An On-Line course in using Maple to study Quantum Computing and Algorithmic Graph Theory

http://sites.math.rutgers.edu/~zeilberg/math640_25.html

Last Update: May 10, 2025

Teacher: Dr. Doron ZEILBERGER ("Dr. Z")

Description

Experimental Mathematics used to be considered an oxymoron, but the future of mathematics is in that direction. In addition to learning the philosophy and methodology of this budding field, students will become computer-algebra wizards, and that should be very helpful in whatever mathematical specialty they are doing (or will do) research in.
We will first learn Maple, and how to program with it. This semester we will learn, from an experimental mathematics point of view, Quantum computing and algorithmic graph theory. People who already took previous editions of this class are welcome to take it again, since except for the basics, there is very little overlap with previous years.
This class is suitable for graduate students in other departments, and the software development skills learned will be useful for doing any quantitative research. Very smart advanced undergraduates are also welcome. In particular, the methods learned should be applicable almost everywhere.
Maple packages produced in this class
- QC.txt, studying and simulating quantum circuits (Expanded with many new procedures by Salman Manzoor and Tijil Kiran)
- AGT.txt, graph theoy algorithms (expanded by Jeff Tang and Matthew Esaia)
- AL.txt, discovering Amitsur-Levitsky identities and other kinds (Greatly expanded by Joseph Koutsoutis)
Added April 6, 2025: Pick at project

Textbooks
- Quantum Mechanics (The Theoretical Minimum) by Leonard Susskind and Art Friedman.
- The Nielsen-Chuang classic
  [solutions]
- Introduction to Classical and Quantum Computing by Thomas G. Wong.
- Peter Shor's seminal paper
- Robin Wilson's wonderful graph theory book
- Don Knuth's half-a-page version of Hao Huang's Proof of the sensitivity conjecture
- the Einstein-Podolsky-Rosen classic
- John Bell's seminal paper
Optional (but Highly recommended) Getting Started Homework
Teach yourself the basics of Maple by reading Frank Garvan's golden-oldie part 1, part 2
Note: a few commands are no longer valid in today's Maple. The most important one is that " has been replaced by %.
For more details, see this 2002 masterpiece.
How to submit homework
- Every class has a homework assignment. Both Mondays and Thursdays' are due 8:00pm Sunday of the following week.
  It should be sent to the email address
  ShaloshBEkhad at gmail dot com
  Subject: HomeWork#X
  and then attach a .txt file(s) called
  hwXYourFirstNameYourLastName.txt
  where X is the number of the assignment
  Except for the first assignment, you should have two attached .txt files.
  For example, when David Hilbert submits homework assignments 2 and 3, he should email ShaloshBEkhad at gmail dot com, with
  Subject: HomeWork#2#3
  and attach the text files (the source code plus human comments (such lines must start with #)
  hw2DavidHilbert.txt and hw3DavidHilbert.txt
- The very first line of the .txt files should have
  #Please do not post homework
  or
  #OK to post homework
  Followed by
  #Your Name, Date, Assignment X
Programs done on Thurs., Jan. 23, 2025
C1.txt , Contains procedures
- Graphs(n): inputs a non-neg. integer and outputs the set of ALL (undirected simple) graphs on {1,...,n} with the data structure [n,E] where it is understood that the set of vertices is {1, ...,n} and E is the set of edges, where an edge joining vertex i and vertex j is denoted by {i,j}
- Tri(G): inputs a graph [n,E] and outputs the set of all triples {i,j,k} such {{i,j},{i,k},{j,k}} is a subset of E
- Comp(G): the complement of G=[n,E] i.e. the graph [n, E1] such E1 is the set of all possible edges minus E. For example Comp([3,{{1,2},{1,3}}]); should give [3,{{2,3}}]
- TotTri(G): the total number of love triangles and hate triangles
Homework for Thurs., Jan. 23, 2025, class (due Sunday Jan. 26, 8:00pm)
Please email ShaloshBEkhad at gmail dot com an email with
Subject: HomeWork#1
and an attachment
hw1FirstNameLastName.txt
and indicate whether it is OK to post. If you do not say "Please do not post", it will be posted.
1. [People who took this class before, or already know Maple, are excempt from this] Read and do all the examples, plus make up similar ones, in the first 30 pages of Frank Garvan's awesome Maple booklet ( part 1, part 2)
  Only record a small sample in hw1YourName.txt .
2. [For everyone!] Read and (try to understand) Chapter 1 of the Susskind-Frieman Quantum Mechanics book book. Do all the exercises.
3. [Optional (and very challenging) for novices, mandatory for experts]
  Write a procedure AM(G) that inputs a graph [n,E] and outputs the adjacency matrix, represented as a list of lengthn of lists of length n, such that M[i][j]=1 if {i,j} belongs to E and 0 otherwise. For example
  AM([2,{{1,2}});
  should output [[0,1],[1,0]] .
4. [Optional (and very challenging) for novices, mandatory for experts] write a procedue
  Image(pi,G)
  that inputs a permutation pi of {1,...,n} and a graph G=[n,E] and outputs the image under pi.
5. [Optional (and very challenging) for novices, mandatory for experts] By definition an unlabelled graph is an equivalene class of labeled graphs under graph isomorphism (i.e. mapping under the symmetric group). Write a procedure
  ULgraphs(n)
  that outputs a set of graphs with ONE member of each equivalence class.
  What is
  [seq(nops(ULgraphs(i)),i=1..6)]
  Is is in the OEIS? What is its A-number.
Added Jan. 26, 2025: Students' homework (that they kindly agreed to be posted) are in this directory

Programs done on Monday, Jan. 27, 2025
C2.txt , Contains procedures (in addition to those of C1.txt listed by Help1();)
- LC(p): inputs a rational number between 0 and 1 and outputs true with prob. p
- RG(n,p)
- Cliques(G,k): inputs a graph G and a pos. integer k outputs the set of k-cliques
Homework for Monday, Jan. 27, 2025, class (due Sunday Feb. 2, 8:00pm)
Please email ShaloshBEkhad at gmail dot com an email with
Subject: HomeWork#2
and an attachment
hw2FirstNameLastName.txt
and indicate whether it is OK to post. If you do not say "Please do not post", it will be posted.
1. [People who took this class before, or already know Maple, are excempt from this] Read and do all the examples, plus make up similar ones, pages 30-60 of Frank Garvan's awesome Maple booklet ( part 1, part 2)
  Only record a small sample in hw1YourName.txt .
2. [For everyone!] Read and (try to understand) Chapter 2 of the Susskind-Frieman Quantum Mechanics book book. Do all the exercises (if they exist)
3. [Optional for novices] Write a procedure
  TotClique(G,k)
  that inputs a graph G=[n,E] and a positive integer k, and outputs the total number of k-cliques and k-anti-cliques.
4. [Optional for novices] Writing a procedure
  CheckRamsey(k,K), that inputs a positive intgeger k and a large integer K that picks K random graphs (using RG(18,1/2)) and finds the minimum of ToClique(G,4);
5. [Optional for novices] Write a procedure
  EstimateAverageClique(n,p,k,K)
  that estimates the average number of cliques of size k in a random graph given by RG(n,p) by picking K of them and averaging.
  What did you get for
  EstimateAverageClique(10,1/2,4,10000)?
6. [Optional challenge, 5 dollars] Find a closed-form expression for the average number of k-cliques in a graph with p vertices where the prob. of every edge showing up is p.
Added Feb. 2, 2025: Students' homework (that they kindly agreed to be posted) are in this directory

