奇点与奇点定理简介（中）

测地线束与共轭点 在雷查德利方程中，如果所考虑的测地线束局部正比于某个梯度场，或者说垂直于某个超曲面，则称该线束是超曲面垂直的。可以证明，对于这样的测地线束来说，     

该测哪些牛顿环

由牛顿环测透镜曲率半径R的公式是 R=(D_m~2-D_n~2)/(4(m-n)λ) (1) 式中m、n是环的表观序数,m>n,D_m、D_n是环径,λ是光源的波长。测牛顿环时该测哪些环呢?国内教材的一些说法是:有人研究过从哪一级环开始测量比较合理的问题,得出的结论是“要根据仪器条件决定”。一般认为,牛顿环中心附近的环宽度变化很大,不易测准。现在从误差分析来解决这个问题,以便使结论更科学、更合理。由(1)式 |ΔR|≤(D_m|ΔD_m|+D_n|ΔD_n|)/(2(m-n)λ)≤2r_m|△r_m|+2r_n|△r_n|/(m-n)λ即 |ΔR|≤(2(r_m|Δr_m|+r_n|Δr_n|))/((m-n)λ) (2) 牛顿环的光强度分布为 I=4I_0sin~2((πr~2)/(Rλ)) (3) 式中I_0为入射光的强度,r为牛顿环半径。I=0的r为暗环半径。取(3)中的I=0可得 r_k=(kRλ)~(1/2) k=0,1,2,3,…(4)     

斜流通风机的准三元设计法

一、子午面流线的确定 由沿准正交线q(见图1)的平衡条件式 dC_m~2/dq+A(q)=B(q) (1)来计算子午面流线。 A(q)=2(1/(r_mcosε)+(sinφsinε)/r+(dφ)/(dq)tgε) (2)式中: B(q)=2[d/(dq)(P_0/ρ-(C_θ)/r (d(rC_θ)/dq] (3) P_0/ρ=(P_(01)/ρ)+(uC_θ-u_1C_(θ1))-ξ_PW_1~2/2 (4) 全压损失系数ξ_p和当量扩散系数D_(cq)分别表示     

Solution of Dirac Equation with Generalized Hylleraas Potential

We employ the parametric generalization of the Nikiforov-Uvarov method to solve the Alhaidari formalism of the Dirac equation with a generalized Hylleraas potential of the form V(r) = V0(a+exp(λr))/(b+ exp(λr)) + V1(d+exp(λr))/(b+exp(λr)).We obtain the bound state energy eigenvalue and the corresponding eigenfunction expressed in terms of the Jacobi polynomials.By choosing appropriate parameter in the potential model,the generalized Hylleraas potential reduces to the well known potentials as special cases.     

一种新型的两态叠加多模Schr(o¨)dinger猫态光场的等阶N次方Y压缩

本文根据量子力学的线性叠加原理,构造了由多模(即q模)相干态的相反态|{-Zj}〉q及多模虚相干态|{iZj}〉q这两者的线性叠加所组成的一种新型的多模Schrdinger猫态光场|Ψ(2)〉q,利用新近建立的多模压缩态理论,研究了态|Ψ(2)〉q的N次方Y压缩效应.结果发现:①当压缩阶数N＝2p且p＝2m(m＝1,2,3,…,…)时,态|Ψ(2)〉q总是恒处于N-Y最小测不准态;②当压缩阶数N＝2p且p＝2m+1(m＝0,1,2,3,…,…)时,如果各模的初始相位φj,态间的初始相位差θ(I)pq-θ(R)nq以及各单模相干态光场的平均光子数之总和等满足一定的量子化条件,则态|Ψ(2)〉q可呈现出周期性变化的、任意阶的N次方Y压缩效应;③当压缩阶数N＝2p'+1时,无论p'＝2m(m＝0,1,2,…,…)还是p'＝2m+1(m＝0,1,2,3,…,…),只要各模的初始相位φj满足一定的量子条件,则当两态叠加几率幅满足r(I)pq＝r(R)nq时,态|Ψ(2)〉q就恒处于N-Y测不准态,始终不呈现N-Y最小测不准态和N次方Y压缩;而当r(I)pq≠r(R)nq时,态|Ψ(2)〉q始终不呈现N-Y测不准态、N-Y最小测不准态和N次方Y压缩效应.     

