Stable blow-up dynamics in the -critical and -supercritical generalized Hartree equation

We study stable blow-up dynamics in the generalized Hartree equation with radial symmetry, which is a Schrödinger-type equation with a nonlocal, convolution-type nonlinearity: First, we consider the -critical case in dimensions and obtain that a generic blow-up has a self-similar structure and exhibits not only the square root blowup rate , but also the log-log correction (via asymptotic analysis and functional fitting), thus, behaving similarly to the stable blow-up regime in the -critical nonlinear Schrödinger equation. In this setting, we also study blow-up profiles and show that generic blow-up solutions converge to the rescaled , a ground state solution of the elliptic equation . We also consider the -supercritical case in dimensions . We derive the profile equation for the self-similar blow-up and establish the existence and local uniqueness of its solutions. As in the NLS -supercritical regime, the profile equation exhibits branches of nonoscillating, polynomially decaying (multi-bump) solutions. A numerical scheme of putting constraints into solving the corresponding ordinary differential equation is applied during the process of finding the multi-bump solutions. Direct numerical simulation of solutions to the generalized Hartree equation by the dynamic rescaling method indicates that the is the profile for the stable blow-up. In this supercritical case, we obtain the blow-up rate without any correction. This blow-up happens at the focusing level , and thus, numerically observable (unlike the -critical case). In summary, we find that the results are similar to the behavior of stable self-similar blowup solutions in the corresponding settings for the nonlinear Schrödinger equation. Consequently, one may expect that the form of the nonlinearity in the Schrödinger-type equations is not essential in the stable formation of singularities.     

基于U-Net的盐体识别方法

卷积神经网络在计算机视觉领域取得重大突破，利用其强大的图像处理能力，将地下沉积盐体的识别问题转化为图像语义分割问题，应用深度卷积神经网络实现盐体地震图像的像素级语义分割.本文在U-Net基础上，增加网络深度并同时引入批归一化和Dropout处理，使得神经网络模型具有更高的可信度和更强的泛化能力.通过实验发现，在卷积层之后引入批归一化处理，并在池化层和叠加层之后引入Dropout可以稳定提升模型对盐体图像的分割性能.     

New convolutions and their applicability to integral equations of Wiener-Hopf plus Hankel type

We propose four new convolutions exhibiting convenient factorization properties associated with two finite interval integral transformations of Fourier-type together with their norm inequalities. Moreover, we study the solvability of a class of integral equations of Wiener-Hopf plus Hankel type (on finite intervals) with the help of the factorization identities of such convolutions. Fourier-type series are used to produce the solution formula of such equations, and a Shannon-type sampling formula is also obtained.     

Generating Artificial Reverberation via Genetic Algorithms for Real-Time Applications

We introduce a Virtual Studio Technology (VST) 2 audio effect plugin that performs convolution reverb using synthetic Room Impulse Responses (RIRs) generated via a Genetic Algorithm (GA). The parameters of the plugin include some of those defined under the ISO 3382-1 standard (e.g., reverberation time, early decay time, and clarity), which are used to determine the fitness values of potential RIRs so that the user has some control over the shape of the resulting RIRs. In the GA, these RIRs are initially generated via a custom Gaussian noise method, and then evolve via truncation selection, random weighted average crossover, and mutation via Gaussian multiplication in order to produce RIRs that resemble real-world, recorded ones. Binaural Room Impulse Responses (BRIRs) can also be generated by assigning two different RIRs to the left and right stereo channels. With the proposed audio effect, new RIRs that represent virtual rooms, some of which may even be impossible to replicate in the physical world, can be generated and stored. Objective evaluation of the GA shows that contradictory combinations of parameter values will produce RIRs with low fitness. Additionally, through subjective evaluation, it was determined that RIRs generated by the GA were still perceptually distinguishable from similar real-world RIRs, but the perceptual differences were reduced when longer execution times were used for generating the RIRs or the unprocessed audio signals were comprised of only speech.     

