Coefficients Definition and 805 Threads

In CG coefficients methodology there is convention \langle J_1 J_2 J_1 J-J_1|J_1 J_2 J J \rangle \geq 0 So if we have ##J_1=1##, ##J_2=1/2## \langle 1 \quad 1/2 \quad 1/2 \quad 1|1 \quad 1/2 \quad 3/2 \quad 3/2\rangle \geq 0 using this convention. However, because ##J=3/2## is maximal value...

Hi, it is well established that octanol is the preferred solvent for partition coefficient studies in environmental chemistry, however, some studies ui.e Noble A. Parition coefficients (n-octanol-water) for pesticides) J Chromat. 642 3-14, mention that estimating toxicity based on solubility in...

Homework Statement I have to determine the sum and difference of the two coefficients of drag between two individual items that make a torque AUT. I'm given the diameter of each bottle (6.35 cm), wind velocity (14.3 m/s), the forces of the lift load cells (+0.0741 N, -0.0741 N), and the force...

$f(x)$ is a degree 10 polynomial such that $f(p)=q$, $f(q)=r$, $f(r)=p$, where $p$, $q$, $r$ are integers with $p<q<r$. Show that not all the coefficients of $f(x)$ are integers.

I'm kinda just hoping someone can look over my work and tell me if I'm solving the problem correctly. Since my final answer is very messy, I don't trust it. 1. Homework Statement We're asked to find the Fourier series for the following function: $$ f(\theta)=e^{−\alpha \lvert \theta \rvert}}...

Homework Statement Given this matrix: I am asked to find values of the coefficient of the second value of the third row that would make it impossible to proceed and make elimination break down. Homework Equations Gaussian elimination methods I used given here...

Hello, I am given 200,000 surface points in terms of x,y and z coordinates which is supposed to represent a circular paraboloid antenna reflector surface and another 200,000 points when the antenna surface is subjected to wind loads and gravity. I am supposed to find the surface RMS error of...

Homework Statement Hello, I am stuck on four equations for which I must find the coefficients A,B,I,J. I have tried using latex but the commands don't seem to work.Homework Equations Four equations: A+B = I+J \frac{\alpha}{k}(J-I) = A - B D = Ie^{ia(\alpha-k)} + Je^{-ia(\alpha + k)} D =...

Homework Statement Find a particular solution yp of the given equation: y''+y=sinx+xcosx Homework EquationsThe Attempt at a Solution 1) First I got the temporary yp: Asinx+Bcosx+(Cx+D)sinx+(Ex+F)cosx 2) Then I got the associated homogenous eq. check for to check for duplicates: C1cosx+C2sinx...

From the Maxwell Equations we know that there are four boundary conditions for an electromagnetic wave crossing an interface between two dielectric media. For the TE polarisation state, these conditions give us that E_{i} + E_{r} = E_{t} B_{i}\,\cos\theta_{i} - B_{r}\,\cos\theta_{r} =...

##\displaystyle \frac{\partial u(x,t)}{\partial t}-p(x) \frac{\partial^2 u(x)}{\partial x^2}-\frac{\partial p(x)}{\partial x}\frac{\partial u(x)}{\partial x}=f(x,t)##. With periodic boundary conditions: ##u(x,t)=u(x+2\pi,t)##. The right hand side is also a periodic function of space, and it can...

Einstein predicted constraints on the coefficients of stimulated emission and absorption of radiation by atoms. He did that by assuming that the gas of atoms had to reach thermal equilibrium. For the gas to reach thermal equilibrium the coefficients had to be related in a certain way, otherwise...

Homework Statement I was going through Sakurai's textbook and I tried to work out by myself the following expression for the Clebsch-Gordan coefficients (Addition of orbital angular momentum and spin 1/2) \begin{equation} \langle m-1/2,1/2|l+1/2,m \rangle = \sqrt{\frac{l+m+1/2}{2l+1}}...

Hello, I have an equation of the form: ##\partial_t f(x,t)+a\partial_x^2 f(x,t)+g(x)\partial_xf(x,t)=0 ## (In my particular case ##g(x)=kx## with ##k>0## and ##a=2k=2g'(x)##) I'd like to know if there is some general technique that i can use to solve my problem (for example: in the first...

Homework Statement I am self studying an introductory quantum physics text by Marvin Chester Primer of Quantum Mechanics. I am stumped at a problem (1.10) on page 11. We are given f(x) = \sqrt{ \frac{8}{3L} } cos^2 \left ( \frac {\pi}{L} x \right ) and asked to find its Fourier...

Homework Statement [/B]Homework EquationsThe Attempt at a Solution i tried to do it by writing it as ## a_{1999} x^{1999} + a_{1998} x^{1998} ... a_0 \pm1 = 0 ## for 1999 different integer values of x i am thinking of writing it as ## a_{1999} x^{1999} = -a_{1998} x^{1998} - a_{1997} x ^...