McMillan T_c公式的解析推导(Ⅱ) μ~*≠0情形

本文把文献[1]的理论以及所得到的T_c公式推广到μ≠0情形,得到 T_c=2r/πω_(log)·(ω_(log)/ω_c)~μ/(λ-μ)·exp{-(1 λ)/(λ-μ)}.nγ=C=0.5772是Euler常数。     

Nd1.85Ce0.15 CuO4-δ单晶c-轴方向电阻的低场行为

采用自助熔剂缓冷法成功地生长出了Nd1.85Ce0.15CuO4-δ单晶,其零场下零电阻温度约为21K.在0-0.5T范围内分别测量了磁场平行和垂直样品表面的电阻转变曲线以及0.5T不同角度下的电阻转变曲线.结果显示磁场平行和垂直样品表面时的转变温度Tp随磁场的变化均服从H=H0(1-Tp(h)/Tp(0))2关系.0.5T不同角度下的转变温度Tp符合有效质量模型,即Tp∝H1/12(cos2θ+sin2θ/γ2)1/4.该单晶的各向异性因子γ约为45.     

McMillan Tc公式的解析推导(Ⅱ)——μ*≠0情形

本文把文献[1]的理论以及所得到的T_c公式推广到μ^*≠0情形,得到T_c=(2γ)/πω_log·(ω_log/ω_c)^{(μ*/(λ-μ*))}·exp{-(1+λ)/(λ-μ^*)}. Inγ=C=0.5772是Euler常数。 关键词：     

螺旋线型迥旋加速器调变磁场曲线形式的选择

本文中对具有调变场形式为f=sin(Φ(γ)-Nφ)的螺旋形迴旋加速器作了研究,讨论了选择调变曲线形式Φ(γ)的方法,从而可求得满足某些条件下的最合适的曲线形式,该形式在加速器中心为托马斯型,在外半径区域可用几个不同参数的阿基米德螺旋线分段接近,采用这曲线后有很多优点,例可使中心区的散焦力减少到最小,非线性共振亦减弱,并且为了克服中心区的散焦力,在相同的滑相条件下,所要求的D盒加速电压亦大为降低,并使外半径区具有最大磁间隙。     

四态叠加多模叠加态光场|Ψ_e~((4)),Ⅲ〉_q的等阶N次方Y压缩

根据量子力学的线性叠加原理 ,构造了由多模偶相干态与多模虚偶相干态组成的第 种四态叠加多模叠加态光场态 |Ψ(4)e , 〉q的等阶 N次方 Y压缩特性 .结果发现 :1 )当压缩阶数 N=4m,( m=1 ,2 ,3 ,… )时 ,态 |Ψ(4)e , 〉q 恒处于等阶数 N-Y最小测不准态 ;2 )当压缩阶数 N =4m′+2 ,( m′=0 ,1 ,2 ,… )时 ,在 (θ1-θ2 ) ,q,Rj,r1,r2 等取不同的组合定值下 ,态|Ψ(4)e , 〉q可分别呈现出等阶 N次方 Y压缩效应与“半相干态”效应 ;3 )当压缩阶数 N为奇数时 ,在 (θ1-θ2 ) ,q,Rj,r1,r2 等取不同的组合定值下 ,态 |Ψ(4)e , 〉q 可呈现出等阶 N次方 Y压缩效应     

Time-like geodesic structure of a spherically symmetric black hole in the brane-world

Recently Malihe Heydari-Fard obtained a spherically symmetric exterior black hole solution in the brane-world scenario, which can be used to explain the galaxy rotation curves without postulating dark matter. By analysing the particle effective potential, we have investigated the time-like geodesic structure of the spherically symmetric black hole in the brane-world. We mainly take account of how the cosmological constant α and the stellar pressure β affect the time-like geodesic structure of the black hole. We find that the radial particle falls to the singularity from a finite distance or plunges into the singularity, depending on its initial conditions. But the non-radial time-like geodesic structure is more complex than the radial case. We find that the particle moves on the bound orbit or stable (unstable) circle orbit or plunges into the singularity, or reflects to infinity, depending on its energy and initial conditions. By comparing the particle effective potential curves for different values of the stellar pressure β and the cosmological constant α, we find that the stellar pressure parameter β does not affect the time-like geodesic structure of the black hole, but the cosmological constant α has an impact on its time-like geodesic structure.     

