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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

(d)

(e)

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

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
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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