Math 640: EXPERIMENTAL MATHEMATICS Spring 2018 (Rutgers University) Webpage

http://sites.math.rutgers.edu/~zeilberg/math640_18.html

Last Update: April 30, 2018.


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 about the Riemann Zeta function, following the terse, but crystal-clear, notes of Professor Henryk Iwaniec, but we will really understand it, since we will program everything, and the only way to really understand something is to program it.

There is no overlap with previous years.

The prerequisites are the standard courses in real and complex analysis.

In particular, no prior knowledge of Maple, or any programming experience, is assumed. Also, no overlap with previous years.


Added Feb. 4, 2018: Read Don Zagier's nice exposition of Donald Newman's short proof of the Prime Number Theory and Normal Levinson's very lucid Elementary Proof of the Prime Number Theorem .


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Note: a few commands are no longer valid in today's Maple. The most important one is that " has been replaced by %.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


Diary and Homework assignments

Programs done on Thurs., Jan. 18, 2018

C1.txt , Contains procedures

Homework for Thurs., Jan. 18, 2018, class (due Sunday Jan. 21, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#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 exempt 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, edited Jan. 21, 2018]. Which of the following sequences

    Conv(1,1), Conv(Conv(1,1),1), Conv(Conv(1,1),Conv(1,1))

    are already in the On-Line-Encyclopedia of Integer sequences (give their serial number), and which are not?

    Hints: For the first 20 terms, Conv(1,1), type, in Maple (after you uploaded C1.txt ),

    seq(Conv(i->1,i->1,n),n=1..20);

    For the first 20 terms, of Conv(Conv(1,1),1), type, in Maple

    seq( Conv(i->1, i->Conv(i->1,i->1,i),n),n=1..20);

    For the first 20 terms, of Conv(Conv(1,1),Conv(1,1)), type, in Maple

    seq( Conv( i->Conv(i->1,i->1,i) , i->Conv(i->1,i->1,i),n),n=1..20);

    Then copy-and-paste these sequences to the window of the OEIS, and see what happens.

  3. [Mandatory for experts, optional for novices] Write another version of divisors(n), call it DivSmart(n), that first factorizes n, using the command ifactor (or otherwise) , and uses the fact that the set of divisors of

    p1^a1 *....* pk^ak is

    the "Cartesian product" of {1,p1,...,p1^a1} x ... x {1,pk,...,pk^ak}

    [Hint: you may use a recursive procedure, find the largest prime divisible by n, then the partest power, for which it appears, let's call it p^a, then apply it recursively to n/p^a and then take the "Cartesian product" of the output with {1, ..., p^a}]

    Added Jan. 20, 2018: Note that the "Cartesian product" is in quotes, I meant readly the "set product", where AxB={ab | a in A, b in B}

  4. [Optional big challenge, I don't know the answer] Using Maple (or human ingenuity) , conjecture the answer to Problem 12014 in Jan. 2018 Monthly problems.

    [Added Jan. 20, 2018: The convergence is so slow, that it is impossible to conjecture an answer. It looks like you need to do it the human way. But once you derived the answer, you can check it with Maple, by taking n very large].
    [Added Jan. 22, 2018: Edna Jones solved it completely (she got 3 dollars), and Matt Charlney and Yukun Yao solved it partially (they each got one dollar).]

  5. [Optional Challenge, 5 dollars to be divided among solvers for each of the problems,] Use Maple to prove as many problems in the Jan. 2018 Monthly problems.

Added Jan. 22, 2018: Read the nice solutions   .


Programs done on Monday Jan. 22, 2018

C2.txt , Contains procedures

Homework for Monday, Jan. 22, 2018, class (due Sunday Jan. 28, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#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 exempt from this]

    Read and do all the examples, plus make up similar ones, in pages 30-60 of Frank Garvan's awesome Maple booklet ( part 1, part 2)
    Only record a small sample in hw2FirstNameLastName.txt .

  2. [Mandaotry for experts, optional but recommended to novices; Typo corrected Jan. 23, 2018 (thanks to Edna Jones)] Write procedure phi2(n) using the formula n*prod(1-1/p, p|n) and procedure phi3(n) using the formula n*sum(mu(d)/d,d|n). Verify that they are both the same as phi(n) for n from 1 to 1000

  3. [Mandaotry for experts, optional but recommended to novices. Modified Jan. 23 (thanks to Edna Jones)] Let f(s):= abs(Zeta(1/2+s*I)), Find, the smallest zero (on the critical line), to ten decimal digits

    Hint: One way is to start with a resolution, say, 0.1, and look at seq(f(14+0.1*n),n=1..100)
    and find the place where it is smallest, then make the resolution smaller, say, 0.001, and explore in the neighborhood of the previous low place. Etc.

  4. [Mandaotry for experts, optional but recommended to novices] Program Equation (1.19) on p. 6 of Iwaniec' book, i.e. write a procedure

    Lkn(k,n) that inputs k and n and outputs Λk(n)

  5. [Mandaotry for experts, optional but recommended to novices] Verify empirically (1.23) for m,n < 100
    Added Jan. 24, 2018: Edna Jones noticed that the above statement is not valid for all m,n, but only if they satisfy a certain condition between them. What is the condition?

