JHR's MA315 page

Discrete and Combinatorial Algebra

MTRF 8 G222
John Rickert, Associate Professor of Mathematics
Office: G-215A, Crapo Hall
Phone: (812) 877-8473

e-mail: john.rickert@rose-hulman.edu

Office hours this week: MTR 7,9, or make an appointment, or drop in.
On some Thursdays I might be proctoring IFYCSEM exams.

The average score on the final excam was 74.7. You may come to my office and pick-up your exam.

A sketch of the answers to Quiz #7 is online.

To Homework ...Questions
Homework for our next class ...Today's questions

Please let me know if I've missed anything.
The main goal in this class is to have you (the student) perform as an active learner. To do this you will need to do the exercises, raise questions about structures that you are studying, create hypotheses and test these hypotheses.
The quizzes, examinations and homework done during the year will be worth 80% of the course grade. The final examination will be worth 20% of the grade.


Homework for our next class ...the top
For Tuesday 12/1: Read Section 4.0 and work the exercises in 4.0.1, 4.0.2.
For Thursday 12/2: Read through section 4.2.2 work the exercises. Think about exercise
The average score on Thursday's quiz was 16.7 out of 20
For Friday 12/3: Read Section 4.2.3
Work the exercises
For Monday 12/7: Read Section 4.2.4
Work the exercises.
Prove or disprove: Rounding to the nearest 10 is an equivalence relation over the integers.
How about : Rounding is an congruence relation?
For Tuesday 12/8: It was observed that the breakdown into "even" and "odd" permutations is a congruence relation over Sn. Can you prove this?
Read through exercise 4.3.2. Work exercises 4.3.1, 4.3.2. Be sure to be prepared to work exercise 4.3.2
For Thursday 12/10: Quiz #3 on Thursday, Proving equivalence.
Read through exercise 4.3.2. Work exercises 4.3.1, 4.3.2. Be sure to be prepared to work exercise 4.3.2
For Friday 12/11: Read through Exercise 4.3.3
Work exercise 4.3.3
Hand in exercise 4.3.2
For Monday, 12/14: Read through exercise 4.3.6 (page 189)
Work through the work in the reading and do exercises 4.3.4,4.3.5 and 4.3.6
For Tuesday, 12/15: Read through Fact 4.6
For Thursday, 12/17: Read Through exercises 4.3.8
Work exercises 4.3.8.
For Friday, 12/18 Get a better understanding of cycle index polynomials.
How many symmetries does a cube have? What is the group of rotational symmetries of a cube?
Hand In Exercise
For Monday, January 4: Re-read Chapter 4 through exercises 4.3.8 to try to gain a better understanding of the concepts and notations used.
Exam #1, Monday, January 11, 1999
For Thursday, January 14: Read section 5.0
Do exercises 5.0.1, 5.0.2
For Friday, January 15: Hand In Exam re-write.
Read through exercise 5.1.5. Do exercises 5.1.1 - 5.1.5, Especially 5.1.5.
For Monday, January 18: Read through exercise 5.1.8. Do exercises 5.1.6, 5.1.7, 5.1.8
For Tuesday, January 19 Read Section 5.2. Do exercises 5.2.1 and 5.2.2
For Thursday, January 21: Read through exercise 5.3.3.
Do the exercises.
For Monday, January 25: Finish Section 5.3. Hand in exercises and
For Tuesday, January 26: Read through section 5.4.1.
For Thursday, January 28: Read through section 5.4.2
Understand how errors are detected and corrected using the code CN.
For Tuesday, February 2 Work some of the exercises in 5.5.
Come to class with questions about the chapter 5 material.
Thursday, February 4: Chapter 5 Exam
For Tuesday, February 9: Hand in Exam re-write.
For Monday, February 15: Read through exercise 6.3.2.
Quiz #7 answers
For Tuesday, February 16: Read through exercise 6.3.6

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Questions from class

today's questions ...the top
Tuesday 12/1: Problem Is 20 the correct number of equivalence classes? If so, why? If not, why not?
Thursday 12/3: Equivalence relations used on the quiz: Two elements are equivalent if...

