# Homework #1

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Homework 1 problem set for Elec301 at Rice University.

Noon, Thursday, September 5, 2002

## Assignment 1

Homework, tests, and solutions from previous offerings of this course are off limits, under the honor code.

## Problem 1

Form a study group of 3-4 members. With your group, discuss and synthesize the major themes of this week of lectures. Turn in a one page summary of yourdiscussion. You need turn in only one summary per group, but include the names of all group members. Please do notwrite up just a "table of contents."

## Problem 2

Construct a WWW page (with your picture ) and email Mike Wakin (wakin@rice.edu) your name (as you want it to appear on theclass web page) and the URL. If you need assistance setting up your page or taking/scanning a picture (both are easy!),ask your classmates.

## Problem 3: learning styles

Follow this learning styles link (also found on the Elec 301 web page ) and learn about the basics of learning styles. Write a short summary of what you learned. Also,complete the "Index of learning styles" self-scoring test on the web and bring your results to class.

## Problem 4

Make sure you know the material in Lathi , Chapter B, Sections 1-4, 6.1, 6.2, 7. Specifically, be sureto review topics such as:

• complex arithmetic (adding, multiplying, powers)
• finding (complex) roots of polynomials
• complex plane and plotting roots
• vectors (adding, inner products)

## Problem 5: complex number applet

Reacquaint yourself with complex numbers by going to the course applets web page and clicking on the Complex Numbers applet (may take a few seconds to load).

(a) Change the default add function to exponential (exp). Click on the complex plane to get a blue arrow, which isyour complex number $z$ . Click again anywhere on the complex plane to get a yellow arrow,which is equal to $e^{z}$ . Now drag the tip of the blue arrow along the unit circle on with $\left|z\right|=1$ (smaller circle). For which values of $z$ on the unit circle does $e^{z}$ also lie on the unit circle? Why?

(b) Experiment with the functions absolute (abs), real part (re), and imaginary part (im) and report your findings.

## Problem 6: complex arithmetic

Reduce the following to the Cartesian form, $a+ib$ . Do not use your calculator!

(a) $\left(\frac{-1-i}{\sqrt{2}}\right)^{20}$

(b) $\frac{1+2i}{3+4i}$

(c) $\frac{1+\sqrt{3}i}{\sqrt{3}-i}$

(d) $\sqrt{i}$

(e) $i^{i}$

## Problem 7: roots of polynomials

Find the roots of each of the following polynomials (show your work). Use MATLAB to check your answer with the roots command and to plot the roots in the complex plane. Mark the root locations with an 'o'. Putall of the roots on the same plot and identify the corresponding polynomial ( $a$ , $b$ , etc. ..).

(a) $z^{2}-4z$

(b) $z^{2}-4z+4$

(c) $z^{2}-4z+8$

(d) $z^{2}+8$

(e) $z^{2}+4z+8$

(f) $2z^{2}+4z+8$

## Problem 8: nth roots of unity

$e^{\frac{i\times 2\pi }{N}}$ is called an Nth Root of Unity .

(a) Why?

(b) Let $z=e^{\frac{i\times 2\pi }{7}}$ . Draw $\{z, z^{2}, , z^{7}\}$ in the complex plane.

(c) Let $z=e^{\frac{i\times 4\pi }{7}}$ . Draw $\{z, z^{2}, , z^{7}\}$ in the complex plane.

## Problem 9: writing vectors in terms of other vectors

A pair of vectors $u\in \mathbb{C}^{2}$ and $v\in \mathbb{C}^{2}$ are called linearly independent if $u+v=0\text{if and only if}==0$ It is a fact that we can write any vector in $\mathbb{C}^{2}$ as a weighted sum (or linear combination ) of any two linearly independent vectors, where the weights  and  are complex-valued.

