Coefficients Definition and 805 Threads

Homework Statement Find coefficients a,b>0 such that a||x||∞≤||x||≤b||x||∞.Homework EquationsThe Attempt at a Solution No idea how to get started. Help will be appreciated.

Homework Statement So i have to calculate the Clebsch-gordan coefficients for the state j1=3/2 and j2=1/2[/B]Homework Equations Recursiom formula, lowering and uppering operator[/B]The Attempt at a Solution I have tried to calculate the first set of the coefficients stating that: L- l2,2> =...

Suppose that we have a 2\pi-periodic, integrable function f: \mathbb{R} \rightarrow \mathbb{R}, whose continuous Fourier coefficients \hat{f} are known. The convolution theorem tells us that: $$\displaystyle \widehat{{f^2}} = \widehat{f \cdot f} = \hat{f} \ast \hat{f},$$ where \ast denotes the...

Homework Statement Assume a two level laser system with no degeneracy (g1 = g2 = 1). If [tex]N_2 + N1 = \text{constant}[\tex], show that [tex]B12 = B21[\tex]. Homework Equations [tex]\frac{\partial N_1}{\partial t}=-B_{12}\rho(v_{21}N_1[\tex] [tex]\frac{\partial N_2}{\partial...

Is there a way of writing summation(s) to obtain the extended binomial coefficients? i.e., Considering the expansion of (1+x+x^2+x^3+...+x^N)^M can we write expressions (presumably involving summation and/or product notation) for the coefficients (on x^j in the expansion of the above, for each...

Homework Statement A block of mass ## M_1 ## is on an inclined plane (the plane is inclined at 30 degrees), and is attached with a light cord over a light, frictionless pulley to another mass ## M_2 ##. ## M_1 ## is fixed at 5.00kg, and ## M_2 ## can be varied. When ## M_2 = 3.20 kg ##, ##...

given :$f(x)=x^3+ax^2+bx+1$ with $a\geq 0 \,\, and \,\, b\geq 0$ and also all of the three roots of $f(x)=0$ are real numbers, prove $f(2)\geq 27$

I managed to get through all of the problems in chapter three of Problem Solving through Problems, and I am now on to chapter 4 in the section on polynomials. A few problems I have encountered so far involve polynomials of the form: P(x) = 1 + x + x2 + x3 +...+xn I noticed that when n, the...

I am reviewing the method of partial fraction decomposition, and I get to the point that I have a matrix equation that relates the coefficients of the the original numerator to the coefficients of the numerators of the partial fractions, with the each column corresponding to a certain polynomial...

Homework Statement Find coefficients ##b_n## \sum^{\infty}_{n=0}\frac{t^n}{n!}b_n=e^{\frac{4at}{3}}e^{-\frac{t^2}{3}} a is constant.Homework Equations e^x=\sum^{\infty}_{n=0}\frac{x^n}{n!}The Attempt at a Solution Here e^{\frac{t}{3}(4a-t)}=\sum^{\infty}_{n=0}\frac{( \frac{t}{3}(4a-t))^n}{n!} I...

Hello, I remember an example of application of the logistic regression to medicine / epidemiology, which said (more or less) that the probability of a person having miocardial infarction was related to some variables such as age, cholesterol level, etc, and the equation included the various...

I'm trying to clarify my terminological understanding of things related to mathematical expressions. Actually, first I want to make sure I understand what an expression is and is not. From my current understanding, an expression is any number, variable, or combination of numbers and...

Homework Statement Homework EquationsThe Attempt at a Solution Very confused by this problem. For one thing, it doesn't specify if there is or isn't any light present to drive the stimulated emission/absorbtion. I guess there's no reason to assume that there is no light - but since the...

What has the highest coefficient of kinetic friction, steel on polyethylene, rubber on polyethylene, or polyethylene on polyethylene?

Hi there, I'm working on a problem right now that relates to least squares error estimate for polynomial fitting. I'm aware of techniques (iterative formulas) for finding the coefficients of a polynomial that minimizes the square error from a data set. So for example, for a data set that I...

Hi all. I'm currently working on a problem where I have to determine individual and overall heat transfer coefficients for a vertical cylinder with water flowing on the inside and steam on the outside (parallel flow), studying the cases of free and forced convection. I only have four...

Hello 1. Homework Statement Find the Fourrier coefficients in the annulus problem of the text. uxx+uyy=0 in 0<a²<x²+y²<b² u=g(θ) for x²+y²=a² u=h(θ) for x²+y²=b² Homework Equations The solution is The Attempt at a Solution I have the solutions but when I solved it for...