Friday, 12/4: Is Rounding to the nearest factor of 10 an equivalence relation on the integers?
Prove your assertion.
Monday, 12/7 Suppose G,+ and G\0,* are groups and ~ is an equivalence relation over G,+ and G\0,*.
Is it possible for ~ to be a congruence relation over one of these (either G,+ or G\0,*) and not the other?
Dennis Lin observes that a~b iff |a|=|b| is an equivalence relation over the integers (and rationals, reals, complex numbers, etc.) so that it is a congruence relation with respect to multiplication, but not addition.
It was observed that the breakdown into "even" and "odd" permutations is a congruence relation over Sn. Can you prove this?

Tuesday, 12/8 How many equivalence relations are there over S3?
How many congruence relations are there over S3 with respect to composition of permutations?
Peter Webb noticed that any function f(x) can be used to define an equivalence relation by: a~b <-> f(a)=f(b).
Can you prove that this is an equivalence relation?
Thursday, 12/10: Jonathan Webster and Dennis Lin came to the conclusion that there are 203 equivalence relations on S3. Is this correct?
Nathan Froyd observed that the sequence begins 1,1,2,5,15,52,203,... and 203=52*1+15*5+5*10+2*10+1*5+1*1.
The number of congruence relations is still an open question.
For Thursday, 12/17 Get a better understanding of cycle index polynomials.
How many symmetries does a cube have? What is the group of rotational symmetries of a cube?
Monday, 1/18: We have devised a code of size 32 which corrects one error. We have seen that a code that corrects one error, consisting of 15 bits, has size no more than 270.
What is the largest error correcting code on 15 bits?
Monday, 1/25: We now have a code of weight 5 - a code that can correct two errors. How do we perform such a correction efficiently?
Tuesday, 1/26: Question: What do we get from the quadratic polynomial in step 4 if (s1)3 = s2?
Here's the (probably inefficient) code that I'm using to get Maple to do the error correction:
c.0:=0: c.1:=1: c.2:=y: c.3:=y+1:
c.4:=y^2: c.5:=y^2+1: c.6:=y^2+y: c.7:=y^2+y+1:
for k from 8 to 15 do c.k:=y^3+c.(k-8) od:
reduce:= poly-> modp(rem(expand(poly),md,y),2);
The matrix, unfortunately, must be typed in. (OK. I admit, there's a sneaky way to do it, but I'd like to leave some extra credit activities for you.)
If the matrix is called en and the received message is called r then we look at evalf(en&*r);
and determine s1 and s2. (Again, Maple coding can be created ...) If there are at least two errors we determine the coefficients of the quadratic, call them ci and cj for now, and let Maple chug through the calculations: seq( [k,reduce(c.k*c.k+ c.3*c.k+ c.2)],k=1..15);
Please inform me if there are any typos that make the code fail to perform.
Monday, February 8: When building a field with eight elements, we must have 1+1=0. This produces a field with elements 0,1,a,a+1,a2,a2+1,a2+a,a2+a+1.
We saw in class that a3 cannot be equal to a, a2 or a2+a. The remaining possibilities were 1,a+1,a2+1,a2+a+1. Which of these really do produce fields?
Thursday, February 11: Are there finite fields of size pn for all natural numbers n and all primes p?
Friday, February 12: What is the value of In, the number of irreducible polynomials of degree n in Z2[y]?

Monday, February 15:
Quiz #7 answers are available.
Can we further streamline the calculation of In through the use of generating functions?
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A question from class: How do we know that the Maclaurin series expansion of s2/((1-3s)(1-s-s2)) has integer coefficients?
Expanding the denominator gives (1-3s)(1-s-s2)=1-4s+2s2+3s3. We note that a sequence defined by an=4an-1-2an-2-3an-3 with integer values for a0,a1 and a2 consists solely of integers and has generating function A(s), where
A(s)-sA(s)-s2A(a) = a0+(a1-4a0)s +(a2-4a1+a0)s2 . Thus, (a0+(a1-4a0)s +(a2-4a1+a0)s2) / (1-s-s2) is the generating function for such a sequence.
s2/((1-3s)(1-s-s2))=s2* 1/((1-3s)(1-s-s2)). 1/((1-3s)(1-s-s2)) is the generating function for the sequence an=4an-1-2an-2-3an-3 with a0=1, a1-4a0=0, and a2-4a1+2a0=0. Therefore, all of the coefficients of sn will be integers.
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