Let \((a, \beta) *(-r, \lambda)=(a t-\beta \lambda, a \lambda+\beta r)\)
Give \(N\left(-N,(X, y)^{N}\right.\) means \((x, y) \neq(x, y) \times \ldots(x, y) \quad(N\) times \()\)
For the set \(S=\left\{\left(\frac{4}{5}, \frac{3}{5}\right)^{N} \mid N=1,2,3 \cdots\right]\) Describe the first few elements in \(S\).
2) Describe the operations * \(n\) terms of geometry of the unit circle
Homework Answers
Observe
1)
...
2) I have plotted these points on a graph along with a unit circle below.
These points all lie on the unit circle and moves anticlockwise
with a fixed angle as 
increases in equation
.
ZILLDIFFEQ9 11.1.016.
Let \(\left\{\varphi_{n}(x)\right\}\) be an orthogonal set of functions on \([a, b]\) such that \(\psi_{0}(x)=1\) and \(\varphi_{1}(x)=x\), Show that \(\int_{a}^{b}(\alpha x+\beta) \varphi_{n}(x) d x=0\) for \(n=2,3, \ldots\) and any constants \(\alpha\) and \(\beta\),First we note that \(\alpha x+\beta=(\square) \Phi_{1}(x)+(\square \quad) \Psi_{0}(x)\).Using this together with the fact that \(\varphi_{0}\) and \(\varphi_{1}\) are orthogonal to \(\varphi_{n}\) for \(n>1\), we have the following.$$ \begin{aligned} \int_{a}^{b}(\alpha x+\beta) \varphi_{n}(x) d x &=\int_{a}^{b} a x \psi_{n}(x) d x+\int_{a}^{b} \beta \varphi_{n}(x) d x \\ &=\int_{a}^{b}\left(\square_{0}\right) \varphi_{1}(x) \varphi_{n}(x) d...
The one-dimensional infinite potential well can be generalized to three dimensions.
(25 marks) The one-dimensional infinite potential well can be generalized to three dimensions. The allowed energies for a particle of mass \(m\) in a cubic box of side \(L\) are given by$$ E_{n_{p} n_{r, n_{i}}}=\frac{\pi^{2} \hbar^{2}}{2 m L^{2}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right) \quad\left(n_{x}=1,2, \ldots ; n_{y}=1,2, \ldots ; n_{z}=1,2, \ldots\right) $$(a) If we put four electrons inside the box, what is the ground-state energy of the system? Here the ground-state energy is defined to be the minimum energy of the system of electrons. You...
The one-dimensional Schrindinger wave equation for a particle in a potential field
The one-dimensional Schrindinger wave equation for a particle in a potential field \(V=\) \(\frac{1}{2} k x^{2}\) is$$ -\frac{h^{2}}{2 m} \frac{d^{2} \psi}{d x^{2}}+\frac{1}{2} k x^{2} \psi=E \psi(x) $$(a) Lsing \(\xi=\alpha x\) and a constant \(\lambda\), we have$$ a=\left(\frac{m k}{A^{2}}\right)^{1 / 4}, \quad A=\frac{2 L}{A}\left(\frac{m}{k}\right)^{1 / 2} $$show that$$ \frac{d^{2} y(\xi)}{d \xi^{2}}+\left(\lambda-\xi^{2}\right) \psi(\xi)=0 $$(b) Substituting$$ \psi(\xi)=y(\xi) e^{2} / 2 $$show that \(y(t)\) satisfies the Hermite di fferential equation.
Let X equal the larger outcome when a pair of 6-sided dice are rolled.
Let X equal the larger outcome when a pair of 6-sided dice are rolled.(a) Assuming the two dice are independent, show that the probability function of \(X\) is \(f(x)=\frac{2 x-1}{36} \quad x=1, \ldots, 6\)(b) Confirm that \(f(x)\) is a probability function.(c) Find the mean of \(X\).(d) Can you generalise \(E(X)\) to a pair of fair \(m\) -sided dice?\(\left[\right.\) Hint: recall that \(\sum_{i=1}^{n} i=n(n+1) / 2\) and \(\left.\sum_{i=1}^{n} i^{2}=n(n+1)(2 n+1) / 6\right]\)
Weibull Likelihood function
\(\mathrm{T}\) is a failure time following a Weibull distribution. Consider \(\mathrm{Y}=\log \mathrm{T}\) where \(\mathrm{Y}\) has an extreme value distribution with survival function$$ S_{Y}(y)=e^{-\epsilon^{\frac{n \mu}{\sigma}}} $$where \(-\infty<\mu<\infty\) is the location parameter and \(\sigma>0\) is the scale parameter. Expressing with parameters \(\mu\) and \(\varphi=\log \sigma\). Assume that failure times of subjects under study arise from Weibull distribution. Let \(x_{1}, \ldots, x_{n}\) be the observed failure or right censoring times for n subjects. Each subject i (i = \(1, \ldots, \mathrm{n}\) ) has...
