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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 z . Now drag the tip of the blue arrow along the unit circle on with z 1 (smaller circle). For which values of z on the unit circle does 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 b . Do not use your calculator!

(a) -1 2 20

(b) 1 2 3 4

(c) 1 3 3



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 4 z

(b) z 2 4 z 4

(c) z 2 4 z 8

(d) z 2 8

(e) z 2 4 z 8

(f) 2 z 2 4 z 8

Problem 8: nth roots of unity

2 N is called an Nth Root of Unity .

(a) Why?

(b) Let z 2 7 . Draw z z 2 z 7 in the complex plane.

(c) Let z 4 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 2 and v 2 are called linearly independent if u v 0 if and only if 0 It is a fact that we can write any vector in 2 as a weighted sum (or linear combination ) of any two linearly independent vectors, where the weights and are complex-valued.

(a) Write 3 4 6 2 as a linear combination of 1 2 and -5 3 . That is, find and such that 3 4 6 2 1 2 -5 3

(b) More generally, write x x 1 x 2 as a linear combination of 1 2 and -5 3 . 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 x 1 x 2 x x

(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 n 1 x Then for each x in the complex plane, we say x J if the sequence g 0 x g 1 x g 2 x does not tend to infinity. Notice that if x 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 g n x remains below some threshold M for all 0 n 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 .

Example image where the x-axis is x and the y-axis is x .

Using the same values for N , M , and x , create a picture of the Julia set for -0.391 0.587 . 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

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Señales y sistemas. OpenStax CNX. Sep 28, 2006 Download for free at http://cnx.org/content/col10373/1.2
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