Programs done on Thurs., Jan. 30, 2025

C3.txt , Contains procedures (in addition to those of C1.txt and C2.txt listed in Help1(); and Help2(); respectively)
- Neis(G): inputs a graph G=[n,E] and outputs a list of length n, N, such that N[i] is the set of neighbors of vertexi
- CC(G,i): The connected component of vertex i in the graph G
- IsCo(G): Is the graph G=[n,E] connected?
- NuCG(n): the first n terms of the sequence enumerating CONNECTED labeled graphs on n vertices, in other words sequence OEIS A001187 , done by brute force
- NuCGc(n): the first n terms of the sequence enumerating CONNECTED labeled graphs on n vertices, in other words sequence OEIS A001187 , done cleverly, using Exponential Generting Function, the fact that
  Sum(CC[i]/i!*z^i,i=0..infinity)= log(Sum(2^binomial(i,2)/i!*z^i,i=0..infinity)
Homework for Thurs. Jan. 30, 2025, class (due Sunday Feb. 2, 8:00pm)
Please email ShaloshBEkhad at gmail dot com an email with
Subject: HomeWork#3
and an attachment
hw3FirstNameLastName.txt
and indicate whether it is OK to post. If you do not say "Please do not post", it will be posted.
1. [People who took this class before, or already know Maple, are excempt from this] Read and do all the examples, plus make up similar ones, pages 60-90 of Frank Garvan's awesome Maple booklet ( part 1, part 2)
  Only record a small sample in hw1YourName.txt .
2. [For everyone!] Read and (try to understand) Chapter 3 of the Susskind-Frieman Quantum Mechanics book book. Do all the exercises (if they exist)
3. [For everyone!] Read and (try to understand) the section on "Exponential generating functions" (pp. 7-9) of Dr. Z.'s essay
4. [For everyone] Convince yourself that the following one-line procedure, for p rational between 0 and 1 and n positive integer
  PC:=proc(n,p,K) local i:evalf(coeff(add(IsCo(RG(n,p)),i=1..K),true,1)/K):end:
  is an estimate, with large enough K (say at least 200), for the probability of coonectivity of a random graph with n edges where every edge is picked with prob. p (the Edros-Renyi model)
  With n=10,20,30, and p=1/10, 2/10, ... and K=200 experiment and try to estimate the threshhold for connectivity, i.e. when it changes being very unlikely to very likely.
5. [Optional for novices]
  Procedure NuCCc(n) outputs the first n terms (starting at n=1) of OEIS sequence A1187 that counts the sequence let's call it C[n], of the number of connected labeled graphs with n vertices. It is based on the formula
  Sum(C[n]*z^n/n!,n=0..infinity)=log(Sum(2^(n*(n-1)/2)*z^n/n!,n=0..infinity)
  Let P[n](x) be the polynomial in x whose coefficient of x^r is the number of labeled connected graphs on n vertices and EXACTLY r edges. Convince yourself (or believe) that using generating functions one has the formula
  Sum(P[n](x)*z^n/n!,n=0..infinity)=log(Sum((1+x)^(n*(n-1)/2)*z^n/n!,n=0..infinity)
  Write a procedure
  NuCGx(n,x)
  that inputs a pos. integer n and a variable x, and outputs a list of length n whose i-th entry is P[i](x)
6. [Optional for novices]
  Convince yourself that the lowest power (lowdegree in Maple) of P[n](x) is n-1. Using NuCGx(n,x) (make sure to put "option remember" there), Write a procedure
  NuCGk(n,k)
  that inputs a positive integer n and a non-negative integer k and outputs the sequence of cofficients of x^(n-1+k) of P[n](x)
  What OEIS number is NuCGk(n,0)?
  What OEIS number is NuCGk(n,1)? (if it exists)
  Keep upping k and see the largest k for which NuCGk(n,k) is in the OEIS
Added Feb. 2, 2025: Students' homework (that they kindly agreed to be posted) are in this directory

Programs done on Monday, Feb. 3, 2025

C4.txt , Contains procedures (in addition to those of C1.txt and C2.txt and C3.txt listed in Help1(); and Help2(); Help3(); respectively)
- PM(): the three famous Pauli matrices [sigma_x,sigma_y, sigma_z]
- RM(n,K): a random n by n matrix with complex entries from -K to K
- CT(A): the conjugate transpose of A
- IsUM(A): Is the matrix A unitary
- STbf: Inputs a graph G=[n,E] and outputs the set of all spanning trees by brute force
- NST(G): The number of spanning trees of G=[n,E] (using the Matrix Three Therorem)
Homework for Monday Feb. 3, 2025, class (due Sunday Feb. 9, 8:00pm)
Please email ShaloshBEkhad at gmail dot com an email with
Subject: HomeWork#4
and an attachment
hw4FirstNameLastName.txt
and indicate whether it is OK to post. If you do not say "Please do not post", it will be posted.
1. [People who took this class before, or already know Maple, are excempt from this] Read and do all the examples, plus make up similar ones, pages 90-end of Frank Garvan's awesome Maple booklet ( part 1, part 2)
  Only record a small sample in hw1YourName.txt .
2. Read section 1.3 of the The Nielsen-Chuang book
3. Write a procedure
  UM(alpha,beta,gamma,delta)
  that implements box 1.1. on p. 20 of the The Nielsen-Chuang book [Note the superflous comma]
  For random alpha, beta, gamma, delta apply it to IsUM(A). Also, a litle bit of challenge, prove it for geneal (symbolic) alpha,beta,gamma, delta
4. [Optional challenge, 5 dollars to be divided among correct answers] Use the matrix tree theorem applied to the complete graph K_n to prove Cayley's formula that there n^n-2 labelled trees
Added Feb. 9, 2025: Students' homework (that they kindly agreed to be posted) are in this directory

Programs done on Thurs., Feb. 7, 2025
Today, due to the inclement weather we had a remote class where we did the following Maple code:
C5.txt , Contains procedures
- PM(): the three famous Pauli matrices sigma_x,sigma_y, sigma_z
- RHM(n,K): a random n by n Hermitian matrix with entriries from -K to K
- EV(L): inputs a Hermitian matrix (say n by n) and outputs the LIST of eigenvalues
- EVC1(L,g): inputs a matrix L and an eigenvalue g finds the NORMALIZED eigenvector
- EVC(L): inputs a Hermitian matrix and outputs the pair ListOfEigenvalues, ListOfEigenvectors
- ProbDist(L,A): Given an obervalble represented by the Hermitian matrix L and a state vector A outputs first all the eigenvectors (pure states) and the prob. of winding in the respective pure states followed by the expected value of A (i.e. if you make the observation L and the state is A the average in the long run
Homework for Thurs. Feb. 6, 2025, class (due Sunday Feb. 9, 8:00pm)
Please email ShaloshBEkhad at gmail dot com an email with
Subject: HomeWork#5
and an attachment
hw5FirstNameLastName.txt
and indicate whether it is OK to post. If you do not say "Please do not post", it will be posted.
1. Write a procedure
  RandomState(n,K)
  that inputs a pos. integer n and a pos. integer K and outputs a random state vector with n complex components whose real and imaginary parts are from -K to K, then normalize it.
2. Experiment with three different random random Hermitian matrices using RHM(n,K) with n=3 and K=10 and check that the eigenvalues are real and that the eigenvectors (pure states) are all orhogonal to each other.
3. Experiment with five different random random Hermitian matrices using RHM(n,K) with n=3 and K=10 and corresponding five random state vector and apply ProbDist(L,A) to them. For each case check that the proba. distribution consists of non-negative numbers that add up to 1.
4. Write a procedure
  GeneralPauli(n)
  that inputs a NORMALIZED vector n=[nx,ny,nz]
  that constructs the 2 by 2 matrix. at the bottm of p. 349 of the Susskind-Frieman Quantum Mechanics book book.
  Find its eigenvalues and eigenvectors.
Added Feb. 9, 2025: Students' homework (that they kindly agreed to be posted) are in this directory

Programs done on Monday., Feb. 10, 2025

C6.txt ,
- Mul(A,B): the product of matrix A and B (assuming that it exists)
- Com(A,B: The commutator of A and B
- RSV(n,K): a random NORMALIAED state vector of length n with complex entries from -K to K
- Del(L,A): the "avergae error", i.e. standard deviation pf the prob. distribution of measuring the observable L when the state is A
- CheckUP(A,B,V): the ratio of Del(A,V)*Del(B,V)/Expectation(Com(A,B),V) (it should be >=1/2, if the (generalized) Heisenberg Uncertainty Principle is True)
- [modified from C5.txt]
  ProbDist(L,A): Given an obervalble represented by the Hermitian matrix L and a state vector A (i) the list if eigenvalues (possible outcomes of observing L regardless of the state A (ii) the respective probabilities of the outcome being that eigenvalue, if the system is at state A (iii) the expected value of the outcome
Homework for Monday Feb. 10, 2025, class (due Sunday Feb. 16, 8:00pm)
Please email ShaloshBEkhad at gmail dot com an email with
Subject: HomeWork#6
and an attachment
hw6FirstNameLastName.txt
and indicate whether it is OK to post. If you do not say "Please do not post", it will be posted.
1. Write a procedure
  AllHM2(K)
  that inputs a positive integer K and outputs ALL 2 by 2 matrices of the form
  [[a,b+I*c],[b-I*c,d]]
  where a,b,c,d, are integers such that 1 ≤ a,b,c,d ≤ K (there are K⁴ of them
2. [clarified Feb. 13, 2025]
  Write a procedure
  RealSV2(K)
  that inputs a positive integer K and outputs all normalizations of the vectors (i.e. make them unit vectors)
  [a,b]
  with
  where a and b are integers 1 ≤ a,b ≤ K (there are K² of them
3. UncertaintyContest(K)
  that inputs a positive integer K and goes all throgh the pairs A, B of matrices in AllMH2(K), and RealSV2(K) (quite a few of them) and outpts the triplet [A,B,V] that make CheckUP(A,B,V) as small as possible and as big as possible [Make sure that you exclude the case when A and B commute, since then the ratio is infinity]
  What are
  UncertaintyContest(2)?, UncertaintyContest(3)?
  Added Feb. 16, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
4. SPECIAL VALENTINE HOMEWORK (also due Sunday, 8:00pm but a different attachments)
  Design a Valentine that says
  I love Algorithmic Graph Theory
  OR
  I love Quantum Computing
  email two attachments
  ValentineFirstLast.jpg
  and ValentineFirstLast.txt (with the source code)
5. (due Monday, 10:00am). Look the valentines in the directory
  Valentines,
  and fill the form
  ValentineBallot.txt
  and email ShaloshBEkhad@gmail.com with
  Subject: Valentine Ballot
  and an attachment
  ValentineBallotFirstLast.txt
  with your ratings (from 1 to 10, please reserve 0 for your own valentine and for those that have no submissions)
  [The winner will get a chocolate box.]
  Added Feb. 17: Congratulations to Lucy Martinez for winning first prize.
  
