照相法光谱分析工作曲线计算机拟合的数学模型

前言在科学实验和统计研究中,常常需要从一组测得的数据,例如N对(x_i,y_i)去求得自变量x和因变量y的一个近似函数关系式y=f(x)。从图形上看,就是从给定的N个点(x_i,y_i)(其中i=1,2,3,…N)去拟合曲线的问题。许多科学上的经验式,例如物质比热与温度的关系式C_p=a+bT+cT~2,就是用曲线拟合法求得的。由于测量数据总是会有误差的,因此曲线拟合的任务,并不要求所求的曲线通过所有的点(x_i,y_i),而是要绘出一条近似的曲     

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:     

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.     

可忽略自变量测量误差的判断条件

在物理实验中,对组合测量而得的n个数据点(x_i,y_i)用直线方程y=A+Bx进行拟合时,通常需假设自变量x为无误差(或小误差)测量,而因变量y为等精度测量。在什么情况下我们只需考虑因变量的测量误差,而可忽略自变量的测量误差呢? 首先,用x、y,各自的测量标准误差σ_x、σ_y作为判断量直接进行比较大概是不行的,因为它们的量纲一般不同,数值大小也可随度量单位的扩大(或缩小)而缩小(或扩大),根本无法直接比较它们的大小。其次,用x、y各自的测量标准误差与平均值之比σ_x/(?)、σ_y/(?)作为判断量进行比较,大概也是不行的,这是因为当度量单位     

Rational Solutions for the Fokas System

Fokas system is the simplest(2+1)-dimensional extension of the nonlinear Schrodinger equation(Eq.(2),Inverse Problems 10(1994) L19-L22).By using the bilinear transformation method,general rational solutions for the Fokas system are given explicitly in terms of two order-N determinants T_n(n = 0,1) whose elements m_(i,j)~(n)(n = 0,1;1≤i,j≤N)are involved with order-n_i and order-n_j derivatives.When N = 1,three kinds of rational solution,i.e.,fundamental lump and fundamental rogue wave(RW) with n_1 = 1,and higher-order rational solution with n_1 2,are illustrated by explicit formulas from T_n(n = 0,1) and pictures.The fundamental RW is a line RW possessing a line profile on(x,y)-plane,which arises from a constant background with at t 0 and then disappears into the constant background gradually at t 0.The fundamental lump is a traveling wave,which can preserve its profile during the propagation on(x,y)-plane.When N ≥2 and n_1 =n_2=...=n_n = 1,several specific multi-rational solutions are given graphically.     

直线拟合

一、最小二乘原理在物理实验中,经常对某一待测量x进行重复测量。设n次重复测量所得测量数据为x_1,x_2…,x_n。如何由此最终表达出待测量x的结果呢?这里并没有唯一的方法。如果选择最佳结果表达值X的准则,是其与测得值x_i的极差最小,那末X应由所得数据组的中列数(midrange estimate),即[(x_i)_(max) (x_i)_(min)]/2     

New Symmetry Reductions,Dromions—Like and Compacton Solutions for a 2D BS（m,n） Equations Hierarchy with Fully Nonlinear Dispersion

We have found two types of important exact solutions,compacton solutions,which are solitary waves with the property that after colliding with their own kind,they re-emerge with the same coherent shape very much as the solitons do during a completely elastic interaction,in the (1 1)D,(1 2)D and even (1 3)D models,and dromion solutions (exponentially decaying solutions in all direction) in many (1 2)D and (1 3)D models.In this paper,symmetry reductions in (1 2)D are considered for the break soliton-type equation with fully nonlinear dispersion (called BS(m,n) equation)ut b(u^m)xxy 4b(u^n δx^-1uy)x=0,which is a generalized model of (1 2)D break soliton equation ut buxxy 4buuy 4buxδx^-1uy=0,by using the extended direct reduction method.As a result,six types of symmetry reductions are obtained.Starting from the reduction equations and some simple transformations,we obtain the solitary wavke solutions of BS(1,n) equations,compacton solutions of BS(m,m-1) equations and the compacton-like solution of the potential form (called PBS(3,2)) ωxt b(ux^m)xxy 4b(ωx^nωy)x=0.In addition,we show that the variable ∫^x uy dx admits dromion solutions rather than the field u itself in BS(1,n) equation.     