Geodesic Structure of a Non-linear Magnetic Charged Black Hole Surrounded by Quintessence

This paper aims to investigate the geodesic motion in the spacetime of a non-linear magnetic charged black hole surrounded by quintessence. By varying the Lagrangian corresponding to the metric, the orbital motion equation has been obtained. The effects of the magnetic charge $Q$, positive normalization factor $C$, angular momentum $b$, and energy $E$ on time-like and null geodesic motion are discussed from three aspects: orbital stability, orbital types, and circular orbits. By comparing the effects of the above parameters $C$, $b$ on the effective potential, it is found that quintessence has an impact on the types and stability of orbits. In addition, for time-like orbital motion, when $3.443113\leq b\leq6.392\,578$ (for fixed $C=0.0002$, $M = 1$, $Q=0.7$), there are bound orbits, and within this range, the stable circular orbits exist, and the radii of the innermost and outermost stable circular orbit are $r=5.912\,654$ and $r=56.745\,933$, respectively. For null orbital motion, the orbital types have only unstable circular orbit which occur at $r=2.951\,072$ ($E^{2}=E_{2}^{2}=0.4$), absorb orbits and escape orbits, but no stable circular orbits, and bound orbits.     

A numerical examination of certain null geodesies of a high angular momentum kerr field

An observer situated anywhere but in the equatorial plane of a high angular momentum Kerr field cannot see the ring singularity. In the visual field of such an observer, what demarcates his own universe from that through the ring? The projections onto a certain submanifold of the null geodesics which pass through a point on the symmetry axis of a specific Kerr field are examined numerically. All the distinct projections are obtained by varying one parameter, essentially the quadratic Killing tensor constant. Various interesting features of the geodesics emerge. Through the ring is a region in which there exist closed time-like curves and which can be used to construct closed time-like curves through any non-singular point of the manifold. Only geodesies of negative angular momentum can enter this region.     

A numerical examination of certain null geodesies of a high angular momentum kerr field

An observer situated anywhere but in the equatorial plane of a high angular momentum Kerr field cannot see the ring singularity. In the visual field of such an observer, what demarcates his own universe from that through the ring?The projections onto a certain submanifold of the null geodesics which pass through a point on the symmetry axis of a specific Kerr field are examined numerically. All the distinct projections are obtained by varying one parameter, essentially the quadratic Killing tensor constant. Various interesting features of the geodesics emerge.Through the ring is a region in which there exist closed time-like curves and which can be used to construct closed time-like curves through any non-singular point of the manifold. Only geodesies of negative angular momentum can enter this region.     

Algebraic classification of gravitational fields in five-dimensional space-time

We consider the algebraic classification of five-dimensional empty space-time (Kalutsa type) with one time-like direction as a generalization of the Petrov algebraic classification of gravitational fields in four-dimensional space-time. We study two special cases: a) zero electromagnetic field and zero scalar field; b) nonzero electromagnetic field and zero scalar field. For the (1+4) separated Kalutsa five-metric we introduce the pentad metric of a tangent five-space, which is mapped together with the curvature tensor into a ten-dimensional real flat vector space. The classification is constructed in local geodesic coordinates for the above two cases. In both cases the characteristic equation can be reduced to a sixth-order equation that can be simplified when certain requirements are satisfied. Our results demonstrate the nontrivial nature of algebraic classification in five dimensions.This work was performed within the framework of the State Science and Technology Astronomy program.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 73–78, March, 1995.     

