Scattering for the fractional magnetic Schrodinger operators

In this paper, we prove the existence of the scattering operator for the fractional magnetic Schrodinger operators. For this, we construct the fractional distorted Fourier transforms with magnetic potentials. Applying the properties of the distorted Fourier transforms, the existence and the asymptotic completeness of the wave operators are obtained. Furthermore, we prove Carleman estimate to prove the absence of positive eigenvalues for Schrodinger operatorsComment: arXiv admin note: text overlap with arXiv:2001.01962 by other author

Dispersive decay estimates for the magnetic Schr\"odinger equations

In this paper, we present a proof of dispersive decay for both linear and nonlinear magnetic Schr\"odinger equations. To achieve this, we introduce the fractional distorted Fourier transforms with magnetic potentials and define the fractional differential operator \arrowvert J_{A}(t)\arrowvert^{s}. By leveraging the properties of the distorted Fourier transforms and the Strichartz estimates of \arrowvert J_{A}\arrowvert^{s}u, we establish the dispersive bounds with the decay rate $t^{-\frac{n}{2}}$. This decay rate provides valuable insights into the spreading properties and long-term dynamics of the solutions to the magnetic Schr\"odinger equations

Geometric Conditions for the Exact Observability of Schr\"{o}dinger Equations with Point Interaction and Inverse-Square Potentials on Half-Line

We provide necessary and sufficient conditions for the exact observability of the Schrodinger equations with point interaction and inverse-square potentials on half-line. The necessary and sufficient condition for these two cases are derived from two Logvinenko-Sereda type theorems for generalized Fourier transform. Specifically, the generalized Fourier transform associated to the Schr\"{o}dinger operators with inverse-square potentials on half-line are the well known Hankel transforms. We provide a necessary and sufficient condition for a subset $\Omega$ such that a function with its Hankel transform supporting in a given interval can be bounded, in $L^{2}$-norm, from above by its restriction to the set $\Omega$, with constant independent of the position of the intervalComment: arXiv admin note: text overlap with arXiv:2003.11263, arXiv:2007.04096 by other author

Observability and unique continuation inequalities for the Schr\"{o}dinger equations with inverse-square potentials

This paper is inspired by Wang, Wang and Zhang's work [ Observability and unique continuation inequalities for the Schr\"odinger equation. J. Eur. Math. Soc. 21, 3513--3572 (2019)], where they present several observability and unique continuation inequalities for the free Schr\"{o}dinger equation in $\mathbb{R}^{n}$. We extend all such observability and unique continuation inequalities for the Schr\"{o}dinger equations on half-line with inverse-square potentials. Technically, the proofs essentially rely on the representation of the solution, a Nazarov type uncertainty principle for the Hankel transform and an interpolation inequality for functions whose Hankel transform have compact support.Comment: 45 page

The impact of building operations on urban heat/cool islands under urban densification: a comparison between naturally-ventilated and air-conditioned buildings

Many cities are suffering the effects of urban heat islands (UHI) or urban cool islands (UCI) due to rapid urban expansion and numerous infrastructure developments. This paper presents a lumped urban-building thermal coupling model which captures the fundamental physical mechanism for thermal interactions between buildings and their urban environment. The benefits of the model are its simplicity and high computational efficiency for practical use in investigating the diurnal urban air temperature change and its asymmetry in a city with both naturally-ventilated (NV) and air-conditioned (AC) buildings. Our model predicts a lower urban heat island and higher urban cool island intensity in a city with naturally-ventilated buildings than for a city with air-conditioned buildings. During the urban densification (from a low-rise, low-density city to a high-rise, high-density one), the increases in the time constant and internal heat gain give rise to asymmetric warming phenomena, which become more obvious in a city with air-conditioned buildings rather than naturally-ventilated ones. Unlike previous studies, we found that a low-rise, low-density city experiences a stronger urban cool island effect than a high-rise, high-density city due to less heat being emitted into the urban atmosphere. The urban cool/heat island effect will firstly increase/decrease, and then rapidly decrease/increase and ultimately disappear/dominate with increasing time constant in the process of urbanization/urban densification

Identification of Anisomerous Motor Imagery EEG Signals Based on Complex Algorithms

Motor imagery (MI) electroencephalograph (EEG) signals are widely applied in brain-computer interface (BCI). However, classified MI states are limited, and their classification accuracy rates are low because of the characteristics of nonlinearity and nonstationarity. This study proposes a novel MI pattern recognition system that is based on complex algorithms for classifying MI EEG signals. In electrooculogram (EOG) artifact preprocessing, band-pass filtering is performed to obtain the frequency band of MI-related signals, and then, canonical correlation analysis (CCA) combined with wavelet threshold denoising (WTD) is used for EOG artifact preprocessing. We propose a regularized common spatial pattern (R-CSP) algorithm for EEG feature extraction by incorporating the principle of generic learning. A new classifier combining the K-nearest neighbor (KNN) and support vector machine (SVM) approaches is used to classify four anisomerous states, namely, imaginary movements with the left hand, right foot, and right shoulder and the resting state. The highest classification accuracy rate is 92.5%, and the average classification accuracy rate is 87%. The proposed complex algorithm identification method can significantly improve the identification rate of the minority samples and the overall classification performance