# m387

Math 387 course resources

# Math 387 homework

Draft Overleaf template

## Week 14

• Reading for Monday: Finish section 5.2
• Group work for Monday: 230-234
• Reading for Wednesday: Secton 5.3
• Group work for Wednesday: 237-240
• Homework due Tuesday 4/27:
• Problem 230
• Problem 232
• Problem 238
• Problem M: Beginning with the result from 238, use a partial fraction decomposition, together with our power series formulas, to find an explicit formula for $a_i$.

## Week 13

• Reading for Monday: Section 5.1
• Group work for Monday: 207-208, 211-213
• Reading for Wednesday: Start Section 5.2
• Group work for Wednesday: 222-228
• Homework due Tuesday 4/20:
• Problem 213
• Problem K: Suppose you can take between 0 and 3 apples, between 2 and 5 pears, and between 4 and 7 bananas. Suppose you want to take 10 fruits in total. How many ways are there to do this? Set up the problem using generating polynomials, you may use a computer to do the algebra.
• Problem 226
• Problem L: Recall the power series for $(1-x)^{-1}$ is $\sum x^k$.
(a) Since $(1-x)^{-2}$ is $(1-x)^{-1}(1-x)^{-1}$, find its power series by evaluating $(\sum x^k)(\sum x^k)$.
(b) Since $(1-x)^{-2}$ is the derivative of $(1-x)^{-1}$, find its power series by evaluating the derivative of $\sum x^k$.

## Week 12

• Reading for Monday: Start The Coloring of Graphs (via blackboard)
• Reading for Wednesday: Finish The Coloring of Graphs
• Group work for Wednesday: CoG exercise 10
• Homework due Tuesday 4/6:
• Problem G: What is the chromatic polynomial of the complete graph on $k$ vertices? Explain your answer.
• Problem H: Use the reduction algorithm to calculate the chromatic polynomial of the pentagon graph (the edges are (1,2),(2,3),(3,4),(4,5),(5,1)).
• Problem I: The degree of the chromatic polynomial is equal to the number of vertices of the graph. Use induction and the deletion/contraction recurrence to prove this is always the case.
• Problem J: The constant term of the chromatic polynomial is always equal to $0$. Explain why this is always the case. [Hint: how many ways are there to color a graph using a palette of $0$ colors? What does this say about $p(0)$?]
• Notebook 2 due Thursday 4/8: Problems 105-110, 114, 116, 118, 120, 122-123, 126, 141-146, 149-156, 158-159, 164-170, 172-178, 191, 193-194, 196-198, 200-202

## Week 11

• Reading for Monday: Section 4.5
• Group work for Monday: 191, 193-194
• Group work for Wednesday: 196-198, 200-202
• Homework due Thursday 4/1:
• Problem 194: Describe Kruskal’s (or Prim’s) algorithm carefully, and briefly explain why it “works”
• Problem 198
• Problem E: Draw a connected weighted graph (not a tree) with at least 7 vertices and at least 14 edges. Run Kruskal’s algorithm for the graph to find the minimum spanning tree.
• Problem F: For the same graph as in Problem E, choose a special vertex $v$ and run Dijkstra’s algorithm to find the tree of shortest paths from $v$. Compare with the tree from part $E$.

## Week 10

• Reading for Monday: Start section 4.3
• Group work for Monday: 164-170
• Reading for Tuesday: Finish section 4.3
• Group work for Wednesday: 172-178
• Homework due Tuesday 3/23:
• Problem A: Explain why the last element $b_{n-1}$ of a Prufer code sequence B is always $n$.
• Problem B: Find the tree for the following prufer code: 1,2,1,3,1,4,1,5,1,6
• Problem C: Write all sixteen possible Prufer codes with $n=4$ (so the codes have length 2) and draw the corresponding trees.
• Problem D: What is the relationship between the degree of a vertex, and the number of times the vertex occurs in the Prufer code? Give a complete explanation (proof) your answer is correct.

## Week 9

• Monday: Catch-up and homework questions
• Tuesday: Exam posted, homework 8 due
• Wednesday: No class
• Sunday 3/14: Exam due, pi day

## Week 8

• Reading for Monday: Section 4.1
• Group work for Monday: 141-146, 149-150
• Group work for Wednesday: 151-156, 158-159
• Homework due Tuesday 3/9: 141, 146, 152, 155

## Week 7

• Reading for Monday: Section 3.3 again
• Group work for Monday: 105-110, 114, 116
• Reading for Wednesday: Section 3.3.1, 3.3.2
• Group work for Wednesday: 118, 120, 122-123, 126
• Homework due Tuesday 3/2: 107, 112, 123, 126

## Week 6

• Holiday on Monday!
• Reading for Wednesday: Preview section 3.3
• Group work for Wednesday: Review, catch up, bonus problems!
• Homework due Tuesday 2/23: Class survey

## Week 5

• Reading for Monday: Secton 3.1
• Group work for Monday: 89-94
• Reading for Wednesday: Section 3.2
• Group work for Wednesday: 96-99, 102
• Homework due Thursday 2/18: 76, 90, 96, 102
• Notebook 1 due Thursday 2/18: Problems 1-24, 31-36, 38-40, 50-54, 57-61, 66-67, 71-72, 74(a), 75, 77, 79, 87-94, 96-99, 102

## Week 4

• Reading for Monday: Section 2.2 again, including 2.2.1
• Group work for Monday: 57-61, 62(read), 66-67
• Reading for Wednesday: Section 2.3
• Group work for Wednesday: 71-72, 74(a), 75, 77, 79, 87-88
• Homework due Tuesday 2/9: 63, 73, 75, 88

## Week 3

• Reading for Monday: Section 1.4, 1.5, 1.6
• Group work for Monday: 31-36, 38-40
• Reading for Wednesday: 2.1, 2.2
• Group work for Wednesday: 50-54
• Homework due Tuesday 2/2: 35, 38, 41, 56

## Week 2

• Holiday on Monday!
• Reading for Wednesday: Section 1.3
• Group work for Wednesday: 17-24
• Homework due Tuesday 1/26: 18(a), 19(a)(b), 22, 29

## Week 1

• Reading for Monday: Sections 1.1, 1.2
• Group work for Monday: 1-7
• Group work for Wednesday: 8-16
• Homework due Thursday 1/21 (end of day): 2, 6, 9, 15