Quantum algorithm for neighborhood preserving embedding

Neighborhood preserving embedding (NPE) is an important linear dimensionality reduction technique that aims at preserving the local manifold structure. NPE contains three steps, i.e., finding the nearest neighbors of each data point, constructing the weight matrix, and obtaining the transformation matrix. Liang et al. proposed a variational quantum algorithm (VQA) for NPE [Phys. Rev. A 101 032323 (2020)]. The algorithm consists of three quantum sub-algorithms, corresponding to the three steps of NPE, and was expected to have an exponential speedup on the dimensionality n. However, the algorithm has two disadvantages: (i) It is not known how to efficiently obtain the input of the third sub-algorithm from the output of the second one. (ii) Its complexity cannot be rigorously analyzed because the third sub-algorithm in it is a VQA. In this paper, we propose a complete quantum algorithm for NPE, in which we redesign the three sub-algorithms and give a rigorous complexity analysis. It is shown that our algorithm can achieve a polynomial speedup on the number of data points m and an exponential speedup on the dimensionality n under certain conditions over the classical NPE algorithm, and achieve a significant speedup compared to Liang et al.'s algorithm even without considering the complexity of the VQA.     

Euler Characteristics on virtually free products

We define Euler characteristics on classes of residually finite and virtually torsion free groups and we show that they satisfy certain formulas in the case of amalgamated free products and HNN extensions over finite subgroups. These formulas are obtained from a general result which applies to the rank gradient and the first L²?Betti number of a finitely generated group.     

压电式微滴按需喷射的过程控制和规律

微滴喷射增材制造技术中沉积微滴的大小与均匀性是影响成型件质量的关键因素.本文设计了一种用于生成均匀微滴的压电驱动式微滴喷射装置,通过压电材料带动柔性膜片振动,将液体从喷嘴中喷出生成微滴,采用数值模拟和实验相结合的方法,研究了不同控制参数下膜片振幅及其对生成微滴尺寸和均匀性的影响.研究结果表明:膜片振幅大小受到驱动电压和压电频率的共同影响,压电频率是导致膜片中心点振幅实验测量值小于理论计算值的主要原因,膜片振动会导致喷嘴内部压力发生变化从而影响微滴生成尺寸.在相同驱动电压条件下,压电频率为10 Hz时存在压电膜片振幅最大值.随着膜片振幅的增大,喷孔处液体速度和液柱长度增大到临界值时可以生成微滴,当喷孔处的液柱长度超过临界值时,会形成卫星液滴. 当膜片振幅区间在30 $\mu$m$\sim $42 $\mu $m可以稳定生成微滴,生成最小微滴尺寸为339.8$\mu$m,直径最大变化率为0.29%,相邻两微滴间距最大变化率为2.67%,生成微滴的尺寸及均匀性较好.研究结果有助于提高压电式微滴喷射装置的液滴生成质量.      

Torsion simple modules over the quantum spatial ageing algebra

Various classes of simple torsion modules are classified over the quantum spatial ageing algebra (this is a Noetherian algebra of Gelfand-Kirillov dimension 4). Explicit constructions of these modules are given and for each module its annihilator is found.     

The global weak solution for the shallow water wave model of moderate amplitude

In this paper, we firstly establish the existence theorem of the global weak solutions of the Cauchy problem for the shallow water wave model of moderate amplitude, then following the idea in Xin and Zhang’s work (see Xin and Zhang, 2002), we prove the uniqueness of global weak solutions using the localization analysis in the transport equation theory. Finally, several travelling wave solutions are derived.     

交错群A_5与二面体群D_6直积的整群环的挠单位

本文利用Luthar-Passi方法,研究了五次交错群A_5与六阶二面体群D_6直积的整群环的挠单位,得到了该群的Zassenhaus猜想成立.     

长链支化聚丙烯的非线性流变行为

通过大幅振荡剪切试验方法, 研究了长链支化聚丙烯的非线性流变学行为, 揭示了长链支化结构与非线性黏弹响应的关系. 利用傅里叶变换流变学方法得到试样的三次倍频相对振幅I_3/1与应变的标度关系, 用于界定线性聚丙烯和长链支化聚丙烯非线性流变行为的差异, 并定义了非线性系数来量度长链支化程度. 在高应变下, I_3/1与应变的变化关系可以进一步描述长链支化在非线性流场下的特性, 并得到了长链支化程度与其非线性响应之间的变化关系. 通过应力波的Lissajous曲线分解, 发现了环内和环间黏弹性的差异, 长链支化结构在大应变流场下的非线性黏弹性响应弱于线性结构, 支化程度越高非线性响应越弱.