Show that a helix of radius R and height h making N complete turns has the parametrization
Let\(\mathbf{r}(t)=\left\langle R \cos \left(\frac{2 \pi N t}{h}\right), R \sin \left(\frac{2 \pi N t}{h}\right), t\right\rangle, \quad 0 \leq t \leq h\)(a) Show that \(\mathbf{r}(t)\) parametrizes a helix of radius \(R\) and height \(h\) making \(N\) complete turns.(b) Guess which of the two springs in Figure 5 uses more wire.(c) Compute the lengths of the two springs and compare.
Consider the heat conduction problem
5. If \(f(x)=\left\{\begin{array}{cc}0 & -2<x<0 \\ x & 0<x<2\end{array} \quad\right.\)is periodio of period 4 , and whose Fourier series is given by \(\frac{a_{0}}{2}+\sum_{n=1}^{2}\left[a_{n} \cos \left(\frac{n \pi}{2} x\right)+b_{n} \sin \left(\frac{n \pi}{2} x\right)\right], \quad\) find \(a_{n}\)A. \(\frac{2}{n^{2} \pi^{2}}\)B. \(\frac{(-1)^{n}-1}{n^{2} \pi^{2}}\)C. \(\frac{4}{n^{2} \pi^{2}}\)D. \(\frac{2}{n \pi}\)\(\mathbf{E}_{1} \frac{2\left((-1)^{n}-1\right)}{n^{2} \pi^{2}}\)F. \(\frac{4}{n \pi}\)6. Let \(f(x)-2 x-l\) on \([0,2]\). The Fourier sine series for \(f(x)\) is \(\sum_{w}^{n} b_{n} \sin \left(\frac{n \pi}{2} x\right)\), What is \(b, ?\)A. \(\frac{4}{3 \pi}\)B. \(\frac{2}{\pi}\)C. \(\frac{4}{\pi}\)D. \(\frac{-4}{3 \pi}\)E. \(\frac{-2}{\pi}\)F. \(\frac{-4}{\pi}\)7. Let \(f(x)\) be periodic...
Answer the following
Let \(\mathscr{F}\) be the set of Pythagorean triples in \(\mathbb{N}^{3}=\mathbb{N} \times \mathbb{N} \times \mathbb{N}\). That is, if \((a, b, c) \in \mathbb{N} \times \mathbb{N} \times \mathbb{N}\) and \(a^{2}+b^{2}=c^{2}\) then \((a, b, c) \in \mathscr{T}\). Withoutloss of generality assume that \(a \leq b \leq c\). Let \((a, b, c)\) and \((e, f, g) \in \mathscr{F} .\) Define \((a, b, c) R(d, e, f) \Leftrightarrow \frac{d}{a}=\frac{e}{b}=\frac{f}{c}\).Note: If \((a, b, c) R(e, f, g)\) then \((a, b, c)\) and \((d, e, f)\) are...
(2 points) The area A of the region S that lies under the graph of the...
(2 points) The area \(A\) of the region \(S\) that lies under the graph of the continuous function \(f\) on the interval \([a, b]\) is the limit of the sum of the areas of approximating rectangles:$$ A=\lim _{n \rightarrow \infty}\left[f\left(x_{1}\right) \Delta x+f\left(x_{2}\right) \Delta x+\ldots+f\left(x_{n}\right) \Delta x\right]=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f\left(x_{i}\right) \Delta x $$where \(\Delta x=\frac{b-a}{n}\) and \(x_{i}=a+i \Delta x\).The expression$$ A=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{8 n} \tan \left(\frac{i \pi}{8 n}\right) $$gives the area of the function \(f(x)=\) on...
