Prerequisites Pass in (MATH1101 and MATH1102 and MATH1201 and MATH1202) or (MATH1111 and MATH1211); and Pass in MATH2201, or already enrolled in this course; and Pass in MATH2301, or already enrolled in this course; and Pass in MATH2401, or already enrolled in this course.

Format

No regular lectures. The student is expected to do approximately 120 hours of independent work and to attend meetings and seminars.

Assessment

By dissertation (70% weighting) and continuous assessment which may include oral presentation (30% weighting)

• Au Tung Kin, Qubit Channels
• Edmond Kwok, Page rank and Voting.
• Yip Ka Wa, Comparison of different method on image compression.
• Zhang Shi Xiao, Minimal polynomial and its applications. [ Reference.]
  Additional project. Matlab code for generalized stationary vectors.

Schedule

• Jan, 17. Introductory meeting.
• Feb. 2. Student presentations.
  • Zhang Shi Xiao, Nonnegative matrices and Markov Chain.
    Question: Extend the methods of finding powers and stationary vectors in the presentation to matrices of higher dimension.
  • Yip Ka Wa, Singular value decompositions and applications.
    Questions: Investigate whether one can describe the kind of images for which low rank SVD approximations can give very good results. (For example, a picture looks like the identity matrix is bad.)
    Compare other image compression methods and compare their adventages/disadventages. Find out which areas (engineering, arts, communications?) are SVD used most?
  • Edmond, Kwok Ki Lung, Page-ranking of Google matrix. Questions: Can we apply page ranking to other areas such as voting?
    Can we have a Google matrix so that the first column has the maximum column sum, but the first entry of the stationary vector has the smallest value?
    Partial answer: And, for the voting idea you give me, I think it would be a interesting system of voting that, a person receiving the greatest amount of voting, doesn't get to the highest position in vote-score. http://en.wikipedia.org/wiki/Pagerank

    Taking the first picture showing in the link above, we can see that page E receiving more vote than page C (page E gets 6 vote, page C only gets one vote). But till now I am still finding the way to construct the example that a person getting the most vote having the vote-score (i.e. the ranking score) less than the highest one.

  • Au Tung Kin, Quantum computing and Quantum gates.
    A concrete problem in quantum computing. Qubits are respresented by 2x2 density matrices, i.e., positive semidefinite matrices with diagonal entries added up to 1. Let A_1, A_2, A_3 and B_1, B_2, B_3, be 2x2 density matrices.
    
    • Determine when will there be a linear map T(A_i) = B_i for i = 1, 2,3.
    • Determine when will the map T satify the extra condition that the 4x4 matrix with T(E_{ij}) as the (i,j) block for (i,j) = (1,1), (1,2), (2,1), (2,2), and E_{ij} is the 2x2 matrix with 1 at the (i,j) position, and 0 elsewhere.
• Feb. 8. Student presentations.
  • Shixiao Zhang, More on stationary vectors.
    I presented some other ways to calculate the stationary vector and the limiting power of P. Actually they are related, but in some sense, obviously, the limiting power of P is just the stationary distribution. And I found out that to use partial fraction decomposition to separate (I-zP)^(-1) may not be always useful considering the high dimension of P. And somehow, to use pi*P=pi is the most fundamental and definition way, but these are just tedious numerical or algebraical calculation.
    Suggestions study the papers, their references, and see whether we can improve their results or proofs.
    • Li, Ng, Ye, Finding Stationary Probability Vector of a Transition Probability Tensor Arising from a Higher-order Markov Chain,
      http://www.math.hkbu.edu.hk/~mng/tensor-research/report1.pdf.
    • Michele Benzi, A direct projection method for Markov chains, Linear Algebra and its Applications 386 (2004) 27–49 10.1.1.133.3838.pdf
  • Yip Ka Wa, Attempt to solve questions on SVD: [ file 1| file 2| file 3]
  • Edmond Kwok, Voting and stationary vectors.
  • Au Tung Kin, Attempt to solve questions on quantum process.
• Feb. 15.
  • Shixiao Zhang, Power of matrices, and stationary vectors.
  • Edmond Kowk, Page rank and voting.
    Question:

    Let P =
    [a 1-a;
     b 1-b].
    Assume you can add one more voter to get
    Q =
    [a 1-a 0;
     b 1-b 0;
    x y 1-x-y]
    Compute the page rank of .85 Q + .15 J, J is the matrix with all entries equal to 1/3.
    Determine the range of x, y so that you can make the first person, second person,
    or the new person having the highest page rank.
    

  • Yip Ka Wa, Latent semantic indexing.
  • Au Tung Kin, Quantum process.
  Feb. 20.
  • Shixiao Zhang, Power of matrices (continued).
  • Edmond Kwok, Changing the page rank.
  • Yip Ka Wa, Image Processing.
  • Au Tung Kin, Quantum process (cont'd).
• Feb. 29.
  • Shixiao Zhang, [ Minimal polynomial 1. ] [ Minimal polynomial 2. ]
  • Edmond Kowk, Changing the page rank (cont'd).
  • Yip Ka Wa, Comparison of image processing methods.
  • Au Tung Kin, Completely positive linear maps.

STUDENT SEMINAR
 
An oral presentation by four students taking
MATH2999  Directed Studies in Mathematics 
will be held on Thursday, May 3, 2012
in Room 208, Run Run Shaw Building
========================================
The schedule is as follows:
 
2:30 – 3:00 Mr. Zhang Shixiao
Stationary Probability Vector of a Higher-order  Markov Chain
 
3:00 – 3:30 Mr. Kwok Ki Lung
Page Rank and Vote
 
3:30 – 4:00Mr. Au Tung Kin
Quantum Process on One Quabit System
 
4:00 – 4:30 Mr. Yip Ka Wa
Singular Value Decomposition and its Applications