Some New Lie Symmetry Groups of Differential-Difference Equations Obtained from a Simple Direct Method

In this paper, based on the symbolic computing system Maple, the direct method for Lie symmetry groups presented by Sen-Yue Lou [J. Phys. A： Math. Gen. 38 （2005） L129] is extended from the continuous differential equations to the differential-difference equations. With the extended method, we study the well-known differential-difference KP equation, KZ equation and （2＋1）-dimensional ANNV system, and both the Lie point symmetry groups and the non-Lie symmetry groups are obtained.     

Spectral flow invariants and twisted cyclic theory for the Haar state on

In [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], we presented a K-theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely SU_q(2) and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes from the generator of the modular group of the Haar state. In contrast to the Cuntz algebras studied in [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], the computations are considerably more complex and interesting, because there are non-trivial ‘eta’ contributions to this index.     

近红外光谱分析中建模校正集的选择

将极大线性无关组的概念及方法引入近红外光谱分析,探讨了在建立定量分析模型时代表性样品,即校正集样品的选择问题。以2 652个烟末样品为实验材料,随机选取1 001个样品构成预测集,其余1 651个样品为代表性样品备选集。用Matlab软件求出代表性样品备选集光谱矩阵的极大线性无关组,以此作为代表性样品,构成建模的校正集。用PLS回归法建立了烟末样品总糖含量定量分析的预测模型,并将模型用于预测集中1 001个烟末样品总糖含量的预测分析。实验结果表明,当选择的校正集包含的样品数量大于32时,所建各模型对预测集样品预测的平均相对误差均小于4%,平均相关系数大于0.96。其中选择32个代表性样品和146个代表性样品所建模型定量分析预测集中各样品的总糖含量,两个结果经统计检验没有显著性差异(α=0.05),说明求极大线性无关组的方法用于校正集样品的选择,可实现“少而精”选择样品的目的。此外,我们用求极大线性无关组选择校正集样品和随机方法选择校正集样品两种方法,选择了同样数目28,32,41,76,146,163个样品建模进行预测效果的对比实验,结果显示,求极大线性无关组法选择校正集建模的预测效果优于随机选择校正集建模的预测效果。     

Low-temperature elastic constants and piezoelectric coefficients of langasite (La3Ga5SiO14)

A complete set of elastic constants C _ij and piezoelectric coefficients e _ij of a La₃Ga₅SiO₁₄ (langasite) single crystal was determined from 5.5 to 275.5?K by resonant ultrasound spectroscopy. Unlike a conventional crystal, the elastic constants C _ij of the langasite crystal showed three types of temperature dependence: (i) monotonic elastic stiffening upon cooling (C ₄₄ and C ₁₄); (ii) monotonic elastic softening (C ₆₆); and (iii) a stiffening-to-softening transition below 150?K (C ₁₁, C ₁₂, C ₁₃ and C ₃₃). In addition, a strong correlation between C ₆₆ and the piezoelectric coefficient e ₁₁ was confirmed. Group theoretical lattice dynamics analysis revealed that the novel phenomena of elastic softening and strong correlation are explained on the basis of two types of optical mode internal displacements which have the totally symmetric A ₁ and doubly degenerated E symmetries in the point group D ₃.     

Hopf-algebraic renormalization of QED in the linear covariant gauge

In the context of massless quantum electrodynamics (QED) with a linear covariant gauge fixing, the connection between the counterterm and the Hopf-algebraic approach to renormalization is examined. The coproduct formula of Green’s functions contains two invariant charges, which give rise to different renormalization group functions. All formulas are tested by explicit computations to third loop order. The possibility of a finite electron self-energy by fixing a generalized linear covariant gauge is discussed. An analysis of subdivergences leads to the conclusion that such a gauge only exists in quenched QED.     

具有若干$c_p$-可补子群的有限群

设$H$为有限群$G$的子群且$p$为整除群$G$的阶的素因子. 我们称$H$在$G$中为$c_p$-可补的,如果$G$中存在$H$的包含$H_G$的补子群$T$使得$H\cap T/H_G$为$p''$-群, 其中$H_G$为$H$在$G$中的核. 群$G$ 称$CS_p$-群, 如果$G$的所有$p$-子群都在$G$中$c_p$-可补. 本文,我们刻画具有若干$c_p$-可补$p$-子群的有限群的$p$-可解性和$p$-超可解性. 此外,我们给出$p$-可解群为$CS_p$-群的若干等价条件. 最后, 我们给出两个$CS_p$-群的直积为$CS_p$-群的判别准则. 我们的结果推广了近期的若干结论.     

基于评价信息随机化的群体评价方法与应用

针对群体评价中共识集结的相关问题,从仿真的视角讨论了评价信息随机化的群体共识聚合求解方法。首先,面向实数类型的评价信息,将精确性的数据给予一定的宽松性处理,进一步结合正态分布的3σ原则,利用随机模拟的方式集结出带有概率特征的可能性排序;其次,面向区间数类型的评价信息,整合出各子区间发生概率不同的区间数评价信息,在充分随机模拟的情况下,给出了带有优胜概率特征的可能性排序。最后,通过相应的算例进行求解分析,说明了该方法的可行性和有效性。基于群体共识视角,针对实数和区间数两种类型的评价信息,分别进行相应的随机化处理,并为进一步探索区间数的分布形式提供了一种新的研究思路。     

Adaptive Evolution Analysis of a Predator-Prey Model With Group Defense北大核心CSCD

Based on the theoretical framework of adaptive dynamics, the evolution of the predator-prey model with functional response of group defense effect on the predator handling time, was investigated. Firstly, in view of the interaction of predator populations with interspecific competition, the evolutionary conditions for a single predator population to split into 2 populations with different strategies through evolutionary branching were given. Secondly, when the ecological equilibrium of the model is unstable and the system has a limit cycle, the population will have strong coexistence under large mutation, but this coexistence will be evolutionarily unstable. Finally, the conclusions for the model with Holling-Ⅱ type functional response were compared. The results indicate that, with a sufficiently large prey carrying capacity, group defense effects can evolutionarily lead to the extinction of predators. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Local Dispersive and Strichartz Estimates for the Schrödinger Operator on the Heisenberg Group

It was proved by Bahouri et al. [9] that the Schrödinger equation on the Heisenberg group $\mathbb{H}^d,$ involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on $\mathbb{H}^d$ for the linear Schrödinger equation, by a refined study of the Schrödinger kernel $S_t$ on $\mathbb{H}^d.$ The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on $\mathbb{H}^d$ derived by Gaveau [19], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results we establish local Strichartz estimates and prove that the kernel $S_t$ concentrates on quantized horizontal hyperplanes of $\mathbb{H}^d.$