On the Bounds of Weak $(1,1)$ Norm of Hardy-Littlewood Maximal Operator with $Llog L({mathbb S^{n-1}})$ Kernels

Let $Omegain L^1{({mathbb S^{n-1}})}$, be a function of homogeneous of degree zero, and $M_Omega$ be the Hardy-Littlewood maximal operator associated with $Omega$ defined by $M_Omega(f)(x) = sup_{r>0}frac1{r^n}int_{|x-y|<r}|Omega(x-y)f(y)|dy.$ It was shown by Christ and Rubio de Francia that $|M_Omega(f)|_{L^{1,infty}({mathbb R^n})} le C(|Omega|_{Llog L({mathbb S^{n-1}})}+1)|f|_{L^1({mathbb R^n})}$ provided $Omegain Llog L {({mathbb S^{n-1}})}$. In this paper, we show that, if $Omegain Llog L({mathbb S^{n-1}})$, then for all $fin L^1({mathbb R^n})$, $M_Omega$ enjoys the limiting weak-type behaviors that $$lim_{lambdato 0^+}lambda|{xin{mathbb R^n}:M_Omega(f)(x)>lambda}| = n^{-1}|Omega|_{L^1({mathbb S^{n-1}})}|f|_{L^1({mathbb R^n})}.$$ This removes the smoothness restrictions on the kernel $Omega$, such as Dini-type conditions, in previous results. To prove our result, we present a new upper bound of $|M_Omega|_{L^1to L^{1,infty}}$, which essentially improves the upper bound $C(|Omega|_{Llog L({mathbb S^{n-1}})}+1)$ given by Christ and Rubio de Francia. As a consequence, the upper and lower bounds of $|M_Omega|_{L^1to L^{1,infty}}$ are obtained for $Omegain Llog L {({mathbb S^{n-1}})}$.

Limiting weak-type behaviors for singular integrals with rough $Llog L(mathbb{S}^n)$ kernels

Let $Omega$ be a function of homogeneous of degree zero and vanish on the unit sphere $mathbb {S}^n$. In this paper, we investigate the limiting weak-type behavior for singular integral operator $T_Omega$ associated with rough kernel $Omega$. We show that, if $Omegain Llog L(mathbb S^{n})$, then $lim_{lambdato0^+}lambda|{xinmathbb{R}^n:|T_Omega(f)(x)|>lambda}| = n^{-1}|Omega|_{L^1(mathbb {S}^n)}|f|_{L^1(mathbb{R}^n)},quad0le fin L^1(mathbb{R}^n).$ Moreover,$(n^{-1}|Omega|_{L^1(mathbb{S}^{n-1})}$ is a lower bound of weak-type norm of $T_Omega$ when $Omegain Llog L(mathbb{S}^{n-1})$. Corresponding results for rough bilinear singular integral operators defined in the form $T_{vecOmega}(f_1,f_2) = T_{Omega_1}(f_1)cdot T_{Omega_2}(f_2)$ have also been established.

IIoT-Enabled Health Monitoring for Integrated Heat Pump System Using Mixture Slow Feature Analysis

The sustaining evolution of sensing and advancement in communications technologies have revolutionized prognostics and health management for various electrical equipment towards data-driven ways. This revolution delivers a promising solution for the health monitoring problem of heat pump (HP) system, a vital device widely deployed in modern buildings for heating use, to timely evaluate its operation status to avoid unexpected downtime. Many HPs were practically manufactured and installed many years ago, resulting in fewer sensors available due to technology limitations and cost control at that time. It raises a dilemma to safeguard HPs at an affordable cost. We propose a hybrid scheme by integrating industrial Internet-of-Things (IIoT) and intelligent health monitoring algorithms to handle this challenge. To start with, an IIoT network is constructed to sense and store measurements. Specifically, temperature sensors are properly chosen and deployed at the inlet and outlet of the water tank to measure water temperature. Second, with temperature information, we propose an unsupervised learning algorithm named mixture slow feature analysis (MSFA) to timely evaluate the health status of the integrated HP. Characterized by frequent operation switches of different HPs due to the variable demand for hot water, various heating patterns with different heating speeds are observed. Slowness, a kind of dynamics to measure the varying speed of steady distribution, is properly considered in MSFA for both heating pattern division and health evaluation. Finally, the efficacy of the proposed method is verified through a real integrated HP with five connected HPs installed ten years ago. The experimental results show that MSFA is capable of accurately identifying health status of the system, especially failure at a preliminary stage compared to its competing algorithms.

