Product Hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type

The aim of this article is to develop the theory of product Hardy spaces associated with operators which possess the weak assumption of Davies--Gaffney heat kernel estimates, in the setting of spaces of homogeneous type. We also establish a Calder\'on--Zygmund decomposition on product spaces, which is of independent interest, and use it to study the interpolation of these product Hardy spaces. We then show that under the assumption of generalized Gaussian estimates, the product Hardy spaces coincide with the Lebesgue spaces, for an appropriate range of~$p$.Comment: Accepted by Math.

Brain mapping and detection of functional patterns in fMRI using wavelet transform; application in detection of dyslexia

Background Functional Magnetic Resonance Imaging (fMRI) has been proven to be useful for studying brain functions. However, due to the existence of noise and distortion, mapping between the fMRI signal and the actual neural activity is difficult. Because of the difficulty, differential pattern analysis of fMRI brain images for healthy and diseased cases is regarded as an important research topic. From fMRI scans, increased blood ows can be identified as activated brain regions. Also, based on the multi-sliced images of the volume data, fMRI provides the functional information for detecting and analyzing different parts of the brain. Methods In this paper, the capability of a hierarchical method that performed an optimization algorithm based on modified maximum model (MCM) in our previous study is evaluated. The optimization algorithm is designed by adopting modified maximum correlation model (MCM) to detect active regions that contain significant responses. Specifically, in the study, the optimization algorithm is examined based on two groups of datasets, dyslexia and healthy subjects to verify the ability of the algorithm that enhances the quality of signal activities in the interested regions of the brain. After verifying the algorithm, discrete wavelet transform (DWT) is applied to identify the difference between healthy and dyslexia subjects. Results We successfully showed that our optimization algorithm improves the fMRI signal activity for both healthy and dyslexia subjects. In addition, we found that DWT based features can identify the difference between healthy and dyslexia subjects. Conclusion The results of this study provide insights of associations of functional abnormalities in dyslexic subjects that may be helpful for neurobiological identification from healthy subject

A Hybrid Mechanism Forming a 2:1 Librating-Circulating Resonant Configuration in the Planetary System

A diversity of resonance configurations may be formed under different migration of two giant planets. And the researchers show that the HD 128311 and HD 73526 planetary systems are involved in a 2:1 mean motion resonance but not in apsidal corotation, because one of the resonance argument circulates over the dynamical evolution. In this paper, we investigate potential mechanisms to form the 2:1 librating-circulating resonance configuration. In the late stage of planetary formation, scattering or colliding among planetesimals and planetary embryos can frequently occur. Hence, in our model, we consider a planetary configuration of two giants together with few terrestrial planets. We find that both colliding or scattering events at very early stage of dynamical evolution can influence the configurations trapped into resonance. A planet-planet scattering of a moderate terrestrial planet, or multiple scattering of smaller planets in a crowded planetary system can change the resonant configuration. In addition, collision or merging can alter the masses and location of the giant planets, which also play an important role in shaping the resonant configuration during the dynamical evolution. In this sense, the librating-circulating resonance configuration is more likely to form by a hybrid mechanism of scattering and collision.Comment: 8 pages, 5 figures, 2 Tables, accepted for publication in MNRA