  Programs done on Thurs., Feb. 13, 2025
  
  C7.txt ,
  - PC1(T): one step in the Pruffer code algorithm. Looks for the smallest leaf outputs the pair [a,T1] where a is the LABEL of the (only) neighboor of the smallest leaf and T1 is the smaller tree where the only edge incident that leaf is removed.
  - PC(T): the Pruffer code of the labeled tree T labeled by {1, ...,n} where n=nops(T)+1
  - Added after class (based on Nick Belov's idea and top of p. 49 of Robin Wilson's wonderful graph theory book
    AntiPC(P): inputs a list of length n-2 with entries from {1, ...,n} outputs the tree T such that PC(T)=P
  Homework for Thurs. Feb. 13, 2025, class (due Sunday Feb. 16, 8:00pm)
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#7
  and an attachment
  hw7FirstNameLastName.txt
  and indicate whether it is OK to post. If you do not say "Please do not post", it will be posted.
  1. Carefully read the code for AntiPC(P) (based on Nick Belov's idea and Robin Wilson's book) and convince yourself that it does the right thing. By using rand, generate several sequences of length 10 with entries in [1,12] call them P, and find out whether PC(AntiPC(P))=P as it should.
  2. Write a procedure
    NumberOfLeaves(T)
    that inputs a labeled tree, T, and outputs the number of leaves (vertices of degree 1). For example
    NumberOfLeaves({{1,2},{1,3},{1,4}}); should output 3
  3. Write a procedure
    RandomTree(n)
    that inputs a positive integer n, and outputs a random labeled tree on {1, ...,n}
    [Hint: using rand(1..n), generate a random list of length n-2, P, and then let T:=AntiPC(P)]
  4. Write a procedure
    EstimateAverageNumberOfLeavs(n,K)
    that uses simulation to esimates the average number of leaves in a random labeled tree with n vertices
    [Hint: run RandomTree(n) K times, for each time find out the number of leaves, add them all up and divide by K.]
    What did you get for
    EstimateAverageNumberOfLeavs(100,1000)?
    EstimateAverageNumberOfLeavs(100,10000)?
    Are they close?
  5. [Optional theoretical challenge, "open book" and "open internet", 5 dollars]. Find the exact expression for the average number of leaves in a random tree with n vertices. Call it a_n. After you found it, prove that
    limit(a_n/n, n goes to infinity)
    exists, and find its value.
  6. Do a preliminary reading of Chapter 6 of Quantum Mechanics (The Theoretical Minimum) by Leonard Susskind and Art Friedman.
  Added Feb. 16, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Monday, Feb. 17, 2025
  
  C8.txt ,
  - ES(): the four fullly entangled states in the Alice-Bob combined system (based on p. 166 of the Susskind-Friedman's book)
  - TP(A,B):the tensor product of matrix A and matrix B (using our data structure of lists-of-lists)
  - [Added after class]: ExpMV(M,V): inputs a Hermitian matrix M and a state vector V finds the expected value of the observable M in state V using (more efficient than ProbDist(M,V)[3])
  Reading Homework for Monday Feb. 17, 2025, in preparation for the next class (2/20/25)
  
  Read Turing and Abel prizes winner Avi Wigderson's essay (section III.24, pp. 196-199) in the Princeton Companion of Mathematics edited by Sir W. Timothy Gowers.
  
  Homework for Monday Feb. 17, 2025, class (due Sunday Feb. 23, 8:00pm)
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#8
  and an attachment
  hw8FirstNameLastName.txt
  and indicate whether it is OK to post. If you do not say "OK to post", it will not be posted.
  1. Read and understand procedure ExpMV(M,V) that is supposed to give you the same ouptut as ProbDist(M,V)[3]. Take three take a random 3 by 3 Hermitian matrices (using RHM(3,10)) and a random 3-state vetor (using RSV(3,10)) and check that if you take evalf(ProbDist(M,V)[3]) you get the same thing as evalf(ExpMV(M,V)). Do it three times.
  2. By taking V:=RSV(2,100); verify that
    add(ExpMV(PM()[i],V)^2,i=1..3)=1
    Do it three times. In other words the sum of the squares of the expectation of the three Pauli matrices adds up to 1 for every state vector.
  3. The Alice-versions of the Pauli matrices are obtained by taking the tensor product with the identity, getting a 4 by 4 Hermitian matrix, in other words
    TP(PM()[i],PM()[4])
    Let's call them Ax,Ay,Az. For a random state vector, obtained by RSV(4,10), say, write a procedure
    EntanglementAlice(V)
    that outputs the sum of the squares of the expected values of Alice's measurement in that state.
  4. Write a procedure
    RandomProductState4(K)
    that outputs a random product state vector of length 4 obtained by taking the tensor product of two RSV(2,K)
  5. what are EntanglementAlice(V) for ES()[i], i=1..4?
    What is it for a random product state?
    By using RSV(4,10) twenty times, find the vector with maximal entanglemt (i.e. the smallest value of EntanglementAlice(V)) and the least (closest to 1)
  6. Do the same with EntanglementBob
Added Feb. 23, 2025: Students' homework (that they kindly agreed to be posted) are in this directory

Programs done on Thurs., Feb. 20, 2025

C9.txt , contains the following procedures:
- Gnd(n,d): the graph whose vertices are {0,...,n-1} and edges {i, i+j mod n} j=1..d (and at the end of the days the vertices are labeled {1, ....n}.
- AM(G): the adjacency matrix of the graph G=[n,E] (the stupid way, but it works for large graphs)
- lam2(G): inputs a graph and returns FAIL if it is not regular, otherwise it returns [degree, Lambda_2] (Lambda_2 is the second largestg eigenvalue of the adjacency matrix) (the spectral gap is d-Lambda2)
- Vk(k): The list of all 0-1 vectors of length k in lex order
- HD1(v): inputs a 0-1 vector and outputs all the vectors where exactly one bit is changed
  [The original name in class was Nei(v) but Pablo suggested changing the name since in C3.txt there was a procedure called Neis(v)]
- Ck(k): the k-dim unit cube as a graph in our format
Homework for Thurs. Feb. 20, 2025, class (due Sunday Feb. 26, 8:00pm)
Please email ShaloshBEkhad at gmail dot com an email with
Subject: HomeWork#9
and an attachment
hw9FirstNameLastName.txt
and indicate whether it is OK to post. If you do not say "OK to post", it will not be posted.
1. [A little bit challenging for novices]
  Using the definition of the Cheeger constant in this wikipedia article, write a procdeure
  CheegerC(G)
  that inputs a graph in our format [n,E] and outpits the Cheeger constant.
  [Hints: 1.with the combinat package loaded, choose(S,i) gives you the set of subsets of S of size i 2. To get the number of edges between S and its complement form the set of potential edges {s1,s2} where s1 is in S and s2 is in its complement and then intersect it with E and then take its nops) ]
2. Write a procedure
  CheckBounds(G)
  that outputs the Cheeger constant and the interval [(d-Lambda_2)/2,sqrt(2*d*(d-Lambda_2))] and checks that the former is inside the latter.
  What did you get for CheckBounds(G) for
  G=Gnd(n,2), 5 ≤ n ≤ 10
  G=Gnd(n,3), 5 ≤ n ≤ 10
3. [Optional challenge]
  1. (3 dollars) Conjecture an explicit expression for
  charpoly(AM(Ck(k)),x)
  2. (5 dollars, I don't know the answer): by searching the intenet find out whether it is well-known, and if possible find a reference for a proof
  Added Feb. 24, 2025: Joseph Koutsoutis won the prize and also Lucy Martinez who found this this interesting thesis by Captain Stanley F. Florkowski III.
  3. (20 dollars, I don't know the answer), Without "peeking" prove this conjecture.
  