The limiting weak type behaviors and The lower bound for a new weak $Llog L$ type norm of strong maximal operators

It is well known that the weak ($1,1$) bounds doesnt hold for the strong maximal operators, but it still enjoys certain weak $Llog L$ type norm inequality. Let $Phi_n(t)=t(1+(log^+t)^{n-1})$ and the space $L_{Phi_n}({mathbb R^{n}})$ be the set of all measurable functions on ${mathbb R^{n}}$ such that $|f|_{L_{Phi_n}({mathbb R^{n}})} :=|Phi_n(|f|)|_{L^1({mathbb R^{n}})}<infty$. In this paper, we introduce a new weak norm space $L_{Phi_n}^{1,infty}({mathbb R^{n}})$, which is more larger than $L^{1,infty}({mathbb R^{n}})$ space, and establish the correspondng limiting weak type behaviors of the strong maximal operators. As a corollary, we show that $ max{{2^n}{((n-1)!)^{-1}},1}$ is a lower bound for the best constant of the $L_{Phi_n}to L_{Phi_n}^{1,infty}$ norm of the strong maximal operators. Similar results have been extended to the multilinear strong maximal operators.

Numerical simulation of pulmonary airway reopening by the EOS-based multiphase lattice Boltzmann method

The aerosol formation is associated with the rupture of the liquid plug during the pulmonary airway reopening. The fluid dynamics of this process is difficult to predict because the rupture involved complex liquid-gas transition. Equation of state (EOS) plays a key role in the thermodynamic process of liquid-gas transition. Here, we propose an EOS-based multiphase lattice Boltzmann model, in which the nonideal force is directly evaluated by EOSs. This multiphase model is used to model the pulmonary airway reopening and study aerosol formation during exhalation. The numerical model is first validated with the simulations of Fujioka et al.(2008). and the result is in reasonable agreement with their study. Furthermore, two rupture cases with and without aerosol formation are contrasted and analyzed. It is found that the injury on the epithelium in the case with aerosol formation is essentially the same that of without aerosol formation even while the pressure drop in airway increases by about 67%. Then extensive simulations are performed to investigate the effects of pressure drop, thickness of liquid plug and film on aerosol size and the mechanical stresses. The results show that aerosol size and the mechanical stresses increase as the pressure drop enlarges and thickness of liquid plug become thicken, while aerosol size and the mechanical stresses decrease as thickness of liquid film is thicken. The present multiphase model can be extended to study the generation and transmission of bioaerosols which can carry the bioparticles of influenza or coronavirus.

A Transfer Learning-based State of Charge Estimation for Lithium-Ion Battery at Varying Ambient Temperatures

Accurate and reliable state of charge (SoC) estimation becomes increasingly important to provide a stable and efficient environment for Lithium-ion batteries (LiBs) powered devices. Most data-driven SoC models are built for a fixed ambient temperature, which neglect the high sensitivity of LiBs to temperature and may cause severe prediction errors. Nevertheless, a systematic evaluation of the impact of temperature on SoC estimation and ways for a prompt adjustment of the estimation model to new temperatures using limited data have been hardly discussed. To solve these challenges, a novel SoC estimation method is proposed by exploiting temporal dynamics of measurements and transferring consistent estimation ability among different temperatures. First, temporal dynamics, which is presented by correlations between the past fluctuation and the future motion, is extracted using canonical variate analysis. Next, two models, including a reference SoC estimation model and an estimation ability monitoring model, are developed with temporal dynamics. The monitoring model provides a path to quantitatively evaluate the influences of temperature on SoC estimation ability. After that, once the inability of the reference SoC estimation model is detected, consistent temporal dynamics between temperatures are selected for transfer learning. Finally, the efficacy of the proposed method is verified through a benchmark. Our proposed method not only reduces prediction errors at fixed temperatures (e.g., reduced by 24.35% at -20{deg}C, 49.82% at 25{deg}C) but also improves prediction accuracies at new temperatures.