(a) Write $\left(\begin{array}{c}3+4i\\ 6+2i\end{array}\right)$ as a linear combination of $\left(\begin{array}{c}1\\ 2\end{array}\right)$ and $\left(\begin{array}{c}-5\\ 3\end{array}\right)$ . That is, find  and  such that $\left(\begin{array}{c}3+4i\\ 6+2i\end{array}\right)=\left(\begin{array}{c}1\\ 2\end{array}\right)+\left(\begin{array}{c}-5\\ 3\end{array}\right)$

(b) More generally, write $x=\begin{pmatrix}x_{1}\\ x_{2}\\ \end{pmatrix}$ as a linear combination of $\left(\begin{array}{c}1\\ 2\end{array}\right)$ and $\left(\begin{array}{c}-5\\ 3\end{array}\right)$ . We will denote the answer for a given $x$ as $(x)$ and $(x)$ .

(c) Write the answer to (a) in matrix form, i.e. find a 22 matrix $A$ such that $A\begin{pmatrix}x_{1}\\ x_{2}\\ \end{pmatrix}=\left(\begin{array}{c}(x)\\ (x)\end{array}\right)()$

(d) Repeat (b) and (c) for a general set of linearly independent vectors $u$ and $v$ .

## Problem 10: fun with fractals

A Julia set $J$ is obtained by characterizing points in the complex plane. Specifically,let $f(x)=x^{2}+$ with  complex, and define ${g}_{0}(x)=x$ ${g}_{1}(x)=f({g}_{0}(x))=f(x)$ ${g}_{2}(x)=f({g}_{1}(x))=f(f(x))$  ${g}_{n}(x)=f({g}_{n1}(x))$ Then for each $x$ in the complex plane, we say $x\in J$ if the sequence $\{\left|{g}_{0}(x)\right|, \left|{g}_{1}(x)\right|, \left|{g}_{2}(x)\right|, \}$ does not tend to infinity. Notice that if $x\in J$ , then each element of the sequence $\{{g}_{0}(x), {g}_{1}(x), {g}_{2}(x), \}$ also belongs to $J$ .

For most values of  , the boundary of a Julia set is a fractal curve - it contains"jagged" detail no matter how far you zoom in on it. The well-known Mandelbrot set contains all values of  for which the corresponding Julia set is connected.

(a) Let $=-1$ . Is $x=1$ in $J$ ?

(b) Let $=0$ . What conditions on $x$ ensure that $x$ belongs to $J$ ?

(c) Create an approximate picture of a Julia set in MATLAB. The easiest way is to create a matrix of complexnumbers, decide for each number whether it belongs to $J$ , and plot the results using the imagesc command. To determine whether a number belongs to $J$ , it is helpful to define a limit $N$ on the number of iterations of $g$ . For a given $x$ , if the magnitude $\left|{g}_{n}(x)\right|$ remains below some threshold $M$ for all $0\le n\le N$ , we say that $x$ belongs to $J$ . The code below will help you get started:

N = 100; % Max # of iterations M = 2; % Magnitude threshold mu = -0.75; % Julia parameter realVals = [-1.6:0.01:1.6]; imagVals = [-1.2:0.01:1.2]; xVals = ones(length(imagVals),1) * realVals + ... j*imagVals'*ones(1,length(realVals)); Jmap = ones(size(xVals)); g = xVals; % Start with g0 % Insert code here to fill in elements of Jmap. Leave a '1' % in locations where x belongs to J, insert '0' in the % locations otherwise. It is not necessary to store all 100 % iterations of g! imagesc(realVals, imagVals, Jmap); colormap gray; xlabel('Re(x)'); ylabel('Imag(x)');

This creates the following picture for $=-0.75$ , $N=100$ , and $M=2$ .

Using the same values for $N$ , $M$ , and $x$ , create a picture of the Julia set for $=-0.391-0.587i$ . Print out this picture and hand it in with yourMATLAB code.

Try assigning different color values to Jmap. For example, let Jmap indicate the first iteration when the magnitudeexceeds $M$ . Tip: try imagesc(log(Jmap)) and colormap jet for a neat picture.

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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