  Added Feb. 23, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Monday, Feb. 24, 2025
  
  C10.txt , contains the following procedures:
  - NAND(x,y): not (x and y)
  - NOT(x): NAND(x,x)
  - AND(x,y): NAND(NAND(x,y),NAND(x,y))
  - OR(x,y): NAND(NAND(x,x) , NAND(y,y)):
  - RSLP(n,k): a random straight-line progran with NAND only
  - TF(n): the set of all true-false vectors of length n
  - EvalSP1(P,IN): inputs a straight-line NAND program and an input true-false vector outputs the value of the last gate
  - EvalSP(P,n): inputs a NAND SLP and outouts the set of vectors of length n that yields true
  Reading Homework for Monday Feb. 24, 2025, class (due Thurs., Feb. 27, 2025)
  
  Read Don Knuth's half-a-page version of Hao Huang's Proof of the sensitivity conjecture
  Homework for Monday Feb. 24, 2025, class (due Sunday March 2, 2025, 8:00pm)
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#10
  and an attachment
  hw10FirstNameLastName.txt
  and indicate whether it is OK to post. If you do not say "OK to post", it will not be posted.
  1. [clarified Feb. 25, 2025]
    Write a progmram
    SimulateNAND(n,k, K)
    that for K times does the following
    - picks a random P=RSLP(n,k)
    - computes nops(EvalSP(P,n)) ( a nunmber from 0 to 2ⁿ)
    - adds all of these numbers up
    - divides by K
    Whad did you get for:
    SimulateNAND(10,5, 1000), SimulateNAND(10,10, 1000), SimulateNAND(10,20, 1000), SimulateNAND(10,50, 1000)?
  2. (Modified Feb. 26, 2025)
    Adapt RSLP(n,k) to write a procedure
    Monotone(n,k,p) where n and k are positive integers and p is a rational number between 0 and 1), that has no NOT, i.e. only using AND and OR, and a typical member is [i,j,AND] or [i,j,OR] (now i ≤ j, i=j is not allowed) and for each non-input gate, the probability of AND is p and the probability of OR is 1-p)
    then write
    SimulateMonotone(n,k, p, K)
    What did you get for
    SimulateMonotone(10,5, 1/2,1000), SimulateMonotone(10,10, 1/2, 1000), SimulateMonotone(10,20, 1/2, 1000), SimulateMonotone(10,50, 1/2, 1000),
    SimulateMonotone(10,5, 3/4,1000), SimulateMonotone(10,10, 3/4, 1000), SimulateMonotone(10,20, 3/4, 1000), SimulateMonotone(10,50, 3/4, 1000),
    SimulateMonotone(10,5, 1,1000), SimulateMonotone(10,10, 1, 1000), SimulateMonotone(10,20, 1, 1000), SimulateMonotone(10,50,1, 1000),
    Hint: In order to simulate a "loaded die" that returns true with probabality p (where p is a rational number between 0 and 1) you can use
    LD:=proc(p):evalb(rand(1..denom(p))()<=numer(p)):end:
  Added March 2, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Thursday, Feb. 27, 2025
  
  C11.txt ,
  Contains (in addition to old procedures from C1.txt, C2.txt and C9.txt that can be viewed via Help1(), Help2(), Help9(), respectively):
  - Mul(A,B): the product of matrix A and B (assuming that it exists), in our favorite format of lists-of-lists
  - NeiList(G): inputs a graph G=[n,E] and outputs a list of length n whose i-th entry is the set of neighbors of vertex i
  - MaxNN(G,H): inputs a graph G=[n,E] and a subset H of {1, ..., n} outputs the maximum number of members that belong to H over all members of H
  - MinMaxNN(G,r): minimum of MaxNN(G,H) over all subsets H of {1, ..., n} with cardinality r
    (Note: the Hao Huang famous sensitivity conjecture (now a theorem) says that MinMaxNN(Ck(n),2^(n-1)+1)>=sqrt(n)
  - MinMaxNNpablo(G,r): A (much more) memory-efficient of MinMaxMN(G,r), suggested by Pablo Blacno (after discussions with Auora Hiveley)
  - An(n): the 2^n by 2^n matrix (given as a list of lists) in Don Knuth's half-a-page version of Hao Huang's Proof of the sensitivity conjecture
  - Bn(n): the 2^(n-1) by 2^n matrix in the above paper
  Homework for Thursday Feb. 27, 2025, class (due Sunday March 2, 2025, 8:00pm)
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#11
  and an attachment
  hw11FirstNameLastName.txt
  and indicate whether it is OK to post. If you do not say "OK to post", it will not be posted.
  1. Using the corrected Mul(A,B), verify that indeed A_n B_n=sqrt(n)B_n for small n. How far were you able to go with in a reasonable amount of time?
  2. As Pablo noticed the largest r such that MaxNN(G,r) is 0 is the independence number of G (the largest subset of vertices that have no edges between them). Use
    MaxNNpablo(G,r)
    to write a procedure
    IndNu(G)
    that finds the independence number of a graph G=[n,E]
    What are the independence numbers of Gnd(n,2) for n from 3 to 15? Can you make a conjecture? [5 dollars to be divided] Can you prove it?
    What are the independence numbers of Gnd(n,3) for n from 3 to 15? Can you make a conjecture? [5 dollars to be divided] Can you prove it?
  3. Write a procedure
    UpperBoundMinMaxNN(G,r,K)
    that uses combinat[randcomb]({seq(i,i=1..n)},r) K times and finds the minimum of MaxNN(G,H) for K random r-element subsets of the set of vertices {1,..,n}.
    What did you get for
    UpperBoundMinMaxNN(Ck(8),129,5000)?
    How far is it from the actual minimum of 3 (according to the sensitivity theorem)?
  Added March 2, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Monday, March 3, 2025
  
  C12.txt ,
  - Verify1(n): checks that An(n) times Bn(n)=sqrt(Bn(n))
  - Findy(n,H): inputs a pos. integer n and a subset H with exacty 2^(n-1)+1 elements and outputs the vector y such that
  - LuckyVertex(n,H): inputs pos. integer n and a subset H of size 2^(n-1)+1 outputs a member that is guaranteed to have at least sqrt(n) neighbors in H
  Homework for Monday March 3, 2025, class (due Sunday March 9, 2025, 8:00pm)
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#12
  and an attachment
  hw12FirstNameLastName.txt
  and indicate whether it is OK to post. If you do not say "OK to post", it will not be posted.
  1. Write a procedure
    AveLuckyVertexFriends(n,K)
    that inputs a positive integer n, and for K random subsets of {1, ..., 2^n} with 2^(n-1)+1 elements gotten with
    combinat[randcomb](2^n,2^(n-1)+1)
    does the following
    1. Finds the lucky vertex (in terms of the row number)
    2. Finds the number of members of H that are joined by an edge to it
    [Hint: The absolute value of An(n) is the adjacency matrix of the n-dim unit cube so you have to sum the absolute value of the entry in the row that correpond to the lucky vertex that belong to columns of H]
    Call that number FriendsOfLucky(n,H)
    3. Make sure that it is always ≥ sqrt(n), according to the sensitivity lemma If you can find an H that violates it, return FAIL.
    4. Average it over all K random chices and output the average.
    What did you get for
    AveLuckyVertexFriends(9,100)?
  Added March 9, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Thurs., March 6, 2025
  
  C13.txt ,
  Contains the following procedures:
  - C1QGnames(): The names of the seven most common quantum 1-gates , namely: [` Identity `, ` PauliX `, `PauliY `, `PauliZ `, ` PhaseS `, `TPi/8` , ` Hadamard `]
  - C1QG(): The list of length 7 containing the above 2 by 2 unitary matrices in our list-of-lists format for matrices
  - BV(n): inputs a non-neg. integer n and outputs the list of length 2^n of all 0-1 vectors in lex. order
  - nBasis(n,X): the basis of n-quantum gates circuits X[e1,e2,.., en] corresponds to |e1>|e2>...|en>
  - UMtoT(M,n,X): inputs a unitary matrix M of size 2^n by 2^n (correpsonding to a quantum circuit with n qbits, outputs the corresponding LINEAR TRANSFORMATION as it acts on the canonical basis
    [The output is a pair T=[Table,Base]. To see how T[1] acts on a a member of the base do T[1][T[2][i]] where i is between 1 and 2^n.
  - TtoUM(T,n): inputs a table T=[T1,Base] of n-gate quantum network and outputs the corresponding 2^n by 2^n unitary matrix (the reverse of UTtoT(M,n,X))
  - [From previous claases: Mul(A,B) (matrix multiplcation AB in list-of-lists format. TP(A,B): Tensor product of A and B, ES(), the four famous Bell states (maximally entangled)
  Homework for Thurs. March 6, 2025, class (due Sunday March 9, 2025, 8:00pm)
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#13
  and an attachment
  hw13FirstNameLastName.txt
  and indicate whether it is OK to post. If you do not say "OK to post", it will not be posted.
  1. On p. 130 of Thomas G. Wong's wonderful book Introduction to Classical and Quantum Computing it is proved that
    HXH=Z
    This can be rephrased, in our program as
    Mul(Mul(C1QG()[7],C1QG()[2]),C1QG()[7])=C1QG()[4]
    Find all triples 2 ≤ i,j,k ≤ 7 (there are 6^3=216 such triples) such that
    Mul(Mul(C1QG()[i],C1QG()[j]),C1QG()[k])
    is equal to a member of C1QG(), say C1QG()[r]. Output the set of all such quadruples [i,j,k,r] (in particular [7,2,7,4] should be a member of that set).
    In other words find all such identities where a product of three (not necessarily distinct) mmembers of C1QG() equals to one of them.
    How big is that set? for some 1 ≤ r ≤ 7
  2. [Added March 8, 2025] Adapt the above problem to find, more generally all quadruples 2 ≤ i,j,k ≤ 7 and c constant (obviously of absolute value 1) such that
    
