Parseval等式在一类无穷级数求和中的应用

本文运用著名的Parseval等式给出了sum from n=1 to ∞( 1/n2k),sum from n=1 to ∞(n2/(n2+a2)(1+k))及sum from n=1 to ∞(n2/(n2-a2)(1+k))(k≥1)等一些无穷级数的求和公式.     

浅谈两个误差公式的证明

在误差理论中,Bessel公式σ_x~2=1/n-1 sum from i=1 to n(x_i-x)~2 (1)是估算测量列x_1,…,x_n方差的基本公式(式中x=1/n sum from i=1 to n x_i为平均值).对于它的证明,一般有两种方法:一种按误差是随机变量的观点,通过求上式的数学期望来证明。这对不熟悉概率论的初学者(例如大学低年级学生)来说,恐怕不易接受。另一种是按方差的下述定义:     

边界形状的变化对偏微分方程本征值的影响

我们考虑下列方程和边界条件: 在区域V中:(~2十E)φ=0, (1) 在V的边界上: φ=0 (2)所定解的最低本征值E,简称为区域V的本征值。 例如V为半径为R的无穷长圆柱,则 E=a~2/R~2,φ=J_0(E~(1/2)r)=J_0((ar)/R)。 (3)此处a=2.405为Bessel函数J_0的最小根。设由于形状变化而使得V改为 r≤R(θ)=R[1+ sum form n=-∞ to ∞ s_ne~(inθ)) (4)其中 s_(-n)=s_n~* (n≠0),s_0=0. (5)我们用(4)的本征值 E=a~2/R~2f(s_n) (6)与等面积圆柱的本征值 a~2/R~2[1+ sum form n=-∞ to ∞ |s_n|~2]~(-1) 的比值 F(s_n)=[1+ sum form n=-∞ to ∞ |s_n|~2]f(s_n) (7)来表示边界形状的变化对柱的本征值的影响。 我们用边界微扰法计算(6)及(7)。为计算方便起见,我们选长度单位使得R=a;     

密度泛函理论研究簇合物(HFGaN_3)_n(n=1-6):尺寸效应

用密度泛函理论DFT方法,计算研究不对称簇合物(HFGaN_3)_n(n=1-6)的几何构型、稳定性、IR谱和热力学性质.结果表明,簇合物(HFGaN_3)_n(n=2-6)的优化构型拥有一个Ga和α-N原子交替的2n元环状结构.通过计算研究(HFGaN_3)_n(n=1-6)的平均结合能、二阶能量差分和能隙与团簇尺寸关系,发现后两者表现出明显的"奇—偶"振荡现象.对计算获得的IR谱进行归属,获得四个特征区.讨论了团簇尺寸和温度对(HFGaN_3)_n(n=1-6)热力学函数的影响.由焓变和吉布斯自由能可知,200-800K温度范围内由单体形成稳定的多聚体(HFGaN_3)_n(n=2-6)在热力学上有利.     

线性参数估计中一个值得注意的问题

测量量(x_i,y_i)满足一定条件的线性函数 y=α+βx (1)的最小二乘参数估计值为■的方差为 其中(?)=1/n sum from 1 to n(x_i),(?)=1/n sum from 1 to n(y_i),σ~2=D(y_i)。普通物理实验的数据处理中有很多这类问题。测量量(x‘,y‘)满足一定条件的线性函数最佳估计量,应改用下式计算y=a+刀x(1)万 乙(y‘一a)x‘一1(5)戈乙1的最小二乘参数估计值为 a=y一刀x(2)D(少(6) X.名i声干(劣一劣)(y‘一y)名(x。一x)“(3)从数学上看,(3)式与(5)式的差刀的方差为 D(刀) az名(x‘一x)2别仅在于解正规方程(4)。a+名二‘刀==兄夕‘习x‘a+习x蓄刀=习x‘…     

(CuSe)_n(n=1-10)团簇结构稳定性与磁性的理论研究

本文采用包含广义梯度近似的密度泛函理论对铜、硒二元团簇(CuSe)_n(n=1-10)的结构稳定性与磁性质进行系统的研究,发现:(CuSe)_n(n=1-10)二元团簇的原子总数为8时几何结构出现由平面向立体的转变行为,且Cu3Se3为"幻数"构型.同时,(CuSe)_n(n=1-10)团簇的平均结合能比同尺寸单质铜团簇的大,因此混合团簇(CuSe)_n(n=1-10)具有比单质铜团簇更强的稳定性;另外,根据(CuSe)_n(n=1-10)团簇的二阶差分能和HOMO-LUMO能隙随尺寸增加的演化趋势可获知,Cu8Se8的结构极其稳定.最后,对团簇磁性的研究发现,铜硒团簇展现出比较特别的磁性,其中,Cu3Se3的总磁矩为3μB,且磁性主要来源于硒原子,而当n=4,6,8和10时,(CuSe)_n(n=1-10)团簇磁性出现淬灭现象.     