Exact solutions of a(2+1)-dimensional extended shallow water wave equation

We give the bilinear form and n-soliton solutions of a(2+1)-dimensional [(2+1)-D] extended shallow water wave(eSWW) equation associated with two functions v and r by using Hirota bilinear method. We provide solitons, breathers,and hybrid solutions of them. Four cases of a crucial φ(y), which is an arbitrary real continuous function appeared in f of bilinear form, are selected by using Jacobi elliptic functions, which yield a periodic solution and three kinds of doubly localized dormion-type solution. The first order Jacobi-type solution travels parallelly along the x axis with the velocity(3k_1~2+ α, 0) on(x, y)-plane. If φ(y) = sn(y, 3/10), it is a periodic solution. If φ(y) = cn(y, 1), it is a dormion-type-Ⅰ solutions which has a maximum(3/4)k_1p_1 and a minimum-(3/4)k_1p_1. The width of the contour line is ln■. If φ(y) = sn(y, 1), we get a dormion-type-Ⅱ solution(26) which has only one extreme value-(3/2)k_1p_1. The width of the contour line is ln■. If φ(y) = sn(y, 1/2)/(1 + y~2), we get a dormion-type-Ⅲ solution(21) which shows very strong doubly localized feature on(x, y) plane. Moreover, several interesting patterns of the mixture of periodic and localized solutions are also given in graphic way.     

用超导转变点和气体三相点温度监测标准铑铁电阻温度计

用 SRM767a 超导转变固定温度点器件中的六个 T_c 点和密封气体三相点瓶复现的三个气体三相点在 0.5—25K 间对标准铑铁电阻温度计做九点监测,并用多项式R=sum from k=0 to 6 bk[ln(T+9)]~k做最小二乘法拟合,发现受验证的16支温度计对分度值的偏差≤2mK.     

用雅可俾函数行列式指导热力学关系

一 、引言1.在本文提出的方法中要用到: 1.雅可俾函数行列式的三个主要性质[1]。设 x=x(u,v) y=y(u,v) u=u(s, t)v=v(s, t)雅可俾函数行列式定义为J的性质如下:令 2.对化学性质没有变化的均匀系,有下面四个麦克斯韦关系[2] 二、方法和例证 1、在热力学关系中只含有体系状态参量T、p、V、S时,首先将偏微分关系写成雅可俾函数行列式的形式,然后乘以D(x,y)/D(x,y),而其中变量(x,y)对绝热过程是o,V)或…。p)对共他过程是(T,D戍iT,P｝。最后利用雅可饶函数行列式的性质和麦克斯韦莱系,即可得到所需要的热力学关系。、_、;_-。。 例如,我们要…     

一种二元溶液折射率的计算方法

在20℃下配制了一系列浓度的丙三醇—乙醇溶液,测定了各溶液的密度ρ与折射率n。用一种电磁感应原理导出的计算方法n=1+ρ(x1D1+x2D2)对这些丙三醇—乙醇溶液的折射率进行了计算,结果表明,此方法之折射率计算值与实测值符合很好。     

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;     

特殊洛仑兹变换的一种简单推导

本文介绍一种简便的推导特殊洛仑兹变换的方法以供初学者参考. 1.根据相对性原理和初始时刻(t=t‘=0)坐标系∑(x’、y’、z’)和∑(x,y,z原点(0’,0)重合可知,变换必须是线性齐次的. 2.如图示,因xoy面与x’o’y’面始终重合,故无论x、y.t、x’、y’、t’取何值,z=0,z’=0总是同时成立.、所以z’=αz其中α为常数.考虑到z与z之间相互交换是对等的.应有z=αz’则有α=±1,因z’轴与z轴指向相同,应取α=1,即得z=z’(1) 同理,考察zOx面与z’O’x’而始终重合,可得 y=y’(2) 3.因yoz面与y’o’z’面始终平行,在某一时刻t,∑’系的原点o’对∑系…     

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.     