$\tiny{17.2.0}$ $\textrm{ Solve the given equation by the method of undetermined coefficients.}$ \begin{align*}\displaystyle y''+7y'+10y&=80\\ \end{align*} $\textrm{auxilary equation}$ \begin{align*}\displaystyle r^2+7r+10&=0\\ (r+2)(r+5)&=0\\ r&=-2,-5 \end{align*} then is it the...

Hi. I have been trying for sometime to solve the following system of DEs analytically(Is it possible?) but no luck so far. $$x''(t)=-z(t)x'(t)-x(t)+y(t),$$ $$y'(t)=-z(t)y(t)+x^2(t)$$ $$z'(t)=-2z^2(t)-x(t)$$. With the initial conditions ##x(0)=1## , ##x'(0)=0## ,##y(0)=0## and ##z(0)=1##...

$\textrm{Solve the given equation by the method of undetermined coefficients.}$ \begin{align*}\displaystyle y'' +6y'&=-4xe^{-6x}\\ y_p&=Ax^2e^{-6x}+Bxe^{-6x} \end{align*}ok just wanted get this posted before I leave campus so assume finding zeros is next

Where do I find a good comprehensive table of friction coefficients between various materials? I'd like, in particular, to be able to compare the coefficient of static friction between tungsten carbide and sapphire and that between sapphire and sapphire. Tables that I have found tend to have...

If a body is moving in one constant direction, is it considered stationary with respect to other directions? For example, consider a table. Let the surface of the table be defined by an x-y Cartesian plane. If the body is moving in the x direction on the table, and a force (< (μs * Fn)) is...

ok going to see if i can do the steps $\textrm{17.2.9 Solve the given equation by the method of undetermined coefficients.}\\$ \begin{align*}\displaystyle y''-y&=2e^{-x}+3e^{2} \end{align*} $\textit{ textbook answer is:}\\$ \begin{align*}\displaystyle y&=c_1e^{-x}+c_2...

Hi there, my understanding of absorption coefficients is that harder materials generally reflect more sound similarly to how a harder material will result a ball to bounce higher than a softer material. What I am having trouble finding research on and understanding is the reasoning behind the...

$\tiny{17.2.02}$ \nmh{1000} $\textrm{Solve the equation by the method of undetermined coefficients.}\\$ \begin{align*}\displaystyle y''-4y'&=\sin{x}\\ y_p&=A\sin{x}\\ \end{align*} $\textit{ answer}$ \begin{align*}\displaystyle y{\left (x \right )}& = C_{1} + C_{2} e^{4 x} - \frac{1}{17}...

$\tiny{17.2.01}\\$ $\textrm{ Solve the given equation by the method of undetermined coefficients.}$ \begin{align*}\displaystyle y''+7y'+10y&=80 \end{align*} $\textit{this is in the form}$ \begin{align*}\displaystyle x^2+7x+10&=0\\ (x+2)(x+5)&=0 \end{align*} $\textit{this is a...

Homework Statement I need to expand this piecewise function f(x) = h for a<x<L and f(x) = 0 for 0<x<a. I am told that this is a square wave so ao and an in the expansion are 0 (odd function). Therefore I only need to worry about bn. The limits on the integral are from a to L, but what about the...

I am aware that there needs to be two materials for there to be a coefficient of friction, but I mean in general. For example, I know synthetic setae are very resistant to slipping on surfaces.

hi. I am using Stata to do a regression on longitudinal data. However, it does not produce numbers for standardized coefficients. How do I make stata produce the standardised coefficients as part of the regression operation? Mons

Specify the degree and the (nonzero) coefficients of each polynomial. (A) 4 (B) 0 Solution: The number 4 can be expressed as 4x^0. Is this correct? If this is right, then the nonzero coefficient must be 4 itself. Is this right? The degree is 0. The whole number 0 can be expressed as 0x^0...

Hi all, I have a trigonometric function series $$f(x)={1 \over 2}{\Lambda _0} + \sum\limits_{l = 1}^\infty {{\Lambda _l}\cos \left( {lx} \right)} $$ with the normalization condition $$\Lambda_0 + 2\sum\limits_{l = 1}^\infty {{\Lambda _l} = 1} $$ and ##\Lambda_l## being monotonic decrescent...

Homework Statement Derive the expression for coefficients of Fourier series in exponential form for the sequence of rectangular pulses (with amplitude A, period T and duration θ) shown in this image: Derive the expression for signal power depending on the coefficients of Fourier series...

Prove, that $\sum_{j=1}^{2n-1}\frac{(-1)^{j-1}j}{{2n \choose j }} = \frac{n}{n+1}$ i have tried with proof by induction, but it is very difficult to use this technique. I should be very glad to see any approach, that can crack this nut.

Homework Statement Let v(0) = [0.5 0.5 0.5 0.5]T, v(1) = [0.5 0.5 -0.5 -0.5]T, v(2) = [0.5 -0.5 0.5 -0.5]T, and z = [-0.5 0.5 0.5 1.5]T. a) How many v(3) can we find to make {v(0), v(1), v(2), v(3)} a fully orthogonal basis? b) What are z's coefficients of expansion αk in the basis found in...