    Mul(Mul(C1QG()[i],C1QG()[j]),C1QG()[k])=c*C1QG()[r]
    Output the set of all such [i,j,k,r,c].
  3. Write a procedure
    ApplyT(T,X,n,u)
    where T=[T1,V], T1 is a table and V is nBasis(n,X), and u is an arbitrary linear combination of of members of nBasis(n,X), that uses the linearity of the table , and outputs the result of applying the linear transformation T (given as a table defined on the canonical basis), as a linear combination of the basis elements.
    If T=UMtoT(TP(H,Z)), what is
    ApplyT(T,X,2,X[[0,0]]+I*X[[0,1]]+2*I*X[[1,0]]+5*X[[1,1]]); ?
  4. Read and understand the "no-cloning theorem", p. 166, of Wong's book.
  5. [Added March 7, 2025, 10:00 am] Optional challenge (ten dollars). The fact that two complex numbers a[1] and a[2] commute, i.e.
    a[1]*a[2]=a[2]*a[1]
    can be rewriiten as
    add(c[pi]*a[pi[1]]*a[pi[2]], pi in permute(2))=0
    where c[12]=1 and c[21]=-1,
    As you know, for 2 by 2 matrices A[1] and A[2] it is usually not true that A[1]A[2]=A[2]A[1] .
    But perhaps something analogous is true for them if you allow something more complicated. Can you find six numbers c[123], ..., c[321] such that
    add(c[pi]*Mul(Mul(A[pi[1]],A[pi[2]]),A[pi[3]]), pi in permute(3))= [[0,0],[0,0]]
    for every three 2 by matrices A[1],A[2],A[3]?
    If not, can you find 24 numbers c[1234], ..., c[4321] such that
    add(c[pi]*Mul(Mul(A[pi[1]],A[pi[2]]),Mul(A[pi[3]],A[pi[4]])), pi in permute(4))=[[0,0],[0,0]]
    for every four 2 by matrices A[1],A[2],A[3], A[4]?
    [Hint (modified March 7, 2025, 2:10pm): You have to look for 24 numbers such that the above relation holds for EVERY choice of four 2 by 2 matrices A[1],A[2],A[3],A[4]. To find 24 unknown numbers you need 24 independent homogeneous equations. By making 6 random choices (using rand(1..10)(), for example, and then writing a little procedure RM() that outputs a random 2 by 2 matrix), pick 6 random choices of quadruple of numerical 2 by 2 matrices. For each random choice you get a 2 by 2 matrix featuring the unknowns c[pi]. Then set them all equal to zero. Then use Maple's solve. If you get a not-all-zero solution it is good news, and you have a conjecture. Finally to prove it rigorously, plug in four SYMBOLIC matrices A[1],A[2],A[3],A[4].]
  Added March 9, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Monday, March 10, 2025
  - C14.txt ,
  - C14pi.txt ,
  C14.txt contains:
  - Bell(A0,A1,B0,B1,V): The expression due to Bell that is the quantum analog of a0*b0+a0*b1+a1*b0-a1*b1 ( the expectation of its abslute value is <=2) The choice of A0,A1,B0,B1 in the wikipedia article is
    Bell(C1QG()[4],C1QG()[2], -(C1QG()[2]+C1QG()[4])/sqrt(2),(C1QG()[2]-C1QG()[4])/sqrt(2),ES()[1]);
  C14pi.txt contains:
  - Leibnitz(D1): Uses Leibnitz' formula to compute Pi with an error less than1/10^D1
  - LeibnitzN(N): using N terms in the Lebnitz formula to appx. Pi
  Special Pi Day Homework due Wed., March 12, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#14pi
  and an attachment
  hw14piFirstNameLastName.txt
  and indicate whether it is OK to post. If you do not say "OK to post", it will not be posted.
  1. Write a procedure
    JesusG(N)
    that uses N terms of Jesus Guillera's amazing formula in his homepage (given in "Ramanujan notation") that computes an approximation to 128/π².
    How close is JesusG(10) to 128/π²?, How close is JesusG(100) to 128/π²?,
  2. [Pi Hacktaton]: Use any algorithm you can find (except "cheating ones" like copying the digits of Pi) to compute Pi. The fasted and/or most elegant entry will win a Pi T-shirt.
    [Hint: Many series for Pi are of hypergeometric type: Sum(p(n)*f(n),n=0..infinity) where p(n) is a polynoial and f(n)/f(n-1) is a rational function. It is most efficient to just have the current value of f(n) and keep adding p(n)*f(n)]
  Added March 13, 2025: See all Students' Pi codes combined in one file
  Students' Pi homework (that they kindly agreed to be posted) are in this directory
  
  Regular Homework due Sunday, March 16, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#14
  and an attachment
  hw14FirstNameLastName.txt
  1. [Modified and simplified, March 11]
    Write a process
    RandomBell1(K)
    (1) After defining ra:=proc(K): rand(-K..K)():end: Keeping the same A0,A1,B0,B1, as in the wikipedia article
    A0:=C1QG()[4],A1:=C1QG()[2],B0:= -(C1QG()[2]+C1QG()[4])/sqrt(2),B1:=(C1QG()[2]-C1QG()[4])/sqrt(2)
    (2) Picks a random state (in the 2-qubit space generated by 00,01,10,11, sets V=[ra(K)+I*ra(K),ra(K)+ra(K)*I,ra(K)+I*ra(K),ra(K)+I*ra(K)]:
    (3) NORMALIZE it! , i.e. divide V by its NORM. rename the new state V.
    Computes Bell(A0,A1,B0,B1,V)
  2. Write a procedure
    RandomBell(K,N)
    that runs RandomBell1(K) N times and returns
    (1) The average value over all N trials.
    (2) The fraction of times it exceeds 2, thereby refuting Albert (and Boris and Nathan)
    What did you get for
    RandomBell(10,1000)?
  Added March 23, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Thurs., March 13, 2025
  - C15.txt ,
  Today we celebrated (one day early) Pi day.
  - A(n): A sequence of bad approximations to Pi using the prob. that if you toss a coin 2n times the prob. of getting exactly n Heads is asymototic to sqrt(n)/Pi
  - IsUD(pi): Is the permutation pi an up-down i.e. pi[1]pi[3]
  - UD(n): the number of up-down permutations of length n the stupid way (pure brute force)
  - [added after class]
    UDset(n): the SET of up-down permutations of length n the stupid way
  Homework due Sunday, March 16, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#15
  and an attachment
  hw15FirstNameLastName.txt
  1. By looking at UDset(7) and Udset(8), observer that the location of "n" the largest entry (7 and 8 respectively) are always at EVEN locations. Why?
  2. Using this fact, explain why the number of up-down permutations of length n (that is computed by brute force using UD(n)), let's call it a(n), satisfies the non-linear recurrence
    a(n)=Sum(binomial(n-1,2*k-1)*a(2*k-1)*a(n-2*k),k=1..trunc(n/2)):
    (subject to the initial condition a(1)=1).
  3. Using this recurrence, with `option remember`, write a procedure UDc(n) (clever version of UD(n)). Make sure that UDc(n) and UD(n) coincide for 1<=n<=8.
    What is
    evalf([seq(2*n*a(n-1)/a(n),n=1000..1010)]) ?
    Does it seem to converge to any famous constant?
  4. [Optional challenge, 5 dollars] Prove the above conjecture rigorously. Hint: You can "cheat" and look at the exponential generating function stated at the OEIS, but you have to prove that it is indeed the exponential generating function.
  Added March 23, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Monday, March 24, 2025
  - C16.txt ,
  Contains procedures:
  - QFT(q): The q by q matrix of the quantum discrete Fourier Transform (that is always with q=2^n in quantum computing.
  Homework for March 24, due Sunday, March 30, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#16
  and an attachment
  hw16FirstNameLastName.txt
  and indicate whether it is OK to post.
  1. Look at the Maple package QC.txt, and play with its many procedures.
  2. Read Peter Shor's seminal paper p. 9 and write procedures
    ToffoliMatrix(): and FredkinMatrix()
    that returns the appropriate 8 by 8 matrices
  3. Write a procedure
    Sjk(j,k)
    that expresses the matrix S_j,k (Eq. (4.3), p. 14) in our notation of lists-of-lists
  4. Write a procedure
    QFTC(l)
    that inputs a pos. integer n and outputs the quantum circuit of Eq. (4.4) (but replace l by n) in the same format at the output of RQC. i,e.
    [n,[H,[n-1]], ....
    where H is the Hadamard matrix C1QG()[7]=[[1/2*2^(1/2), 1/2*2^(1/2)], [1/2*2^(1/2), -1/2*2^(1/2)]]
  Added March 30, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Thurs., March 27, 2025
  - C17.txt ,
  Contains procedures:
  - Ord(x,n): inputs ae pos. integer n and an integer x < n . Outputs (by brute force) the order of x mod n, i.e. the smallest pos. integer r such that x^r mod n=1
  - FindFact1(n): tries to find a non-trivial factor of n or returns FAIL
  - FindFact(n,K): tries to find a factor of n using FindFact1(n) K times or returns FAIL (very unlikely if K is big)
  - Factorize(n,K): inputs a pos. integer n and outputs the list of prime factors
  - ModExp(x,r,n): x^r mod n the fast way where r is given in reverse binary [2^0,2^1,2^2,...]
  Homework for March 27, due Sunday, March 30, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#17
  and an attachment
  hw17FirstNameLastName.txt
  and indicate whether it is OK to post.
  1. Tijil Kiran noticed that if gcd(x,n)=1, then the order of x mod n always exists, since, thanks to Euler, for every such x relatively prime to n,
    x^phi(n) mod n =1 .
    Hence the order must be a divisor of phi(n). The Maple package numtheory has both phi and divisors. Write a procedure
    MyOrd(x,n)
    that (i) computes phi(n) (ii) finds its set of divisors, (iii) sorts them (iv) keeps trying x^r mod n until it gets 1.
    Take random large x and n and compare the time it takes.
    Why is even this more (possibly more) clever way of computing the order of x mod n not break (in realistic time) the RSA with a classical computer?
  2. [Modified March 29, 2025]
    A weighted directed graph with n vertices can be described a pair [n,E] (like the one generated by procedure RG(n,p) of C2.txt , together with a table T defined on E where for any member e of E T[e] is the distance. Write a procedure, RWG(n,p,K) whose output is a triple [n,E,op(T)] where T is the table of distances. where the distances are random integers from 1 to K.
    For example the graph with vertices {1,2,3} and set of edges {{1,2},{1,3}} and the distance between 1 and 2 is 5 and between 1 and 3 is 2 is expressed as:
    [3, {{1, 2}, {1, 3}}, table([{1, 3} = 2, {1, 2} = 5])]
  3. [Optional challenge, 5 dollars to be divided] Implement Dijksra's algorithm that inputs a weighted directed graph (given as above [n,E,op(T)]) and a starting vertex a, and outputs the list of length n where L[i] is the shortest distance from the starting vertex a to i.
    Also write a procedure where L[i] is not the shortest distance but the actual path from a to i.
  Added March 30, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Monday, March 31, 2025
  - C18.txt ,
  Contains procedures:
  - RWG(n,p,K): inputs a pos. integer n and outputs a triple [n,E,DT] where the vertices are {1,...,n} the edges are of the form {i,j} and DT is table such DT[{i,j}] is the weight of the edge {i,j}
  - Dij(G,a): inputs a weighted graph G=[n,E,DT] and outputs and a starting vertex a outputs a list L of length n such that L[i] is the MINIMAL DISTANCE min. distance from a to i. In particular L[a]=0
  - [completed after class]
    DijP(G,a): same input as Dij(G,a) but outputs in addition to the list of shortest distance from 1 to i another list with the corresponding actual paths.
  Homework for March 31, due Sunday, April 6, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#18
  and an attachment
  hw18FirstNameLastName.txt
  and indicate whether it is OK to post.
  1. Write a procedure
    AveLengthSP(G,a)
    That inputs a weighted graph and a starting vertex a and uses GijP(G,a)[2] to compute the average length of a shortest path from a.
  2. Write a procdeure
    AveLengthStat(n,p,K,M)
    that picks a random weigted graph according to RWG(n,p,K) M times, finds for each AveLengthSP(G,a) and finds the average and variance for each run.
    Run
    AveLengthStat(100,1/4,2,10000)
    five times. What did you get? Are they similar?
  Added April 6, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Pablo Blanco's explanation why you need op(T) rather than T for a table in Maple
  