A two-point boundary value problem on a Lorentz manifold arising in A. Poltorak’s concept of reference frame

Abstract

In A. Poltorak’s concept, the reference frame in General Relativity is a certain manifold equipped with a connection. The question under consideration here is whether it is possible to join two events in the space-time by a time-like geodesic if they are joined by a geodesic of the reference frame connection that has a time-like initial vector. This question is interpreted as whether an event belongs to the proper future of another event in the space-time in case it is so in the reference frame. For reference frames of two special types some geometric conditions are found under which the answer is positive.     

Stability analysis and quasinormal modes of Reissner–Nordstrøm space-time via Lyapunov exponent

We explicitly derive the proper-time (τ) principal Lyapunov exponent (λ_p) and coordinate-time (t) principal Lyapunov exponent (λ_c) for Reissner–Nordstrøm (RN) black hole (BH). We also compute their ratio. For RN space-time, it is shown that the ratio is \(({\lambda _{p}}/{\lambda _{c}})={r_{0}}/{\sqrt {{r_{0}^{2}}-3Mr_{0}+2Q^{2}}}\) for time-like circular geodesics and for Schwarzschild BH, it is \(({\lambda _{p}}/{\lambda _{c}})={\sqrt {r_{0}}}/{\sqrt {r_{0}-3M}}\). We further show that their ratio λ_p/λ_c may vary from orbit to orbit. For instance, for Schwarzschild BH at the innermost stable circular orbit (ISCO), the ratio is \(({\lambda _{p}}/{\lambda _{c}})|_{r_{\text {ISCO}}=6M}=\sqrt {2}\) and at marginally bound circular orbit (MBCO) the ratio is calculated to be \(({\lambda _{p}}/{\lambda _{c}})|_{r_{\mathrm {m}\mathrm {b}}=4M}=2\). Similarly, for extremal RN BH, the ratio at ISCO is \(({\lambda _{p}}/{\lambda _{c}})|_{r_{\text {ISCO}}=4M}={2\sqrt {2}}/{\sqrt {3}}\). We also further analyse the geodesic stability via this exponent. By evaluating the Lyapunov exponent, it is shown that in the eikonal limit, the real and imaginary parts of the quasinormal modes of RN BH is given by the frequency and instability time-scale of the unstable null circular geodesics.     

Influence of dark energy on time-like geodesic motion in Schwarzschild spacetime

In this paper we investigate the influence of the dark energy on the time-like geodesic motion of a particle in Schwarzschild spacetime by analysing the behaviour of the effective potential which appears in an equation of motion. For the non-radial time-like geodesics, we find a bound orbit when the particle energy is in an appropriate range, and also find another possible orbit, which is that the particle drops straightly into the singularity of a black hole or escapes to infinity. For the radial time-like geodesics, we find an unstable circular orbit when the particle energy is the critical value, in which case it is possible for the particle to escape to infinity.     

Geodesic motions in extraordinary string geometry

The geodesic properties of the stationary vacuum string solution in (4+1) dimensions with momentum flow along the string direction are analyzed by using Hamilton-Jacobi method. The geodesic motions show distinct properties from those of the static one. Especially, any freely falling particle can not arrive at the horizon or singularity. To get into the horizon, a particle need to follow a non-geodesic trajectory. There exist stable null circular orbits and bouncing timelike and null geodesics. In the asymptotic geometry, some geodesics will be repelled by the string contrary to the case of Kerr-Neumann black hole. The light bending effect will be minimized at an impact parameter determined by the angular momentum and energy.     

光纤光栅非线性对色散特性的影响

通过在能量较高、考虑非线性时,求解光纤光栅非线性耦合模方程连续波条件下的解,得到光纤光栅失谐量δ与光脉冲传播常量q的非线性色散关系和光栅致群速度色散系数βg2与δ的关系.用MATLAB绘图,得到非线性参量γ和光脉冲能量P0的乘积γP0对色散和βg2的影响.结果表明:随着βg2的增加,非线性色散曲线的上、下两支向δ的负值区移动,当超过某一临界值时,曲线上支开始形成环,这时光纤光栅引起的群速度色散中的反常色散区消失,全部变成正常色散.