Hello! I am taking a course in differential equations and the book I am using is "Elementary Differential Equations" - E. Boyce & R. DiPrima (tenth edition) My question is about guessing the form of a particular solution to a nonhomogenous system of equations. Given a nonhomogenous system of...

Homework Statement I need to calculate the coefficients for open circuit voltage (Voc) and short circuit current (Isc). I know that the higher temperature = lower Voc, higher Isc. I've done a bit of searching but I can't seem to find an equation or any theory about the coefficients though...

I have two questions on fine points of thermal physics. 1. Is an isotropic solid's volume-expansion coefficient beta exactly equal to 3 times its linear-expansion coefficient alpha, or is it only approximately equal? Some textbooks (Fischbane, Walker, Young) say the relation is exact, while...

Let $a$ be the smallest root of the equation $x^2-9x+10=0$. Find $a^4-549a$. Extra credit if the solution does not find the actual roots of the equation.

I am reading Segei Winitzki's book: Linear Algebra via Exterior Products ... I am currently focused on Section 1.6: Dual (conjugate) vector space ... ... I need help in order to get a clear understanding of the notion or concept of coefficients of a vector v as linear functions (covectors) of...

I am reading Segei Winitzki's book: Linear Algebra via Exterior Products ... I am currently focused on Section 1.6: Dual (conjugate) vector space ... ... I need help in order to get a clear understanding of the notion or concept of coefficients of a vector v as linear functions (covectors)...

Why are ##<j_1j_2m_1m_2|jm>## and ##<j_2j_1m_2m_1|jm>## negative of each other when ##j_1+j_2-j## is odd as given below? I would expect ##<j_1j_2m_1m_2|jm>## and ##<j_2j_1m_2m_1|jm>## to always have the same sign since nature doesn't care which particle we label as particle 1 and which as...

I'm reading through some lecture notes for QM in a subsection about stationary states where the definition of orthonormality involving a kronecker delta \int_{-\infty}^{\infty}\psi_n(x) \ \psi_m^*(x)dx=\delta_{m,n}and the formula for some wavefunction that is a superposition of energy...

Homework Statement Find all real numbers k such that x^2+kx+k is reducible in ℝ[x]. Homework EquationsThe Attempt at a Solution This seems like it is simple, but it is new to me so I am looking for confirmation. We know we can find the roots of a polynomial with b^2-4ab. We want b^2-4ab to be...

Homework Statement Hello all, Im asked to construct the state | \frac{5}{2} , \frac{3}{2} \rangle from the eigenfunctions | L, L_z\rangle and the electron states | \uparrow \rangle and | \downarrow \rangle . Homework Equations Clebsch Gordon Coefficient's table The Attempt at a...

Homework Statement In my PDE course we have a homework question stating the following: Let ϑ(x) = x in the interval [-pi, pi ]. Find its Fourier Coefficients. Homework Equations From my notes on this type of question: a_o = 2c_o = 1/pi * integral from -pi to pi [f(x) dx] a_n = c_n + c_(-n)...

If I have a 60-40 beam splitter, I have a semitransparent plate that will classicaly reflect 60% of the light and transmit 40%. On quantum level, photons take both trajectories at the same time, but with different probability coefficients. So is quantum superposition of trajectories for...

Use the method of undetermined coefficients to find a general solution of the ODE: $y''+3y'+2y=2x^{2}+4x+5$$r^{2}+3r+2=0$ $r=-2$ and $r=-1$ $y_{h}=c_{1}e^{-2x}+c_{2}e^{-x}$ I'm not sure how to get $y_{p}$ here So here's what I've done so far. I have my final exam tomorrow and I have a few...

Is there a way to find the following sum in closed form: ∑K(N,n) , where K(N,n) is the binomial coefficient and the sum can extend over any interval from n=0..N. I.e. not necessarily n=0 to N in which case on can just use the binomial theorem.

Evaluate \sum\limits_{k=1}^{12} {12\choose{k}}k^2 The answer is 159744.

If you were to calculate work done by rolling friction of a wood sphere rolling down an incline and compare it to work done by friction of a wood block sliding down an incline, are the values for the coefficient of rolling and kinetic friction somehow proportional to each other? This is for a...

Homework Statement Solve the following differential equation and compare computer solutions \begin{equation*} 4y''+12y'+9=0 \end{equation*} Homework Equations None The Attempt at a Solution [/B] First of all looking at this equation, even though it is in the section where we learned about...

Hi there. I've been struggling with this problem for days now (4 days, no joke) and I feel like I have a mental block and really cannot get any further. I have a system that's described by f(t) = g''(t) + 15g'(t) + 1600g(t) Where the input is g(t) The problem is to, with this information...