  Programs done on Thurs., April 3, 2025
  - C19.txt ,
  Contains procedures:
  - MST1(G,partT,AvE): inputs a weighted graph G and performs ONE step in the Kruskal algoritm by looking at the cheapest member of AvE and trying to join it to partT if it does NOT create a cycle, either way we remove it from AvE
  - MST(G): the minimal spanning tree of the weighted graph G=[n,E,DT] followed by its cost (the sum of the costs of the edges of the cheapest tree)
  - CC(G,a): inputs a graph G=[n,E] and a vertex a and outputs the set of vertices connected to a.For example CC([5,{{1,2},{2,3},{4,5}}],1) is {1,2,3}.
  Homework for April 3, due Sunday, April 6, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#19
  and an attachment
  hw19FirstNameLastName.txt
  and indicate whether it is OK to post.
  1. Write a procedure
    EstimateAverageMinCost(n,p,K,N)
    that runs MST(RWG(n,p,K)) N times (where N is a large pos. integer) and outputs the fraction of those that are connected, and among them the average cost of a minimal spanning trees (both in decimals)
  2. Using the above, write a procedure
    EstimateTable(n,N,a,b)
    that inputs a pos. integer n (not too large), a large (at least 1000) pos. integer N, pos. integer a and pos. integer b and outputs the list
    L[i][K] (i beteen 1 and a) (K between 1 and b)
    such that L[i][K] is the estimated average of a minimal spanning tree of a random weighted spanning tree with prob. of picking an edge with prob. i/a and picking a weight of an edge from 1 to K.
    What did you get for
    EstimateTable(10,3000,5,10)?
    Run it a few times and see whether the results are close (of course they shouldn't be the same, this is simulation)
  3. [Big challenge, I have no clue, 50 dollars if you do it yourself, 20 dollars if you can find a reference]. Find a closed-form expression, in terms of n, p, and K to the expected cost of a minimal spanning tree if you pick a weighted random graph according to RWG(n,p,K) (i.e. you pick each edge (independently) with prob. p and for each edge you pick an integer weight, uniformly from 1 to K.
    Added April 6, 2025: Lucy Martinez found the following references: On the value of a random minimum spanning tree problem by Alan Frieze and Minimal Spanning Trees for Graphs with Random Edge Lengths by J. Michael Steele.
  4. [Optional Challenge, 10 dollars to be divided]
    Write a procedure
    EulerianPath(G)
    that inputs a Eulerian graph G=[n,E] (see section 6 of Robin Wilson's graph theory book), that (i) checks that it is indeed Eulerian (ii) outputs a Eulerian trail using either Fleuri's algorithm (see there) or any other correct method.
  5. Get ready for next class by reading, and trying to understand, section 5 of Peter Shor's seminal paper. This is very challenging (and not so well-written), and it took me five times to understand it, and we will go over it next class, but it would help if you will make an effort to try and understand it, at least at some level.
  Added April 6, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Monday, April 7, 2025
  - C20.txt ,
  Contains procedures:
  - Eq55(q,k,n,x): the value of Eq. (5.5) in Peter Shor's seminal paper
  - Eq56(q,k,n,c,x): the value of Eq. (5.5) in Shor's paper
  - Eq56(q,k,n,c,x): the value of Eq. (5.6) in Shor's paper
  - Eq57(q,k,n,c,x): the value of Eq. (5.7) in Shor's paper
  - Eq58(q,k,n,c,x): the value of Eq. (5.8) in Shor's paper (w/o the error term)
  - Eq59(q,k,n,c,x): the value of Eq. (5.8) in Shor's paper (w/o the error term)
  - [Added after class]
    Histogram56(q,k,n,c): the plot of [c,Eq56(q,k,n,c,x)] for c from 1 to q-1. For example
    Histogram56(128,86,4,3);
  Homework for April 7, due Sunday, April 13, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#20
  and an attachment
  hw20FirstNameLastName.txt
  and indicate whether it is OK to post.
  1. Pick one of the projects in project or propose one of your own
  2. Write procedures Eq510(q,r,c)
  3. Denoting {rc}_q/r by x, find a closed form expression for Eq. 5.10 in Shor's paper. Using this as an approximation for the true inside of Eq. 5.5 What is the approximation for the probability of observing [c,k] (note that it is independent of k].
    Hint: first use Maple's int command. Then do evalc then take the absolute value-squared (the sum of the square of the real part and the square of the imaginary part)
  4. Plot
    Histogram56(128,86,4,3);
    Histogram56(256,86,4,3);
    and come up with five more examples like this.
  Added April 13, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Thurs., April 10, 2025
  - C21.txt ,
  Contains procedures:
  - Histogram510(q,n,x) : a histogram of Eq, 5.10 in Shor's paper. To get Fig. 5.1 do Histogram(256,33,5);
  - BA1(f,r): the closest fraction d/r to f, where r is fixed
  - BA(f): the best fraction d/r where r is between 1 and sqrt(denom(f)) by brute force
  - CF(f): inputs an integer or fraction outputs the list consisting of the continued-fraction of
  - EvalCF(L): inputs a list of pos. integers and outputs the fraction whose cont. fraction it is (reverse of CF(f))
  - CVG(f): inputs a rational number f and outputs its list of CONVERGENTS
  Homework for April 10, due Sunday, April 13, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#21
  and an attachment
  hw21FirstNameLastName.txt
  and indicate whether it is OK to post.
  1. Write a procedure
    CFx(x,k)
    that inputs any positive number (given in decimals) and outputs the first k terms of its infinite continued fraction
  2. Write a procedure
    GuessPCF(x)
    that inputs a quadratic irrationality x (e.g. sqrt(n) where n is not a square-free) and outputs two lists [L1,L2]
    such that the continued fraction of x is [L1,op(L2)^infinity].
    Find GuessPCF(n) for all non-square integers beween 2 and 100
  3. Is the sequence of lengths of their periods in the OEIS? Which is the largest period.
  4. Find the best rational approximations to e and Pi with denominator less than 10000000
  5. [Optional challenge, 10000 dollars]. Prove or disprove that the terms of the continued fraction of the cubic-root of 2 are bounded.
  Added April 13, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Monday, April 14, 2025
  - C22.txt ,
  contains procedures:
  - BAc(f): a clever version of BA(f): the best relative approximation to the fraction f
  - Likelyc(q,r): All c less than q such that |{rc}_q| <=r/2
  - LikelycC(q,r): All c less than q such that |{rc}_q| <=r/2 the clever way #i.e. all integer multioles of q/r rounded
  - ConfirmR(q,r): confirms that all the good c's give you the same r
  - ALnm(X,n,m,K): Tries to find a relation between m n by n matrices. Try: ALnm(X,1,2); Alnm(X,2,4); Alnm(X,3,6);
  Homework for April 14, due Sunday, April 20, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#22
