关于薛定谔算子的极点散射: 基于狄利克雷级数的视角

本文讨论了在实轴上具有紧支集的势的薛定谔算子的极点散射问题. 本文旨在将狄利克雷级数理论与散射理论相结合, 文中运用了Littlewood的经典方法得到关于极点个数的新的估计. 本文首次将狄利克雷级数方法用于极点估计, 由此得到了极点个数的上界与下界, 这些结果改进和推广了该论题的一些相关结论.     

A new non-perturbative approach in quantum mechanics for time-independent Schr?dinger equations

A new non-perturbative approach is proposed to solve time-independent Schr?dinger equations in quantum mechanics.It is based on the homotopy analysis method(HAM)that was developed by the author in 1992 for highly nonlinear equations and has been widely applied in many fields.Unlike perturbative methods,this HAM-based approach has nothing to do with small/large physical parameters.Besides,convergent series solution can be obtained even if the disturbance is far from the known status.A nonlinear harmonic oscillator is used as an example to illustrate the validity of this approach for disturbances that might be one thousand times larger than the possible superior limit of the perturbative approach.This HAM-based approach could provide us rigorous theoretical results in quantum mechanics,which can be directly compared with experimental data.Obviously,this is of great benefit not only for improving the accuracy of experimental measurements but also for validating physical theories.     

富里叶级数的收敛速度

对正弦和余弦富立叶级数,通过合并相邻同号项,使其重排成交错级数.讨论了重排形成的交错级数的敛散性.指出根据自变量x的不同取值,该交错级数可能是单调递减或周期递减的级数.按照莱布尼茨判定法提出了不同精度要求的级数项数的计算公式.选取一到三阶收敛的富立叶级数计算了不同比值精度及差值精度要求的级数项数.计算表明,在x的取值为2π的等分点时,富立叶级数的部分和随项数的增加单调地逼近其收敛值.在x的取值为其它点时,富立叶级数的部分和随项数的增加围绕收敛值上下变动,周期地逼近其收敛值.低收敛阶富立叶级数的收敛速度较慢.要达到0.01%的精度,一收敛阶富立叶级数需要数万项,二收敛阶富立叶级数也需要数百项.在不同计算点处,要达到相同的计算精度,需要的级数项数差别较大.     

Analytical and approximate solutions of(2+1)-dimensional time-fractional Burgers-Kadomtsev-Petviashvili equation

In this paper, we applied the sub-equation method to obtain a new exact solution set for the extended version of the time-fractional Kadomtsev-Petviashvili equation, namely BurgersKadomtsev-Petviashvili equation(Burgers-K-P) that arises in shallow water waves.Furthermore, using the residual power series method(RPSM), approximate solutions of the equation were obtained with the help of the Mathematica symbolic computation package. We also presented a few graphical illustrations for some surfaces. The fractional derivatives were considered in the conformable sense. All of the obtained solutions were replaced back in the governing equation to check and ensure the reliability of the method. The numerical outcomes confirmed that both methods are simple, robust and effective to achieve exact and approximate solutions of nonlinear fractional differential equations.     

A note on Herglotz’s theorem for time series on function spaces

In this article, we prove Herglotz’s theorem for Hilbert-valued time series. This requires the notion of an operator-valued measure, which we shall make precise for our setting. Herglotz’s theorem for functional time series allows to generalize existing results that are central to frequency domain analysis on the function space. In particular, we use this result to prove the existence of a functional Cramér representation of a large class of processes, including those with jumps in the spectral distribution and long-memory processes. We furthermore obtain an optimal finite dimensional reduction of the time series under weaker assumptions than available in the literature. The results of this paper therefore enable Fourier analysis for processes of which the spectral density operator does not necessarily exist.     

