There are many nonlinear partial differential equations (NPDEs) for noise problems. In particular, the heat equation (low-pass filter) is an important partial differential equation that deals with noise problems. This chapter deals with the NPDEs from the heat equation and describes the relationship between the NPDEs from the heat equation and the NPDEs from total variation. Therefore, various NPDEs were studied. However, they are not sufficient to solve various noise problems in the industry. This chapter focuses on several methods based on NPDEs that attempt to solve the various noise problems.

In boundary layer meteorology, surface layer similarity theory plays a critical role in measuring and modeling biospheric fluxes. In stable boundary layer, surface layer similarity called z-less stratification has been one of main research topics for over than two decades and the issue has yet to be settled in micrometeorology. In this scientific discussion on z-less turbulence, different turbulence statistics were used inconsistently and it was argued that z-less turbulence was valid if only any turbulence statistics were constant with different atmospheric stabilities. Consequently, such inconsistently tested turbulence statistics and misconception on z-less turbulence hinder us from correctly understanding turbulence structure in the stable boundary layer. This note revisits z-less turbulence and emphasizes that different dimensionless turbulence statistics generally do not exhibit a common behavior in the limit of z-less stratification.

Conductivity imaging based on the current-injection MRI technique has been developed in magnetic resonance electrical impedance tomography. Current injected through a pair of surface electrodes induces a magnetic flux density distribution inside an imaging object, which results in additional magnetic field inhomogeneity. We can extract phase changes related to the current injection and obtain an image of the induced magnetic flux density. Without rotating the object inside the bore, we can measure only one component B_z of the magnetic flux density B = (B_x, B_y, B_z). Based on a relation between the internal conductivity distribution and B_z data subject to multiple current injections, one may reconstruct cross-sectional conductivity images. As the image reconstruction algorithm, we have been using the harmonic B_z algorithm in numerous experimental studies. Performing conductivity imaging of intact animal and human subjects, we found technical difficulties that originated from the MR signal void phenomena in the local regions of bones, lungs and gas-filled tubular organs. Measured B_z data inside such a problematic region contain an excessive amount of noise that deteriorates the conductivity image quality. In order to alleviate this technical problem, we applied hybrid methods incorporating ramp-preserving denoising, harmonic inpainting with isotropic diffusion and ROI imaging using the local harmonic B_z algorithm. These methods allow us to produce conductivity images of intact animals with best achievable quality. We suggest guidelines to choose a hybrid method depending on the overall noise level and existence of distinct problematic regions of MR signal void.

 Considering the feedback in radiation, temperature, and soil moisture with alterations in rainfall patterns, the influence of the changing monsoon on Net Ecosystem CO₂ Exchange (NEE) can be critical to the estimation of carbon balance in Asia. In this paper, we examined CO₂ fluxes measured by the eddy covariance method from 2004 to 2008 in two major ecosystems in the KoFlux sites in Korea, i.e., the Gwangneung Deciduous forest (GDK) and the Haenam Farmland (HFK). Our objectives were to identify the repeatability of the mid-season depression of NEE encountered at the two sites based on the single-year observation, and to further scrutinize its cause, effect, and interannual variability by using multi-year observations. In both GDK and HFK sites, the mid-season depression of NEE was reproduced each year but with different timing, magnitude, and mechanism. At the GDK site, a predominant factor causing the mid-season depression was a decreased solar radiation and the consequent reduction in Gross Primary Productivity (GPP) during the summer monsoon period. At the HFK site, however, the monsoonal effect was less pronounced and the apparent mid-season depression was mainly a result of the management practices such as cultivation of spring barley and rice transplantation. Other flux observation sites in East Asia also showed a decline in radiation but with a lesser degree during the monsoon season, resulting in less pronounced depression in NEE. In our study, the observed depressions in NEE caused both GDK and HFK sites to become a weaker carbon sink or even a source in the middle of the growing season. On average, the GDK site (with maximum leaf area index of ~5) was a weak carbon sink with NEE of −84 gC m⁻² y⁻¹. Despite about 20% larger GPP (of 1321 gC m⁻²y⁻¹) in comparison with the GDK site, the HFK site (with maximum leaf area index of 3–4) was a weaker carbon sink with NEE of −58 gC m⁻² y⁻¹ because of greater ecosystem respiration (of 1263 gC m⁻² y⁻¹). These NEE values were near the low end of the ranges reported in the literature for similar ecosystems in mid-latitudes. With the projected trends of the extended length of monsoon with more intensive rainfalls in East Asia, the observed delicate coupling between carbon and hydrological cycles may turn these key ecosystems into carbon neutral.

Turbulence statistics such as flux-variance relationship are critical information in measuring and modeling ecosystem exchanges of carbon, water, energy, and momentum at the biosphere-atmosphere interface. Using a recently proposed mathematical technique, the Hilbert-Huang transform (HHT), this study highlights its possibility to quantify impacts of non-turbulent flows on turbulence statistics in the stable surface layer. The HHT is suitable for the analysis of non-stationary and intermittent data and thus very useful for better understanding the interplay of the surface layer similarity with complex nocturnal environment. Our analysis showed that the HHT can successfully sift non-turbulent components and be used as a tool to estimate the relationships between turbulence statistics and atmospheric stability in complex environments such as nocturnal stable boundary layer.

Bright source catalogues based on the new mid- and far-infrared all-sky survey by the infrared astronomical satellite AKARI were released into the public domain in March 2010. The mid-infrared catalogue contains more than 870 thousand sources observed at 9 and 18 μm, and the far-infrared catalogue provides information of about 427 thousand sources at 65, 90, 140, and 160 μm. The AKARI catalogues will take over the IRAS catalogues and will become one of the most important catalogues in astronomy. We present the characteristics of the AKARI infrared source catalogues as well as current activity for the future versions.

We study the asymptotic flocking dynamics for the Cucker–Smale-type second-order continuous-time dynamical system with the Rayleigh friction. For mean-field communications with a positive lower bound, we show that an asymptotic flocking occurs for any compactly supported initial configuration in a large coupling regime. In contrast, in a small coupling regime, an asymptotic flocking is possible for a restricted class of initial configurations near complete flocking states. We also present several numerical simulations and compare them with our analytical results.

In previous works (Nakao et al., Reliab. Comput., 9(5):359–372, 2003; Watanabe et al., J. Math. Fluid Mech., 6(1):1–20, 2004), the authors considered the numerical verification method of solutions for two-dimensional heat convection problems known as Rayleigh-Bénard problem. In the present paper, to make the arguments self-contained, we first summarize these results including the basic formulation of the problem with numerical examples. Next, we will give a method to verify the bifurcation point itself, which should be an important information to clarify the global bifurcation structure, and show a numerical example. Finally, an extension to the three dimensional case will be described.

In this note, we present a Cucker-Smale type flocking model with nonlinear velocity couplings, and derive sufficient conditions for the formation of flocking in terms of communication weight and initial spatial, velocity standard deviations.