Dear Everyone, I would like to ask a question regarding the evaluation of the coefficients in Linear Combination of Atomic Orbitals (LCAO). In Molecular Quantum Mechanics book authored by Peter Atkins and Ronald Friedman (Fourth Edition ISBN 0199274983), we are trying to calculate the...

Homework Statement The question asks me to calculate the non-zero Clebsch Gordan coefficients ##\langle j_1+j_2, j_s-2|j_1,m_1;j_2,m_2\rangle## Where ##j_s=j_1+j_2##. Homework EquationsThe Attempt at a Solution The ##j_s-2## part is the ##m## of a ##|j,m\rangle## and I know that m has to equal...

What exactly is the difference between something like (XY)2 and 2XY? I get that the 2XY would indicate 2 moles of XY, but I don't see why (XY)2 wouldn't be the same thing, so long as the whole formula is surrounded by parenthesis? thanks for any help

Oiy. It is my 6th year teaching physics and early on, I diverged from the textbook (Holt Modern Physics) regarding how they handled "g". To me, it made more sense that "g" was -9.81m/s2 , while the text handled it as "g" = 9.81 m/s2, and negatives are assigned directionally. Overall, my...

Is there a general way to express, for instance, the coefficient of the order $x^j$ term in the expression $$\frac{\Sigma_{n}^{\infty}a_nx^n}{\Sigma_{m}^{\infty}b_mx^m}$$ ? Basically I am working with a quotient of two infinite power series and I want to know the term in this quotient that is...

Homework Statement A transmission line has the primary coefficients as given below. ##R=2\Omega/m## ##L=8 nH/m## ##G=0.5 mS/m## ##C=0.23 pF/m## Determine the lines secondary coefficients ##Z0##. ##\alpha## and ##\beta## at a frequency of ##1 GHz## Homework Equations [/B] In my notes I am...

Homework Statement z2-(3+i)z+(2+i) = 0 Homework EquationsThe Attempt at a Solution [/B] Does the quadratic formula work in this case? Should you deal with the real and complex parts separately?

Homework Statement I have ## f(t) = \sum\limits^{\infty}_0 a_{n} e^{2 \pi i n t} ## [1] and ## g(t) = \sum\limits^{\infty}_0 b_{n} e^{ \pi i n t} ## [2] I want to show that ##b_n = a _{2n} ## Homework Equations see above. The Attempt at a Solution [/B] So obviously you want to use the...

I am building a business cycle index, which include 4 variables that drive the index. Each variable I also include autoregessive coefficients that are all significant and negative. I was wondering what is the significance of this? In other words, what is the significance of have autoregessive...

Homework Statement Link: http://i.imgur.com/klFmtTH.png Homework Equations a_0=\frac{1}{T_0}\int ^{T_0}_{0}x(t)dt a_n=\frac{2}{T_0}\int ^{\frac{T_0}{2}}_{\frac{-T_0}{2}}x(t)cos(n\omega t)dt \omega =2\pi f=\frac{2\pi}{T_0} The Attempt at a Solution Firstly, x(t) is an even function because...

Homework Statement The system is a spring with constant 3k hanging from a ceiling with a mass m attached to it, then attached to that mass another spring with constant 2k and another mass m attached to that. So spring -> mass -> spring ->mass. Find the normal modes and characteristic system...

sin(x) * y''(x) + 2cos(x) * y(x) = 0 y(0) = 0 y'(0) = 1 how do I solve it? (I know the solution because I have created the diff. equation starting from a simple function). -- lightarrow

Homework Statement Find the general solution y"+3y'+2y=0 Homework Equations y(t) =c_1e^r_1t + c_2e^r_2t The Attempt at a Solution a=1 b=3 c=2 r^2+3r+2=0 (r+2)(r+1)=0 r_1=-2 r_2=-1 General solution: y(t) =c_1e^(-2t)+c_2e^(-t)I was wondering if the order mattered. The answer in the book is...

Homework Statement Expand the brackets to get a cubic containing the unknowns. This is an example in the textbook but I don't see how they've expanded the brackets to get their answer: (x-3)(px^2+qx+r)=px^3+(q-3p)x^2+(r-3q)x-3r Homework Equations (x-3)(px^2+qx+r) The Attempt at a Solution I...

Hi folks, I´m trying to derivate the reflection and transmission factor between solid-solid interface according to prof. Rose´s book "ultrasonic waves in solid media". In this chapter, I´ve found there a few mistakes. however the first two rows in matrix M are correct. Can you please give me...

Hi PF! I have a system of 4 first order linear ODE's, call each ODE ##\psi_1,...,\psi_4 : \psi = \psi(x,y,z,t)##. However, there are three algebraic variables (not ##x,y,z,t##, let's call them ##c_1,c_2,c_3##) that must be solved for as well, and I have three different (non-differential)...

Ok. I will try to do my best to explain you what is my doubt. I'm not a native English Speaker and the book I was reading is not in English Lang, but I've translated it to English. In a spin-1/2 Stern Gerlach-like experiment, we can express the ket representing the spin component of the...