  and an attachment
  hw22FirstNameLastName.txt
  and indicate whether it is OK to post.
  1. Run ConfirmR(2^n,19) for n from 10 to 20. Are all the denominators always 19?
  2. Redicover
    - The fact that multiplication of numbers is commutative by running ALnm(X,1,2,10);
    - That there is no Amitsur-Levitsky relation between three 2x2 matrices by running ALnm(X,2,3,10);
    - The Amitsur-Levitsky relation between four 2x2 matrices by running ALnm(X,2,4,10);
    - That there is no Amitsur-Levitsky relation between five 3x3 matrices by running ALnm(X,3,5,10);
    - The Amitsur-Levitsky relation between four 3x3 matrices by running ALnm(X,3,6,10);
      [Warning: be patient, Maple has to solve a system with more than 720 equations and 720 unknowns]
  3. Work on your project
  Added April 20, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Thurs., April 17, 2025
  - C23.txt ,
  contains procedures:
  - BT(n): the SET of full binary trees with n leaves
  - T(N): the first N terms of the sequence enumerating the number of full binary trees with n leaves
  - Images(T): all the images of the full binary tree T under reflection at ALL levels.
  - UBT(n): the set of all unlabeled full binary trees (one representative each)
  - UT(N): the first N terms of the sequence enumerating the number of UNLABELED full binary trees with n leaves generating function satisfies the functional equation f(x)=x+(f(x)^2+f(x^2))/2
  Homework for April 17, due Sunday, April 20, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#23
  and an attachment
  hw23FirstNameLastName.txt
  and indicate whether it is OK to post.
  1. A rooted ternary tree is either a single vertex, or an ordered tree where every vertex is either a leaf (has no children) or has exactly three children. Write ternary analogs for BT(n), T(N), Images(T), UBT(n), and UT(N), calling them:
    TT(n), T3(N), Images3(T), UBT3(n), UT3(N), respectively. What is UT3(40)? Is it in the OEIS? What is its A number.
    [Hint, the functional equation, according to Polya is
    r(x)=x+1/6*(r(x)^3+ 3*r(x)*r(x^2)+ 2*r(x^3))
    Note that the number of leaves is always odd so only extract the odd coefficients]
  2. If you type
    ALnmG(X,3,4,10,{[1,3],[2,1],[2,3],[3,1],[3,2]});
    In the Maple package AL.txt you would get six identities that any four 3x3 matrices A1,A2,A3,A4 where the only the diagonal entries as well as the [1,2] entry are non-zero, i.e. matrices of the form
    [[a11,a12,0],[0,a22,0],[0,0,a33]]
    One of them is:
    X[[1, 2, 3, 4]]-X[[1, 2, 4, 3]]-X[[2, 1, 3, 4]]+X[[2, 1, 4, 3]]
    Convince yourself that this is stating that for any four such matrices
    (A1A2-A2A1)(A3A4-A4A3)= 0 matrix
    Or in our language
    Mul(Mul(A1,A2)-Mul(A2,A1), Mul(A3,A4)-Mul(A4,A3))=[[0,0,0],[0,0,0],[0,0,0]] .
    Prove it rigorously.
  3. [Optional challenge] (5 dollars)
    A full k-ary tree is either a single vertex, called a root, or an ordered tree where every vertex is either a leaf or else has exactly k children.
    Let f(x) be the generating function
    Convince yourself that
    f(x)=x+f(x)^k
    Note that such a tree must have 1+(k-1)m leaves for some m. Can you conjecture, and if possible, prove, an explicit expression for
    A_k(m):=Number of full k-ary trees with 1+(k-1)m leaves?
  4. [Option challenge, 10 dollars] Write a procedure
    UTk(k,N)
    that inputs a positive integer k ≥ 2, and a positive integer N and outputs the first N terms of the sequence
    Number of UNLABELED full k-ary trees with 1+(k-1)m leaves for m=1..N
    [Hint: You need to find a general expression (or look it up) for the "cycle index polynomial" of the symmetric group on {1,...,k}, and then generlize UT(N) and your UT3(N) with the appropriate functional equation for r(x)=r_k(x)]
  Added April 20, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Mon., April 21, 2025
  - C24.txt ,
  contains procedures:
  - FibMod(INI,m):the trajectory of the mapping [a,b]->[b,a+b mod m] starting with INI
  - RF(n): A random function from {1,...,n} to {1,...,n}
  - FindPer(f,i): inputs a function from {1,..,n} to {1,...,n} where n:=nops(f): and outputs the pair [L1,L2] where L1 is the beginning segment and L2 is the cycle
  - FindPer(f,i)
  - AllF(m,n): all lists of length n with entries from {1,..,m}
  - GFperiod(n,x):the polynomial whose coeff. of x^j is the prob. that if you take a random function from [1,n] to [1,n] and a random starting point the length of the ultimate cycle is j
  - GFperiodPer(n,x):the polynomial whose coeff. of x^j is the prob. that if you take a random permutation from [1,n] to [1,n] and a random starting point the length of the ultimate cycle is j, followed by the average and standard-deviation
  - EstimateGFperiod(n,x,K): samples K random functions from [1,n] to [1,n] and a random starting point. and finds the prob. gen. function for this run followed by the sample average and standard-deviation. Try:EstimateGFperiod(20,x,100);
  Homework for April 21, due Sunday, April 27, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#24
  and an attachment
  hw24FirstNameLastName.txt
  and indicate whether it is OK to post.
  1. Briefly describe the progress on your team's project
  2. Write a procedure
    CompareFibMod(INI,M)
    that for m from 2 to M compares FibMod(INI,m) to EstimateGFperiod(m^2,x,10000)[2][1]. Is the former ever larger than the latter? when INI=[0,1] and M=300?
  3. [Optional, 5 dollars, I know the answer, but it is a fun challenge] Prove rigorously, that for every positive integer n:
    GFperiodPer(n,x)[1] is (x+x^2+...+x^n)/n .
  4. [Optional challenge, 20 dollars, to be divided (I don't know the answer)]
    seq(GFperiod(n,x)[2][1]*n^(n+1),n=1..5);
    coincides with the first five terms of OEIS sequence A262970, and according to the OEIS equals it for all n.
    It is stated there that the E.G.F.of this sequene is T/(1-T)^4, where
    T:=Sum(n^(n-1)/n!*x^n,n=1..infinity)
    Prove it!
  Added April 27, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Thurs., April 24, 2025
  - C25.txt
  Contains procedures:
  - EC(pi,i): inputs a permutation pi and finds the cycle that belongs to i
  - Mult(a): inputs an integer partion of n, a in frequency-notation, and outputs
    n!/(1!^a1*2!^a2*...*n!^an)/(a1*a2!*..*an!)*0!^a1*1!^a2*...(n-1)!^an
    (the number of permutations of length n, with a[1] 1-cycles, a[2] 2-cycles, ..., a[n] n-cycles
    [This formula is due to Cayley]
  - PtoC(pi): inputs a permutation in one-line notation outputs its cycle decomposition as a set if cycles
  - WtP(pi,x): The weight of the permutation pi according to the weight
    x[1]^NumberOfOneCycles*x[2]^NumberOfTwoCycles...*x[n]^NumberOf n cycles
  - CIP(G,x): The Polya cycle-index polynomial of the gp G of permutations of length n
    