Time-Series Regeneration with Convolutional Recurrent Generative Adversarial Network for Remaining Useful Life Estimation

For health prognostic task, ever-increasing efforts have been focused on machine learning-based methods, which are capable of yielding accurate remaining useful life (RUL) estimation for industrial equipment or components without exploring the degradation mechanism. A prerequisite ensuring the success of these methods depends on a wealth of run-to-failure data, however, run-to-failure data may be insufficient in practice. That is, conducting a substantial amount of destructive experiments not only is high costs, but also may cause catastrophic consequences. Out of this consideration, an enhanced RUL framework focusing on data self-generation is put forward for both non-cyclic and cyclic degradation patterns for the first time. It is designed to enrich data from a data-driven way, generating realistic-like time-series to enhance current RUL methods. First, high-quality data generation is ensured through the proposed convolutional recurrent generative adversarial network (CR-GAN), which adopts a two-channel fusion convolutional recurrent neural network. Next, a hierarchical framework is proposed to combine generated data into current RUL estimation methods. Finally, the efficacy of the proposed method is verified through both non-cyclic and cyclic degradation systems. With the enhanced RUL framework, an aero-engine system following non-cyclic degradation has been tested using three typical RUL models. State-of-art RUL estimation results are achieved by enhancing capsule network with generated time-series. Specifically, estimation errors evaluated by the index score function have been reduced by 21.77%, and 32.67% for the two employed operating conditions, respectively. Besides, the estimation error is reduced to zero for the Lithium-ion battery system, which presents cyclic degradation.

Limiting weak-type behavior for rough bilinear operators

Let $Omega_1,Omega_2$ be functions of homogeneous of degree $0$ and $vecOmega=(Omega_1,Omega_2)in Llog L(mathbb{S}^{n-1})times Llog L(mathbb{S}^{n-1})$. In this paper, we investigate the limiting weak-type behavior for bilinear maximal function $M_{vecOmega}$ and bilinear singular integral $T_{vecOmega}$ associated with rough kernel $vecOmega$. For all $f,gin L^1(mathbb{R}^n)$, we show that $$lim_{lambdato 0^+}lambda |big{ xinmathbb{R}^n:M_{vecOmega}(f_1,f_2)(x)>lambdabig}|^2 = frac{|Omega_1Omega_2|_{L^{1/2}(mathbb{S}^{n-1})}}{omega_{n-1}^2}prodlimits_{i=1}^2| f_i|_{L^1}$$ and $$lim_{lambdato 0^+}lambda|big{ xinmathbb{R}^n:| T_{vecOmega}(f_1,f_2)(x)|>lambdabig}|^{2} = frac{|Omega_1Omega_2|_{L^{1/2}(mathbb{S}^{n-1})}}{n^2}prodlimits_{i=1}^2| f_i|_{L^1}.$$ As consequences, the lower bounds of weak-type norms of $M_{vecOmega}$ and $T_{vecOmega}$ are obtained. These results are new even in the linear case. The corresponding results for rough bilinear fractional maximal function and fractional integral operator are also discussed.

Invariant learning based multi-stage identification for Lithium-ion battery performance degradation

By informing accurate performance (e.g., capacity), health state management plays a significant role in safeguarding battery and its powered system. While most current approaches are primarily based on data-driven methods, lacking in-depth analysis of battery performance degradation mechanism may discount their performances. To fill in the research gap about data-driven battery performance degradation analysis, an invariant learning based method is proposed to investigate whether the battery performance degradation follows a fixed behavior. First, to unfold the hidden dynamics of cycling battery data, measurements are reconstructed in phase subspace. Next, a novel multi-stage division strategy is put forward to judge the existent of multiple degradation behaviors. Then the whole aging procedure is sequentially divided into several segments, among which cycling data with consistent degradation speed are assigned in the same stage. Simulations on a well-know benchmark verify the efficacy of the proposed multi-stages identification strategy. The proposed method not only enables insights into degradation mechanism from data perspective, but also will be helpful to related topics, such as stage of health.