Multiplicative Stochastic Resonance for a Linear System Driven by O-U Noise

In the paper, we study a linear system driven by O-U noise and give a method which is different from the one stated in Europhys. Lett. 40 （1997） 117. We find the same phenomenon of multiplicative stochastic resonance for the response of the system to the signal as the one found in Europhys. Left. 40 （1997） 117. The merit of our method is that it prevents the complex formulas when making sum from n = 0 to n →∞ as in Europhys. Lett. 40 （1997） 117, which leads to the approximate results of the figures.     

Absorption of a massless scalar wave from a Schwarzschild black hole surrounded by quintessence

By using the partial wave method,we investigate the absorption of a massless scalar wave from a Schwarzschild black hole surrounded by quintessence.We obtained the expression of absorption cross section σabs(ω)=π/ω2sum from l=0 to ∞(2l+1)|Tωl(ω)|2=π/ω2sum from l=0 to ∞(2l+1)Γωl(ω).Then we numerically carry out the absorption cross section and we find that the larger the angular momentum quantum number l is,the smaller the corresponding maximum value of the partial absorption cross section is,and that the total absorption cross section tends to the geometric-optical limit σ hf abs ≈πb 2 c.We also find that higher value of ω q(state parameter of quintessence) corresponds to the higher value of absorption cross section σ abs.     

同轴圆柱系统微变问题的静电解

我们从一种较为特殊但又是十分典型的同轴结构的探讨开始,它的截面在极坐标系统中(原点置于轴心)具有 ρ_1=r_l(1+ε_1 cosnφ),ρ_2=r_2(1+ε_2cos(φ-φ_0))。 (1)其中r_1和r_2系变形前外的导体因截面半径;ε_1和ε_2则为表征其变形特点的小参数;φ_0表示相似变形后内外导体叉开的一个角度;而n为某一正整数。象这样的同轴系统,不妨形象化称为“n波”同轴系统。 我们知道,二维拉普拉斯方程的环内问题,在极坐标系统中的一般解可以用圆谐函数来组合:     

水合金属团簇M~+(H_2O)_n(M=Cu,Ag,Au;n=1-4)的结构和稳定性研究

利用密度泛函理论B3LYP方法优化了M~+(H_2O)_n(M=Cu,Ag,Au;n=1-4)团簇的几何结构,并对它们的稳定性和电子性质进行了系统研究.计算结果表明,Cu~+(H_2O)_n和Ag~+(H_2O)_n团簇的水合骨架结构由三个氧原子与金属离子相连接而成,而Au~+(H_2O)_n团簇则由两个氧原子与金属离子形成的线性结构而得到.通过计算的结合能和HOMO-LUMO能隙也发现:三种水合金属离子体系的稳定性大小为:Eb(Cu)=Eb(Au)Eb(Ag),但是Ag~+(H_2O)_n团簇具有较高的动力学稳定性.这些研究对于今后过渡金属离子水溶液性质的研究和应用提供了重要的理论参考.     

产生零功率的热核聚变堆第二判据

把Lawson判据(第一判据)应用到聚变堆时,应该增添一个重要参数η_n——聚变堆加热总效率。这样由等离子体温度T,密度n和约束时间τ的乘积与η_n组成了三维空间,用它来描述聚变堆的收支平衡条件(Breakeven Condition),并由此空间中一个曲面来表示,称之为第二判据。在nτ～T二维空间上的Lawson判据只是它的一个特例。第二判据更全面地展示出在收支平衡下,主要参数数值及它们之间的关系,它更符合于实际聚变堆所要求的数值。     

Investigation of the inhomogeneous barrier height of an Au/Bi4Ti3O12/n-Si structure through Gaussian distribution of barrier height

<正>A Au/Bi₄Ti₃O₁₂/n-Si structure is fabricated in order to investigate its current-voltage（I-V） characteristics in a temperature range of 300 K-400 K.Obtained I-V data are evaluated by the thermionic emission（TE） theory.Zero-bias barrier height(Φ_B0) and ideality factor（n） calculated from I-V characteristics,are found to be temperature-dependent such thatΦ_B0 increases with temperature increasing,whereas n decreases.The obtained temperature dependence ofΦ_B0 and linearity inΦ_B0 versus the n plot,together with a lower barrier height and Richardson constant values obtained from the Richardson plot,indicate that the barrier height of the structure is inhomogeneous in nature.Therefore,I-V characteristics are explained on the basis of Gaussian distribution of barrier height.     

近红外区的光泵钠分子激光振荡

本文报道了以准分子激光泵浦的染料激光激发钠分子,产生由钠分子A~1 sum from u to +态到X~1 sum from g to +态的众多激光输出谱线.当加以谐振腔后,这些谱线的强度可增加一个数量级.     

