#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:

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



Question 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.


Answer 1:

He was born in 1931 in Philadelphia, PA.




Question 2:

Let m be the age of Donald Trump. N be the 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? 


Answer 2:

m= 74
n= 77

(0,0) to (77,74)

NuGPaths(77,74)/NuPaths(77,74)

2/39



Question 3:

What is a Wilf-Zeilberger pair?



Answer 3:

In mathematics, specifically combinatorics, a Wilf–Zeilberger pair, or WZ pair, is a pair of functions that can be used to certify certain combinatorial identities. WZ pairs are named after Herbert S. Wilf and Doron Zeilberger, and are instrumental in the evaluation of many sums involving binomial coefficients, factorials, and in general any hypergeometric series. 

Source: https://en.wikipedia.org/wiki/Wilf%E2%80%93Zeilberger_pair




Question 4:


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


Answer 4:

Eugène Charles Catalan was Belgian. 


The Catalan's constants numerical value is approximately

G = 0.915965594177219015054603514932384110774



Question 5:


Apply cyclic shifts to [1,1,-1,-1,1] and get a set of 10 such lists. Verify that is whole of Paths(3,2). 


Answer 5:

{[-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]}




Question 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 on 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 in general. 

EXTRA CREDIT: Write a maple function 

	    INPUT: a list of lists with location and amount of gas
                   Location are in increasing order and go from 0 to 1
		   amount of gasoline adds up to 1
		   



Answer 6:

Method:

Since the input lists the locations in increasing order paired with the amount of gasoline received at that location, adding the location and amount of gasoline should be greater than or equal to the next location. 

Property:
[[location l, gasoline g1],[location l+1, gasoline g2]]

Here, l + g1 >= l+1.

Thus, an algorithm to solve this problem would be to generate the set of cycle shifts of the input list based on the location.
Then choose then set which satisfies the above property for all locations in the input list.