    [Added after class]
  - Pars1(n,m): The set of lists [a1,a2,...,an], of non-neg. integers such that 1*a1+...+n*an=m
  - CIPsn(n,x): Same output as CIP(permute(n),x), done cleverly, using Mult(a), much faster!
  Homework for April 24, due Sunday, April 27, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#25
  and an attachment
  hw25FirstNameLastName.txt
  and indicate whether it is OK to post.
  
  Homework for April 24, due Sunday, April 27, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#25
  and an attachment
  hw25FirstNameLastName.txt
  and indicate whether it is OK to post.
  1. Briefly describe the progress on your team's project
  2. Read and understand p. 10 and the first quarter of p. 11 of this masterpiece
  3. According to Polya theory (see above essay) The number of sequences of length n with entries in {1,...,c} up to the action of the symmetric group (i.e. the number of equivalence classes) is obtained by plugging in x[i]=c for all i=1..n. Write a procedure
    NuEqC(n,c)
    that uses CIPsn(n,x) to find that number.
    What is
    [seq(NuEqC(i,2),i=1..12)]
    [seq(NuEqC(i,3),i=1..12)]
    [seq(NuEqC(i,4),i=1..12)]
    Can you conjecture a closed-form expression? Can you prove it WITHOUT Polya theory (at least the first one with c=2)
  Added April 27, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Monday., April 28, 2025
  - C26.txt
  Contains procedures (Note, you need to have AGT.txt in the same directory)
  - MultP(pi,sig): the product of the permutation pi and sig
  - GenGp(S): inputs a set of permutations (all of the same length, nops(S[1])) outputs the subset of S_n generated by S
  - CtoP(C): Given a permutation in CYCLE notation (a set of list) converts it to one-line notation
  - IndPer(pi): inputs a permuation pi on the set of VERTICES of K_n outputs the member of S_binomial(n,2) , a certaion permutation on the set of edges INDUCED by pi
  - CIPkn(n,x): the cycle-index polynomial for K_n done stupidly
  - ParToPer(P): Given a partion P of an integer n (a member of Pars1(n,n)) outputs the "canononical permuation" of that type
  - CIPknC(n,x): the cycle-index polynomial for K_n done cleverly
  Homework for April 28, due Sunday, May 4, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#26
  and an attachment
  hw26FirstNameLastName.txt
  and indicate whether it is OK to post.
  1. Briefly describe the progress on your team's project
  2. Let me know whether you are coming to the Annual Princeton Cemetery tour, Sat., May 10, 2025, 6:00pm, followed by a free dinner at a restaurant tbd. We meet Outside the Princeton Public Library. More details later.
  3. A graph may be viewed as a way of coloring the edges of the complete graph with two colors. By Polya's theorem, to obtain the number of ways of coloring unlabeled objects with c colors is to plug-in x[i]=c into the cycle index polynomial for all x[i]. Use CIPknC(13,x) to find the exact number of unlabelled graphs on 13 vertices.
  4. According to Polya theory, in order to find the weight-enumerator, according to the weight, t^NumberOfEdges, for the set of unlabeled graphs on n vertices, you replace x[i] by 1+t^i, in CIPknC(n,x) for every x[i] that shows up, and expand. How many unlabled graphs are there with 13 vertices and 30 edges?
  5. What are the OEIS numbers (if they exist) for
    - The sequence enumerating unlabeld graphs with n vertices and n-1 edges?
    - The sequence enumerating unlabeld graphs with n vertices and n edges?
    - [Correcting (May 3, 2025) a typo pointed out by Kaylee Weatherspoon]
      The sequence enumerating unlabeld graphs with n vertices and n+1 edges?
  Added May 4, 2025: Students' homework (that they kindly agreed to be posted) are in this directory
  
  Programs done on Thurs., May 1, 2025
  - C27.txt
  Contains procedures
  - NuULGe(N,r): The first N terms of the sequence enumerating UNLABELED GRAPHS with n vertices and n+r edges
  - LkT(k,N): the first N terms of the number of ORDERED full k-ary trees with (k-1)*n+1 leaves
  - [Added after class]
    LkUT(k,N): the first N terms of the number of UNORDERED full k-ary trees with (k-1)*n+1 leaves
  Optional Homework for May 1, due Sunday, May 4, 2025, 8:00pm
  
  Please email ShaloshBEkhad at gmail dot com an email with
  Subject: HomeWork#27
  and an attachment
  hw27FirstNameLastName.txt
  and indicate whether it is OK to post.
  1. If you are not yet a registered user of the OEIS, sign up, and become a member!
  2. What is the smallest r such that NuULGe(20,r) is NOT in the OEIS? Submit the next two ones.
  3. What is the smallest k such that LkT(k,20) is NOT in the OEIS? Submit the next two ones.
  4. What is the smallest k such that LkUT(k,20) is NOT in the OEIS? Submit the next two ones.
  Added May 4, 2025: Joseph Koutsoutis' homework (that he kindly agreed to be posted) is in this directory
  
  Programs done on Monday, May 5, 2025
  - C28.txt
  Today we had a "hackaton", there are several versions of the procedurs below with the coder's names.
  - Pri(pi,k,j) If prisoner j is allowed to look at k boxes, following the "follow the box" strategy, does he succeed? Currently set to spit out true or false. You can have it return k0 to see how many attempts it actually took. (By Omar Garcia)
  - SUC(pi,k): If there are n prisoners 1...n, and there is a permutation pi of {1,...n} placed in n lockers (unknown to them) and each prisoner is allowed to examine k boxes, and if they all succeed in finding their number they are all set free, but even if one fails, then they all stay in prison. true if they succeed with that permutatiion.
  - EstSUC(n,k,K), estimates the prob. of success with n prisoners and k tries doing it K times
  Optional Homework for May 5, no due date
  
  Please email DoronZeil at gmail dot com an email with
  Subject: HomeWork#28
  and an attachment
  hw28FirstNameLastName.txt
  and indicate whether it is OK to post.
  1. Prove that the probaility of SUC(pi,k) if pi is a random permutation of length n and k is larger than n/2 is
    1- (1/(k+1)+ 1/(k+2)+ ...+1/n)
    In particular for 2n prisoners and n locker-openings it is 1-(1/(n+1)+...+1/(2n)) that converges to 1-log(2)
    [Hint: prove that if k is larger than n/2, the probability of a random permutation having a cycle of length k is 1/k]
  Added May 8, 2025: Here are the amazing final project and here are the final presentations
  
  Here is Group picture at the Mint restaurant after the field trip to two Princeton cemeteries.
  
  Dr. Z.'s home page

An On-Line course in using Maple to study Quantum Computing and Algorithmic Graph Theory

Description

Textbooks

Optional (but Highly recommended) Getting Started Homework

How to submit homework

Programs done on Thurs., Jan. 23, 2025

Homework for Thurs., Jan. 23, 2025, class (due Sunday Jan. 26, 8:00pm)

Programs done on Monday, Jan. 27, 2025

Homework for Monday, Jan. 27, 2025, class (due Sunday Feb. 2, 8:00pm)

Programs done on Thurs., Jan. 30, 2025

Homework for Thurs. Jan. 30, 2025, class (due Sunday Feb. 2, 8:00pm)

Programs done on Monday, Feb. 3, 2025

Homework for Monday Feb. 3, 2025, class (due Sunday Feb. 9, 8:00pm)

Programs done on Thurs., Feb. 7, 2025

Homework for Thurs. Feb. 6, 2025, class (due Sunday Feb. 9, 8:00pm)

Programs done on Monday., Feb. 10, 2025

Homework for Monday Feb. 10, 2025, class (due Sunday Feb. 16, 8:00pm)

Programs done on Thurs., Feb. 13, 2025

Homework for Thurs. Feb. 13, 2025, class (due Sunday Feb. 16, 8:00pm)

Programs done on Monday, Feb. 17, 2025

Reading Homework for Monday Feb. 17, 2025, in preparation for the next class (2/20/25)

Homework for Monday Feb. 17, 2025, class (due Sunday Feb. 23, 8:00pm)

Programs done on Thurs., Feb. 20, 2025

Homework for Thurs. Feb. 20, 2025, class (due Sunday Feb. 26, 8:00pm)

Programs done on Monday, Feb. 24, 2025

Reading Homework for Monday Feb. 24, 2025, class (due Thurs., Feb. 27, 2025)

Homework for Monday Feb. 24, 2025, class (due Sunday March 2, 2025, 8:00pm)

Programs done on Thursday, Feb. 27, 2025

Homework for Thursday Feb. 27, 2025, class (due Sunday March 2, 2025, 8:00pm)

Programs done on Monday, March 3, 2025

Homework for Monday March 3, 2025, class (due Sunday March 9, 2025, 8:00pm)

Programs done on Thurs., March 6, 2025

Homework for Thurs. March 6, 2025, class (due Sunday March 9, 2025, 8:00pm)

Programs done on Monday, March 10, 2025

Special Pi Day Homework due Wed., March 12, 2025, 8:00pm

Regular Homework due Sunday, March 16, 2025, 8:00pm

Programs done on Thurs., March 13, 2025

Homework due Sunday, March 16, 2025, 8:00pm

Programs done on Monday, March 24, 2025

Homework for March 24, due Sunday, March 30, 2025, 8:00pm

Programs done on Thurs., March 27, 2025

Homework for March 27, due Sunday, March 30, 2025, 8:00pm

Programs done on Monday, March 31, 2025

Homework for March 31, due Sunday, April 6, 2025, 8:00pm

Programs done on Thurs., April 3, 2025

Homework for April 3, due Sunday, April 6, 2025, 8:00pm

Programs done on Monday, April 7, 2025

Homework for April 7, due Sunday, April 13, 2025, 8:00pm

Programs done on Thurs., April 10, 2025

Homework for April 10, due Sunday, April 13, 2025, 8:00pm

Programs done on Monday, April 14, 2025

Homework for April 14, due Sunday, April 20, 2025, 8:00pm

Programs done on Thurs., April 17, 2025

Homework for April 17, due Sunday, April 20, 2025, 8:00pm

Programs done on Mon., April 21, 2025

Homework for April 21, due Sunday, April 27, 2025, 8:00pm

Programs done on Thurs., April 24, 2025

Homework for April 24, due Sunday, April 27, 2025, 8:00pm

Homework for April 24, due Sunday, April 27, 2025, 8:00pm

Programs done on Monday., April 28, 2025

Homework for April 28, due Sunday, May 4, 2025, 8:00pm

Programs done on Thurs., May 1, 2025

Optional Homework for May 1, due Sunday, May 4, 2025, 8:00pm

Programs done on Monday, May 5, 2025

Optional Homework for May 5, no due date