Synthesis and characterization of new homologue multinary selenides,M2Sb5Bi5Se17 (M = Sn,Pb)

New quaternary selenides M₂Sb₅Bi₅Se₁₇ (M = Sn, Pb) were synthesized using solid-state sintering reactions that crystallize in the monoclinic system with C2/m (No. 12) space group with lattice parameters a = 27.914(7) Å, b = 4.0804(11) Å, c = 15.512(4) Å, and β = 114.881(9)° for M = Sn, and a = 27.987(3) Å, b = 4.1062(5) Å, c = 15.6372(19) Å, and β = 115.318(3)° for M = Pb, respectively. The crystal structure is related to a homologous series [A⁺²_2x−4B⁺³₄ Se⁻²_2x−2][B⁺³_2y−2Se⁻²_3y−3] with (x, y) = (3, 4) that contains building units of two-dimensional slabs of NaCl¹¹¹-type [Sb₂Bi₄Se₁₁] separated by 1D ribbons NaCl¹⁰⁰-type [Pb₂Sb₃BiSe₆]. The NaCl¹¹¹ unit contains edge-shared octahedra filled with Sb³⁺ and Bi³⁺ cations, which are parallel and overlapped to form a step-layer 2D network stacking alone [001]. The NaCl¹⁰⁰ type ribbons containing Pb²⁺ and Sb³⁺ in square or trigonal pyramidal environments with the general formula [M₆Se₆] filled in the space between 2D layers of NaCl¹¹¹ units. The conductivity measurement revealed semiconducting property with band gaps of ~0.1 eV. Pb₂Sb₅Bi₅Se₁₇ exhibits low thermal conductivity 3,000 μW cm⁻¹ K⁻¹ in a temperature range of 300–480 K.     

In silico study of novel niclosamide derivatives,SARS-CoV-2 nonstructural proteins catalytic residue-targeting small molecules drug candidates

The severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2)-mediated coronavirus disease 2019 (COVID-19) infection remains a global pandemic and health emergency with overwhelming social and economic impacts throughout the world. Therapeutics for COVID-19 are limited to only remdesivir; therefore, there is a need for combined, multidisciplinary efforts to develop new therapeutic molecules and explore the effectiveness of existing drugs against SARS-CoV-2. In the present study, we reported eight (SCOV-L-02, SCOV-L-09, SCOV-L-10, SCOV-L-11, SCOV-L-15, SCOV-L-18, SCOV-L-22, and SCOV-L-23) novel structurally related small-molecule derivatives of niclosamide (SCOV-L series) for their targeting potential against angiotensin-converting enzyme-2 (ACE2), type II transmembrane serine protease (TMPRSS2), and SARS-COV-2 nonstructural proteins (NSPs) including NSP5 (3CLpro), NSP3 (PLpro), and RdRp. Our correlation analysis suggested that ACE2 and TMPRSS2 modulate host immune response via regulation of immune-infiltrating cells at the site of tissue/organs entries. In addition, we identified some TMPRSS2 and ACE2 microRNAs target regulatory networks in SARS-CoV-2 infection and thus open up a new window for microRNAs-based therapy for the treatment of SARS-CoV-2 infection. Our in vitro study revealed that with the exception of SCOV-L-11 and SCOV-L-23 which were non-active, the SCOV-L series exhibited strict antiproliferative activities and non-cytotoxic effects against ACE2- and TMPRSS2-expressing cells. Our molecular docking for the analysis of receptor-ligand interactions revealed that SCOV-L series demonstrated high ligand binding efficacies (at higher levels than clinical drugs) against the ACE2, TMPRSS2, and SARS-COV-2 NSPs. SCOV-L-18, SCOV-L-15, and SCOV-L-09 were particularly found to exhibit strong binding affinities with three key SARS-CoV-2’s proteins: 3CLpro, PLpro, and RdRp. These compounds bind to the several catalytic residues of the proteins, and satisfied the criteria of drug-like candidates, having good adsorption, distribution, metabolism, excretion, and toxicity (ADMET) pharmacokinetic profile. Altogether, the present study suggests the therapeutic potential of SCOV-L series for preventing and managing SARs-COV-2 infection and are currently under detailed investigation in our lab.