Free control of far-field scattering angle of transmission terahertz wave using multilayer split-ring resonators’ metasurfaces

To enhance transmission efficiency of Pancharatnam–Berry (PB) phase metasurfaces, multilayer splitring resonators were proposed to develop encoding sequences. As per the generalized Snell’s law, the deflection angle of the PB phase encoding metasurfaces depends on the metasurface period’s size. Therefore, it is impossible to design an infinitesimal metasurface unit; consequently, the continuous transmission scattering angle cannot be obtained. In digital signal processing, this study introduces the Fourier convolution principle on encoding metasurface sequences to freely control the transmitted scattering angles. Both addition and subtraction operations between two different encoding sequences were then performed to achieve the continuous variation of the scattering angle. Furthermore, we established that the Fourier convolution principle can be applied to the checkerboard coded metasurfaces.     

基于PCAU-Net的快速多通道磁共振成像方法

多通道磁共振成像方法采用多个接收线圈同时欠采样k空间以加快成像速度,并基于后处理算法重建图像,但在较高加速因子时,其图像重建质量仍然较差．本文提出了一种基于PCAU-Net的快速多通道磁共振成像方法,将单通道实数U型卷积神经网络拓展到多通道复数卷积神经网络,设计了一种结构不对称的U型网络结构,通过在解码部分减小网络规模以降低模型的复杂度．PCAU-Net网络在跳跃连接前增加了1×1卷积,以实现跨通道信息交互．输入和输出之间利用残差连接为误差的反向传播提供捷径．实验结果表明,使用规则和随机采样模板,在不同加速因子时,相比常规的GRAPPA重建算法和SPIRiT重建方法,本文提出的PCAU-Net方法可高质量重建出磁共振复数图像,并且相比于PCU-Net方法,PCAU-Net减少了模型参数、缩短了训练时间．     

A generalized Duffing equation with the Coulomb’s friction law and Signorini–type contact conditions

This work provides mathematical and numerical analyses for a spring–mass system, in which Signorini–type contact conditions and Coulomb’s friction law with thermal effects are taken into consideration. The motion of a mass attached to a viscoelastic (Kelvin–Voigt type) nonlinear spring is described by a generalized Duffing equation. Signorini contact conditions are understood as extended complementarity conditions (CCs), where convolution is incorporated, allowing to consider thermal aspects of an obstacle. We prove the existence of global weak solutions for the highly nonlinear differential equation system with all the conditions, based on the regularized differential equation and the normal compliance condition with the standard mollifier. In addition, we investigate what side effects produce higher singularities of contact forces in dynamic contact problems, which is also supported by numerical evidences. Numerical schemes are proposed and then several groups of data are selected for the display of our numerical simulations.     

非线性动力学常微分方程组高精度数值积分方法

建立了一种求解非线性动力学常微分方程组初值问题的新方法．若非线性函数一阶导数存在，则给出解的积分方程表达式，计算得到按规定误差要求的高精度数值解．引入一般自治或非自治非线性系统的首次近似Jacobi矩阵，不作任何假设重构等价的非线性常微分方程组，简捷而有广泛的适应性，不改变方程的本质，但其主项构成线性化方程组，其它项则代表非线性函数高阶余项而不涉及Taylor级数展开计算，给出该方程组初值问题的Duhamel卷积分解析表达式，在时间步长内进行数值积分选代求解，在指定误差内快速收敛，逐步递推获得非线性常微分方程的瞬态响应和全时域高精度数值解．积分解连续满足微分方程组而不是在离散的步长端点上满足代数方程组，打破了传统用增量法在离散点上建立的代数方程组迭代求解，从而使传统Euler型逐步积分法的各种差分格式算法改变成真正的积分格式算法．数值计算中给出指数矩阵递增展开式，变矩阵乘法为乘积系数的加法，避免了大量矩阵自乘而大大提高计算效率．算法验证为无条件稳定，则保证对线性常微分方程而言，计算中舍入误差的传播不会扩散，不出现计算机字长有限而引起舍入误差导致计算不确定性问题．基于以上理论和数值方法，计算了线性非线性算例并进行了分析，验证了本方法简捷而有广泛的适应性，可以有足够的精确性．     

卷积型加权残值法求解薄壳的动力学问题

板壳等弹性体,受到动荷载作用时,其动力分析问题是比较重要且难于解决的．利用卷积型加权残值法,推导出Gurtin变分原理,并应用卷积型加权残值法计算了薄壳的动力学问题,为计算薄壳的结构动力学问题提供了一种有效的方法．