Suppose that the functions
Suppose that the functions \(f: \mathbb{R}^{3} \rightarrow \mathbb{R}, g: \mathbb{R}^{3} \rightarrow \mathbb{R}\), and \(h: \mathbb{R}^{3} \rightarrow \mathbb{R}\) are continuously differentiable and let \(\left(x_{0}, y_{0}, z_{0}\right)\) be a point in \(\mathbb{R}^{3}\) at which$$ f\left(x_{0}, y_{0}, z_{0}\right)=g\left(x_{0}, y_{0}, z_{0}\right)=h\left(x_{0}, y_{0}, z_{0}\right)=0 $$and$$ \left\langle\nabla f\left(x_{0}, y_{0}, z_{0}\right), \nabla g\left(x_{0}, y_{0}, z_{0}\right) \times \nabla h\left(x_{0}, y_{0}, z_{0}\right)\right\rangle \neq 0 $$By considering the set of solutions of this system as consisting of the intersection of a surface with a path, explain why that in a...
Let \(\left\{\varphi_{n}(x)\right\}\) be an orthogonal set of functions on \([a, b]\) such that \(\psi_{0}(x)=1\) and \(\varphi_{1}(x)=x\), Show that \(\int_{a}^{b}(\alpha x+\beta) \varphi_{n}(x) d x=0\) for \(n=2,3, \ldots\) and any constants \(\alpha\) and \(\beta\),First we note that \(\alpha x+\beta=(\square) \Phi_{1}(x)+(\square \quad) \Psi_{0}(x)\).Using this together with the fact that \(\varphi_{0}\) and \(\varphi_{1}\) are orthogonal to \(\varphi_{n}\) for \(n>1\), we have the following.$$ \begin{aligned} \int_{a}^{b}(\alpha x+\beta) \varphi_{n}(x) d x &=\int_{a}^{b} a x \psi_{n}(x) d x+\int_{a}^{b} \beta \varphi_{n}(x) d x \\ &=\int_{a}^{b}\left(\square_{0}\right) \varphi_{1}(x) \varphi_{n}(x) d...
Let \(\mathscr{F}\) be the set of Pythagorean triples in \(\mathbb{N}^{3}=\mathbb{N} \times \mathbb{N} \times \mathbb{N}\). That is, if \((a, b, c) \in \mathbb{N} \times \mathbb{N} \times \mathbb{N}\) and \(a^{2}+b^{2}=c^{2}\) then \((a, b, c) \in \mathscr{T}\). Withoutloss of generality assume that \(a \leq b \leq c\). Let \((a, b, c)\) and \((e, f, g) \in \mathscr{F} .\) Define \((a, b, c) R(d, e, f) \Leftrightarrow \frac{d}{a}=\frac{e}{b}=\frac{f}{c}\).Note: If \((a, b, c) R(e, f, g)\) then \((a, b, c)\) and \((d, e, f)\) are...
(2 points) The area \(A\) of the region \(S\) that lies under the graph of the continuous function \(f\) on the interval \([a, b]\) is the limit of the sum of the areas of approximating rectangles:$$ A=\lim _{n \rightarrow \infty}\left[f\left(x_{1}\right) \Delta x+f\left(x_{2}\right) \Delta x+\ldots+f\left(x_{n}\right) \Delta x\right]=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f\left(x_{i}\right) \Delta x $$where \(\Delta x=\frac{b-a}{n}\) and \(x_{i}=a+i \Delta x\).The expression$$ A=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{8 n} \tan \left(\frac{i \pi}{8 n}\right) $$gives the area of the function \(f(x)=\) on...
Most questions answered within 3 hours.
-
MATLAB
write a MATLAB function (1) output a third-order polynomial
function with the coefficients as the...
asked 2 minutes ago -
A z-score
communicates a raw score’s "relative standing"
under the normal curve in relation to:
asked 11 minutes ago -
An object is vibrating on a spring with the following equation
of motion:
?=(30 ??)cos((2?)/(160)?)
a)...
asked 10 minutes ago -
Vulcan Flyovers offers scenic overflights of Mount St. Helens,
the volcano in Washington State that explosively...
asked 13 minutes ago -
If organizations do not adapt fast enough and move incrementally
from the twentieth-century model to the...
asked 13 minutes ago -
A helium balloon with 2.5L of gas has a gauge pressure of 10,000
Pa. The balloon...
asked 18 minutes ago -
What is responsible for Jupiter's enormous magnetic field?
asked 33 minutes ago -
At the end of the year, a company offered to buy 5,000 units of
a product...
asked 34 minutes ago -
Implement C++ program for each of the following.
Let D = [-48, -14, -8, 0, 1,...
asked 54 minutes ago -
Consider a labor market in which LD = 400 – 4W and LS = 250 +...
asked 1 hour ago -
Let’s say you are teaching a typically-developing young child
how to eat with a spoon. Describe...
asked 52 minutes ago -
Ask Your Teacher A toy gun uses a spring to project a 5.6-g soft
rubber sphere...
asked 1 hour ago