I'm setting up a thermal resistance network to determine the heat loss through the insulated wall of a heated fuel/crude oil tank but i can't seem to find values for the convective heat transfer coefficient of the fuel oil (kerosene) or the crude oil at elevated temperatures. I have only found...

Homework Statement I am trying to understand how to obtain the required wind coefficients of 0.60, -0.30 and -0.60 for the problem, by using the given figure Homework EquationsThe Attempt at a Solution for the windward side, does it correspond to theta = 0 since the question says "normal...

Homework Statement z^6+(2i-1)z^3-1-i=0 Homework EquationsThe Attempt at a Solution I know that I must put k=z^3 and solve the quadratic. But I'm not able to simplify the quadratic. I get the square root of (-8i+1) What am I supposed to do ?

Homework Statement a particle is in a linear superposition of two states with energy E_0 \ and\ E_1 |\phi> = A|E_0> + \frac{A}{(3-\epsilon)^{1/2}}|E_1> where: A \ > \ 0, \ 0\ <\ \epsilon \ <\ 3 What is the value of A expressed as a function of epsilon Homework Equations P(E_0) \ +\...

I haven't taken a course on quantum mechanics yet, but I was asked to solve (numerically) ##[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)]\phi(x)=E\phi(x) ## ##V(x)=2000(x-0.5)^2## by supposing the solution is ##\sum_{0}^{\infty} a_n \phi_n(x)## and ##\phi_n(x)## is the typical solution to the a...

Hey! :o Each element of the ring $\mathbb{C}[z, e^{\lambda z} \mid \lambda \in \mathbb{C}]$ is of the form $\displaystyle{\sum_{k=1}^n α_kz^{d_k}e^{β_kz}}$. A differential equation in this ring is of the form $$Ly = \sum_{k=0}^m \alpha_k y^{(k)}(z)=\sum_{l=1}^n C_lz^{d_l} e^{\beta_l z} , \ \...

Homework Statement How to use binomial theorem for finding sums with binomial coefficients? Example: S={n\choose 1}-3{n\choose 3}+9{n\choose 5}-... How to represent this sum using \sum\limits notation (with binomial theorem)? Homework Equations (a+b)^n=\sum\limits_{k=0}^{n}{n\choose...

Homework Statement A brass rod (linear expansion coef 19*10-6/°C) has a radius of 8139 cm and a copper (linear expansion coef 17*10-6/°C) sheet has a hole or radius 8139+8.86cm. What temperature change (K) is required to close the gap? Homework Equations ΔL/Lo = αΔT? The Attempt at a Solution...

Homework Statement A block is projected up a 30° slope at 4m/s. When it is moving the coefficient of friction is 0.3. When the block is stationary the maximum coefficient of friction is 0.5. a. Calculate how far up the block will go. b. Show that the block will not remain there. c. Calculate...

Homework Statement Find the expansion coefficients of a generic vector (λ, μ) ∈ ℂ^2 in the orthonormal basis: { i/sqrt(2) (1, 1), 1/sqrt(2) (1, -1) } Homework Equations λ_i = |v_i> |v> The Attempt at a Solution I don't think this should be difficult. But, clearly, as I am posting here, I do...

Homework Statement When ##f## and ##g## are ##2\pi##-periodic Riemann integrable functions define their convolution by ##(f*g)(x) = \frac{1}{2\pi} \int_0^{2\pi} f(y)g(x-y)dy## Denoting Fourier coefficients by ##c_n(f)## show that ##c_n(f * g) = c_n(f)c_n(g)##. Homework Equations ##c_n =...

I'm currently working through chapter 7 on Riemannian geometry in Nakahara's book "Geometry, topology & physics" and I'm having a bit of trouble reproducing his calculation for the metric compatibility in a non-coordinate basis, using the Ricci rotation coefficients...

I've run into a couple of problems with exercise 12.6 in MTW (attached). At this stage, all I'm asking is for someone to give me answers to the following questions: 1. What are the Rα'β'γ'δ' in the primed co-ordinate system? and 2. What are the non-zero connection coefficients in the...

Homework Statement In the machine shown in the attached picture, all pulleys have negligible mass and rotational inertia. The coefficient of static friction μs between the table and either mass is larger than the coefficient of kinetic friction μk. The two masses 3m and m originally were held...

Discontinuous Coefficients Problem Hello, I need some help for part d. The book's problem stated, "Occasions arise when the coefficient P(x) in a linear equation fails to be continuous because of the jump discontinuous. Fortunately, we may still obtain a 'reasonable' solution. For example...