#ATTENDANCE QUIZ FOR LECTURE 14 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p14
#with an attachment called
#p14FirstLast.txt
#(e.g. p14DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Oct. 24, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 6

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH, BY THE ANSWER

1. Where and when was Herbert S. Wilf born?
Look up his paper about "random generation of combinatorial objects"
Advances in Mathematics ca. 1980 and at least skim it.

Herbert S. Wilf was born in Philadelphia, Pennsylvania in 1931.

2. Let m:= age of Donald Trump
Let n:= age of Joe Biden
What is the probability that if you take a random lattice walk from (0,0) to (n,m) you will always stay in the region x>=y?

m=74, n=77

NuGPaths(77,74)/NuPaths(77,74) = 2/39

3. What is a Wilf-Zeilberger pair?

A Wilf-Zeilberger pair is a pair of functions that can be used to certify certain combinatorial identities.

4. What nationality was Catalan? What is the constant named after him?

Catalan was French and Belgian. Catalan's constant is named after him, which is defined by the sum of (-1)^n/(2n+1)^2 for n=0..infinity.

5. Do the same for [1,1,-1,-1,1] and get a set of 10 such lists that is the whole of Paths(3,2)

[-1, -1, 1, 1, 1], [-1, 1, -1, 1, 1], [-1, 1, 1, -1, 1], [-1, 1, 1, 1, -1], [1, -1, -1, 1, 1], [1, -1, 1, -1, 1], [1, -1, 1, 1, -1], [1, 1, -1, -1, 1], [1, 1, -1, 1, -1], [1, 1, 1, -1, -1]

6. You are in a circular track in the desert. At random places there are gas containers with random amounts of gas a1,a2,...,ak such that a1+a2+...+ak gallons are exactly what you need to drive in this track.
ex. [[0,0.1],[0.2,0.5],[0.7,0.4]]
Prove that there exists a location on the circular track such that if you start there you don't run out of gas
Devise a method for doing it

Given that there are random places there are gas containers with random containers with amounts a1,...,ak,
the distribution of a1,...,ak among the places on the track may change the starting locations for which it is possible to go fully around the track,
but ultimately there must always be at least one starting location in which it is possible.
One way of doing it is to simulate the car starting at different starting locations, and then keep track of the gas tank.
If it ever goes becomes negative, then that starting location is not viable.