction time possible, opening new avenues for high-dynamic robotics applications. These event cameras’ output is dependent on both motion and texture. However, the event camera fails to capture object edges that are parallel to the camera motion. This is a problem intrinsic to the sensor and therefore challenging to solve algorithmically. Human vision deals with perceptual fading using the active mechanism of small involuntary eye movements, the most prominent ones called microsaccades. By moving the eyes constantly and slightly during fixation, microsaccades can substantially maintain texture stability and persistence. Inspired by microsaccades, we designed an event-based perception system capable of simultaneously maintaining low reaction time and stable texture. In this design, a rotating wedge prism was mounted in front of the aperture of an event camera to redirect light and trigger events. The geometrical optics of the rotating wedge prism allows for algorithmic compensation of the additional rotational motion, resulting in a stable texture appearance and high informational output independent of external motion. The hardware device and software solution are integrated into a system, which we call artificial microsaccade–enhanced event camera (AMI-EV). Benchmark comparisons validated the superior data quality of AMI-EV recordings in scenarios where both standard cameras and event cameras fail to deliver. Various real-world experiments demonstrated the potential of the system to facilitate robotics perception both for low-level and high-level vision tasks. 

We consider the periodic review dynamic pricing and inventory control problem with fixed ordering cost. Demand is random and price dependent, and unsatisfied demand is backlogged. With complete demand information, the celebrated [Formula: see text] policy is proved to be optimal, where s and S are the reorder point and order-up-to level for ordering strategy, and [Formula: see text], a function of on-hand inventory level, characterizes the pricing strategy. In this paper, we consider incomplete demand information and develop online learning algorithms whose average profit approaches that of the optimal [Formula: see text] with a tight [Formula: see text] regret rate. A number of salient features differentiate our work from the existing online learning researches in the operations management (OM) literature. First, computing the optimal [Formula: see text] policy requires solving a dynamic programming (DP) over multiple periods involving unknown quantities, which is different from the majority of learning problems in OM that only require solving single-period optimization questions. It is hence challenging to establish stability results through DP recursions, which we accomplish by proving uniform convergence of the profit-to-go function. The necessity of analyzing action-dependent state transition over multiple periods resembles the reinforcement learning question, considerably more difficult than existing bandit learning algorithms. Second, the pricing function [Formula: see text] is of infinite dimension, and approaching it is much more challenging than approaching a finite number of parameters as seen in existing researches. The demand-price relationship is estimated based on upper confidence bound, but the confidence interval cannot be explicitly calculated due to the complexity of the DP recursion. Finally, because of the multiperiod nature of [Formula: see text] policies the actual distribution of the randomness in demand plays an important role in determining the optimal pricing strategy [Formula: see text], which is unknown to the learner a priori. In this paper, the demand randomness is approximated by an empirical distribution constructed using dependent samples, and a novel Wasserstein metric-based argument is employed to prove convergence of the empirical distribution. This paper was accepted by J. George Shanthikumar, big data analytics.