Scattering for defocusing mass sub-critical NLS

In this paper, we consider the $L_x^2$-scattering of defocusing mass sub-critical nonlinear Schr\"odinger equations with low weighted initial condition. It is known that the scattering holds with $\mathcal{F} H^1$-data, while the continuity of inverse wave operator breaks down with $L^2$-data. Moreover, for large $\mathcal{F} H^s$-data with $s<1$, there only exists the wave operator result, but scattering results are lacking. Our subject is to study the scattering in low weights space. Our results are divided into two parts. Our first result presents a systematic study on the scattering on $\mathcal{F} H^s$ for certain $s<1$, without any restrictions on smallness or radial symmetry. This extends the previous results to spaces with lower weights. Our second result is the almost sure scattering on $L^2$ by introducing a ``narrowed'' Wiener randomization in physical space. For mass subcritical NLS when $d\ge 2$, this result represents the first scattering result without imposing any conditions related to smallness, radial symmetry, or weighted properties on the initial data.Comment: 86 page

Instability of the solitary waves for the Generalized Benjamin-Bona-Mahony Equation

In this work, we consider the generalized Benjamin-Bona-Mahony equation $$\partial_t u+\partial_x u+\partial_x( |u|^pu)-\partial_t \partial_x^{2}u=0, \quad(t,x) \in \mathbb{R} \times \mathbb{R},$$ with $p>4$. This equation has the traveling wave solutions $\phi_{c}(x-ct),$ for any frequency $c>1.$ It has been proved by Souganidis and Strauss \cite{Strauss-1990} that, there exists a number $c_{0}(p)>1$, such that solitary waves $\phi_{c}(x-ct)$ with $1<c<c_{0}(p)$ is orbitally unstable, while for $c>c_{0}(p),$ $\phi_{c}(x-ct)$ is orbitally stable. The linear exponential instability in the former case was further proved by Pego and Weinstein \cite{Pego-1991-eigenvalue}. In this paper, we prove the orbital instability in the critical case $c=c_{0}(p)$.Comment: 32 pages, 1 fictur

High-Brightness Image Enhancement Algorithm

In this paper, we introduce a tone mapping algorithm for processing high-brightness video images. This method can maximally recover the information of high-brightness areas and preserve detailed information. Along with benchmark data, real-life and practical application data were taken to test the proposed method. The experimental objects were license plates. We reconstructed the image in the RGB channel, and gamma correction was carried out. After that, local linear adjustment was completed through a tone mapping window to restore the detailed information of the high-brightness region. The experimental results showed that our algorithm could clearly restore the details of high-brightness local areas. The processed image conformed to the visual effect observed by human eyes but with higher definition. Compared with other algorithms, the proposed algorithm has advantages in terms of both subjective and objective evaluation. It can fully satisfy the needs in various practical applications

Acceleration compensation of a novel piezoelectric balance for the short duration impulse measurement: a time series analysis approach

A novel piezoelectric balance was developed to measure the six-component forces for the complex aircraft scaled model in the impulse combustion wind tunnel at a short duration airloads Mach number of 5. The piezoelectric balance using four triaxial piezoelectric load cells yields the high stiffness, sensitive and good dynamic response characteristics. The dynamic model-balance system was built to analyze the vibration characteristic. The time series analysis approach was developed on the basis of the system transfer function and the natural frequency, and the accelerated forces which induce the airloads overshooting oscillations had been obtained by the second order derivatives function. The experimental results have shown that the problem of overshooting oscillations effect of the impulse can be effectively solved by the acceleration compensation technology for the complex test model with the novel piezoelectric balance

Four-dimensional Cone Beam CT Reconstruction and Enhancement using a Temporal Non-Local Means Method

Four-dimensional Cone Beam Computed Tomography (4D-CBCT) has been developed to provide respiratory phase resolved volumetric imaging in image guided radiation therapy (IGRT). Inadequate number of projections in each phase bin results in low quality 4D-CBCT images with obvious streaking artifacts. In this work, we propose two novel 4D-CBCT algorithms: an iterative reconstruction algorithm and an enhancement algorithm, utilizing a temporal nonlocal means (TNLM) method. We define a TNLM energy term for a given set of 4D-CBCT images. Minimization of this term favors those 4D-CBCT images such that any anatomical features at one spatial point at one phase can be found in a nearby spatial point at neighboring phases. 4D-CBCT reconstruction is achieved by minimizing a total energy containing a data fidelity term and the TNLM energy term. As for the image enhancement, 4D-CBCT images generated by the FDK algorithm are enhanced by minimizing the TNLM function while keeping the enhanced images close to the FDK results. A forward-backward splitting algorithm and a Gauss-Jacobi iteration method are employed to solve the problems. The algorithms are implemented on GPU to achieve a high computational efficiency. The reconstruction algorithm and the enhancement algorithm generate visually similar 4D-CBCT images, both better than the FDK results. Quantitative evaluations indicate that, compared with the FDK results, our reconstruction method improves contrast-to-noise-ratio (CNR) by a factor of 2.56~3.13 and our enhancement method increases the CNR by 2.75~3.33 times. The enhancement method also removes over 80% of the streak artifacts from the FDK results. The total computation time is ~460 sec for the reconstruction algorithm and ~610 sec for the enhancement algorithm on an NVIDIA Tesla C1060 GPU card.Comment: 20 pages, 3 figures, 2 table