In this paper, we first find a type of viscosity solution of $G$-heat equation under degenerate case, and then obtain the related $G$-capacity $c({B_{T}in A})$ for any Borel set $A$. Furthermore, we prove that $I_{A}(B_{T})$ is not quasi-continuous when it is not a constant function.
This short note provides a new and simple proof of the convergence rate for Pengs law of large numbers under sublinear expectations, which improves the corresponding results in Song [15] and Fang et al. [3].
In this paper, we study a stochastic recursive optimal control problem in which the value functional is defined by the solution of a backward stochastic differential equation (BSDE) under $tilde{G}$-expectation. Under standard assumptions, we establi
sh the comparison theorem for this kind of BSDE and give a novel and simple method to obtain the dynamic programming principle. Finally, we prove that the value function is the unique viscosity solution of a type of fully nonlinear HJB equation.
Recent work showed neural-network-based approaches to reconstructing images from compressively sensed measurements offer significant improvements in accuracy and signal compression. Such methods can dramatically boost the capability of computational
imaging hardware. However, to date, there have been two major drawbacks: (1) the high-precision real-valued sensing patterns proposed in the majority of existing works can prove problematic when used with computational imaging hardware such as a digital micromirror sampling device and (2) the network structures for image reconstruction involve intensive computation, which is also not suitable for hardware deployment. To address these problems, we propose a novel hardware-friendly solution based on mixed-weights neural networks for computational imaging. In particular, learned binary-weight sensing patterns are tailored to the sampling device. Moreover, we proposed a recursive network structure for low-resolution image sampling and high-resolution reconstruction scheme. It reduces both the required number of measurements and reconstruction computation by operating convolution on small intermediate feature maps. The recursive structure further reduced the model size, making the network more computationally efficient when deployed with the hardware. Our method has been validated on benchmark datasets and achieved the state of the art reconstruction accuracy. We tested our proposed network in conjunction with a proof-of-concept hardware setup.
After endowing with a 3-Lie-Rinehart structure on Hom 3-Lie algebras, we obtain a class of special Hom 3-Lie algebras, which have close relationships with representations of commutative associative algebras. We provide a special class of Hom 3-Lie-
Rinehart algebras, called split regular Hom 3-Lie-Rinehart algebras, and we then characterize their structures by means of root systems and weight systems associated to a splitting Cartan subalgebra.
In this paper, we define a class of 3-algebras which are called 3-Lie-Rinehart algebras. A 3-Lie-Rinehart algebra is a triple $(L, A, rho)$, where $A$ is a commutative associative algebra, $L$ is an $A$-module, $(A, rho)$ is a 3-Lie algebra $L$-modul
e and $rho(L, L)subseteq Der(A)$. We discuss the basic structures, actions and crossed modules of 3-Lie-Rinehart algebras and construct 3-Lie-Rinehart algebras from given algebras, we also study the derivations from 3-Lie-Rinehart algebras to 3-Lie $A$-algebras. From the study, we see that there is much difference between 3-Lie algebras and 3-Lie-Rinehart algebras.