Momentum dependence of pion-induced φ meson production on nuclei near threshold

We study the near-threshold pion-induced production ofφmesons off nuclei in the kinematical conditions of the HADES experiment,recently performed at GSI.The calculations have been performed within a collision model based on the nuclear spectral function.The model accounts for both the primaryπ~-meson–protonπ~-p→φn and the secondary pion–nucleonπN→φNφproduction processes as well as the effects of the nuclearφand nucleon mean-field potentials.We find that the primary reaction channelπ~-p→φn dominates in theφproduction off~(12)C and~(184)W target nuclei in the HADES acceptance window at incident pion momentum of 1.7 Ge V/c.We calculate the momentum dependence of the absolute and relative(transparency ratio)φmeson yields from the above direct channel.The calculations have been performed for this initial pion momentum allowing for different options for theφN absorption cross sectionσ_(φN )and different scenarios for the in-medium mass shifts of theφmeson and secondary neutron,produced together withφin this channel.We demonstrate that the transparency ratio for theφmesons has,contrary to the absolute cross sections,an insignificant sensitivity to theφmeson and secondary neutron in-medium mass shifts atφmomenta studied in the HADES experiment.On the other hand,we show that there are measurable changes in the transparency ratio due to theφN absorption cross section,which means that such a relative observable can be useful to help determine this cross section from the data taken in the HADES experiment.     

The N-mode squeezed state with enhanced squeezing

It is known that exp [iλ (Q1P1i/2)] is a unitary single-mode squeezing operator,where Q1,P1 are the coordinate and momentum operators,respectively.In this paper we employ Dirac’s coordinate representation to prove that the exponential operator S n ≡ exp [iλ sum((QiPi+1+Qi+1Pi))) from i=1 to n ],(Qn+1=Q1,Pn+1=P1),is an n-mode squeezing operator which enhances the standard squeezing.By virtue of the technique of integration within an ordered product of operators we derive S n ’s normally ordered expansion and obtain new n-mode squeezed vacuum states,its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.     

第一性原理对(NaP_3)_n(n=1～5)团簇的几何结构和性质的研究

本文采用第一性原理研究了(NaP_3)_n(n=1~5)团簇的几何结构、能隙、电荷分布以及态密度.研究结果表明:NaP_3团簇为线型结构,是(NaP_3)_n(n=1~5)团簇中的基本单元,随着n增大,团簇转变为环状结构和空间结构;(NaP_3)_3团簇的能隙出现峰值,表明该团簇较其他团簇有较高的稳定性;(NaP_3)_3团簇中对于高能量区域的态密度是由Na 3s和P 3p轨道贡献,其中在费米附近的能量主要由P 3p轨道贡献,对于低能量区域的态密度主要由Na 3s轨道贡献;(NaP_3)_n(n=1~5)团簇中的Na原子电荷分布均为正值,表明电荷总体上是从Na原子转移到P原子.     

(CoO)_n(n≤6)合金团簇的结构和磁性理论研究

采用密度泛函理论中的广义梯度近似(DFT-GGA)对(CoO)_n(n≤6)合金团簇进行了系统的几何、电子结构和磁性质研究.研究结果表明(CoO)n(n≤6)团簇最稳定结构除(CoO)6团簇为三维立体结构外,其余团簇均呈二维平面结构,且(CoO)_n(n=1,2,3,4和6)结构均表现明显的钴氧分离特征,而(CoO)5团簇表现明显的Co—Co聚合和O—O分离特征.团簇的总磁矩在n=1,3,4时,以3μB为单元成倍增长,(CoO)5团簇显著降低,减小至1μB,(CoO)6团簇又有所增加,增大至6μB.(CoO)n(n≤6)团簇磁性变化的起因也从电荷转移、磁性耦合、电子差分密度和态密度进行了详细阐释.     

Average kinetic energy of ions ejected from pure Coulomb explosions of methane clusters

We use an electrostatic model to study the average kinetic energy of ions ejected from the pure Coulomb explosions of methane clusters(CA_4)_n(light atom A=H and D).It is found that the ratio of the average kinetic energy of the ions to their initial average electrostatic potential energy is irrelevant to the cluster size.This finding implies that as long as the ratio is given,the average kinetic energies of the ions can be simply estimated from their initial average electrostatic potential energies,rather than from the time- consuming simulations.The ratios for the different charge states of carbon ions are presented.     

关于一级近似T_c级数解的收敛问题

本文通过一个实例说明:μ~* =0时,一级近似T_c级数解的收敛半径既不是由z_(ph)=-integral from n=0 to (ω_(ph)) (dωg(ω)·ω＇/ω_(ph)~2-ω~2),也不是由z_(ph)=integral from n=0 to (ω_(ph)) (dωg(ω)ω~2/ω_(ph)~2 ω~2)决定的。     

关于n平面线性相干光处理系统的讨论(Ⅱ)

依据傅里叶变换的相似性定理和空间频率的标度变换,研究了n平面线性相干光处理系统的一般性质。假定原原统的间距参数为{z_i}_n-1。首先证明:这一系统等价于任意参数为{zo_i}_n-1的系统。其次讨论上述两个n平面系统的积分核之间的关系。 关键词：