V_2Ga_5的晶体结构

本文利用单晶与粉末衍射方法测定了V_2Ga_5的晶体结构。V_2Ga_5属四方晶系,其单相区约为VGa_2—V_2Ga_5。在18℃的点阵常数是a=8.9540,c=2.6892,每个晶胞含有二个化合式量,空间羣为D_(4h)~5—P4/mbm。V原子与Ga原子分别占据在4(h)与8(i),2(d)的等效位置上。参数为:x_h=0.180与x_i=0.070,y_i=0.210。     

Fusion,fission, and annihilation of complex waves for the(2+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff system

With the help of the symbolic computation system, Maple and Riccati equation( ξ= a0+ a1ξ+ a22ξ), expansion method, and a linear variable separation approach, a new family of exact solutions with q = lx + my + nt + Γ(x, y,t) for the(2+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff system(GCBS) are derived. Based on the derived solitary wave solution, some novel localized excitations such as fusion, fission, and annihilation of complex waves are investigated.     

Explicit Relations of Physical Potentials Through Generalized Hypervirial and Kramers' Recurrence Relations

Based on a Hamiltonian identity,we study one-dimensional generalized hypervirial theorem,Blanchardlike(non-diagonal case) and Kramers'(diagonal case) recurrence relations for arbitrary x~κ which is independent of the central potential V(x).Some significant results in diagonal case are obtained for special κ in x~κ(κ≥ 2).In particular,we find the orthogonal relation(n_1|n_2) = δ_(n_1n_2)(κ = 0),(n_1|V'(x)\n_2) =(E_(n_1)-E_(n_2))~2〈n_1x|n_2)(κ = 1),E_n =(n/V'(x)x/2|n) +(n|V(x)|n)(κ = 2) and-4E_n(n|x|n) +(n|V'(x)x~2\n〉 +4〈n|V(x)x|n〉 = 0(κ = 3).The latter two formulas can be used directly to calculate the energy levels.We present useful explicit relations for some well known physical potentials without requiring the energy spectra of quantum system.     

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.     

Hydrogen storage capabilities of the most stable isomers of NanBm (m+n=6) clusters

The most stable isomers of NanBm(m+n=6) clusters and their hydrogen storage properties are investigated by means of density functional theory with a 6-311+G(d) basis set. To study the hydrogen storage properties,all of the stable structures of Na n BmHx (m+n=6) clusters have been optimized. It shows that boron atoms of Na n B m are separated from the other boron atoms,and form satellite BHx (x=3,4) clusters around the centre,which attach to the system by a bridging bond of a hydrogen atom or an Na atom. Compared with the hydrogen storage capabilities,the Na3B3 has the highest hydrogen storage capacity among Na n B m clusters. The binding energies,interaction energies of hydrogen atom with Na n B m clusters and second difference in energy of Na3B3Hx clusters have been calculated. The results show that the stability of the Na n B m H x clusters present an odd-even oscillatory effect,as the number of H atoms increases.     

N维势箱函数量子特性的可视化研究

运用量子理论推导和数值计算相结合的方法,本文首先得到了一维势箱函数的示意图及其模型.接着,全面、系统地研究了量子理论中N维势箱函数的波函数、能级和概率密度.最后,运用MATLAB软件对势箱函数的所有特性进行了仿真模拟.我们发现:N维势箱中粒子的能量是量子化的、不连续;量子数n不能为零,且n越大对应的能级越高,而质量m越大,对应的能级越低.一般条件下,一维势箱长度a越大(粒子运动范围越大),对应的能级越低;节点数为n-1,节点越多,波长越短,频率越高,能级越高.二维势箱函数波函数的峰值个数为n x 0x0E?SymboltB@0x0Fn y,且与Ψ=0平面的交线数也为n x 0x0E?SymboltB@0x0Fn y;概率密度分布的极大值个数也为n x 0x0E?SymboltB@0x0Fn y.对于简并度,一般情况下,二维势箱模型下的粒子的简并度是不确定的;但对于二维正方势箱函数模型,其箱内微观粒子的能级简并度分为特殊和一般两种情况.三维势箱函数的简并度为n x+n y+n z.最后,首次借助MATLAB软件的色彩实现了四维表现,得到了三维势箱函数的四维空间切片图.这种可视化的结果与理论结果完全一致,这对于抽象性概念的理解具有重要意义.