  6. [Mandaotry for experts, optional but recommended to novices] Verify empirically (1.24) approximately for s=11, by truncating the infinite sums at 1000, and using evalf.

  7. [For every one] Do a first reading of Chapter 2 of H. Iwaniec's great book

Added Jan. 29, 2018: Read the nice solutions   .

Stuff done on Thurs. Jan. 25, 2018

Today we were fortunate to have a guest-lecture by guru Neil Sloane

Homework for Thurs., Jan. 25, 2018, class (due Sunday Jan. 28, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#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 exempt from this]
    Read and do all the examples, plus make up similar ones, in pages 61-90 of Frank Garvan's awesome Maple booklet ( part 1, part 2)
    Only record a small sample in hw3FirstNameLastName.txt .

  2. [For everyone] Verify (1.1) for n all less than 1000 of the inequality (1.1) in Jeff Lagarias' nice paper, make sure Digits is high enough, say 50.

  3. [Mandaotry for experts, optional but recommended for novices] Write a Maple program (without peeking in the OEIS entry) that inputs a positive integer n and outputs the first N terms of sequence A64413

  4. [Optional Challenge, no deadline] To do more with coordination sequences (and with no deadline), see Dr. Sloane's notes on coordination sequences. Please contact Dr. Sloane directly with any progress.

Added Jan. 29, 2018: Read the nice solutions   .


Programs done on Monday Jan. 29, 2018

C4.txt , Contains procedures

Homework for Jan. 29, 2018 (due Sunday Feb. 4, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#4

and an attachment

hw4FirstNameLastName.txt

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#YourName, Math 640 (Spring 2018) homework assignment 4

  1. [People who took this class before, or already know Maple, are exempt from this]

    Read and do all the examples, plus make up similar ones, in pages 90-end of Frank Garvan's awesome Maple booklet ( part 1, part 2)

  2. [For everyone] Go over Chapter 2, and convince yourself that every step is not hard.

  3. [For everyone] Find a typo in the second line above Eq. (2.10) on p. 10 of the book

  4. [Mandatory for experts, optional but recommended for novices] Write a program

    CheckI212(g,x,a,b)   ,

    that inputs an expression g and a variable x, such that g is a function of x, and real numbers a and b (such that a < b) and finds the ratio of the left side to the right side. Check that it is indeed ≤ 1 for

    g=x^3, a=1, b=5  ;     g=exp(x), a=2, b=3   ;   g=1/x, a=1, b=4   .

  5. [Mandatory for experts, optional but recommended for novices.] Write a program

    CheckI213(h,g,x,a,b)   ,

    that inputs expressions h and g and a variable x, such that h and g are functions of x, and real numbers a and b (a < b) and finds the ratio of the left side to the right side. Check that it is indeed ≤ 1 for

    h=x^2, g=x^3, a=1, b=5   ;   h=x^3, g=exp(x), a=2, b=3  ;   h=1/x^2, g=1/x, a=1, b=4   .

  6. [Mandatory for experts, optional but recommended for novices. Corrected Feb. 3, 2018 thanks to Yukun Yao and Matthew Charnely] Write a program

    CheckI218(h,g,x,a,b)   ,

    that inputs expressions h and g and a variable x, such that h and g are functions of x, and real numbers a and b (a < b) and finds the ratio of the left side minus the first term of the right hand side divided by H/(1-θ). You must first first find θ given by equation (2.15) Check that it is indeed ≤ 1 for

    h=x, g=C*x^3, a=1, b=5   ;   h=x^4, g=C*exp(x), a=2, b=3   ;   h=x^3, g=C*1/x, a=1, b=4   ,
    where C is a constant (that you should find) that would make the hypothesis ,(2.15), correct.

  7. Read and understand Chapter 3.


Added Feb. 5, 2018: Read the nice solutions   .


Programs done on Feb. 1, 2018

C5.txt ,

Homework for Feb. 1, 2018 (due Sunday Feb. 4, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#5

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hw5FirstNameLastName.txt

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#YourName, Math 640 (Spring 2018) homework assignment 5

  1. Use Maple to verify (both parts) of Problem 12022 in Feb. 2018 Monthly problems. for n ≤ 100 (and ALL x except x=1)

    [Challenge (5 dollars to be divided): prove it for all n]

  2. Use Maple to verify Problem 12023 in Feb. 2018 Monthly problems. for many randon α s.

  3. [Challenge (5 dollars to be divided): prove it for all positive real α]

  4. Write procedures CheckI36(x), CheckI37(x), CheckI38(x), to check Equations (3.6), (3.7), (3.8) respectively in H. Iwaniec's great book .

  5. Write a procedure M(x) using (3.9) and write a procedure CheckI311(x) that checks empirically Eq. (3.11)

  6. Read pp. 15-16 in H. Iwaniec's great book .

  7. [Optional challenges, open until solved] Prove as many problems as you can from the Feb. 2018 Monthly problems.

Added Feb. 5, 2018: Read the nice solutions   .


Programs done on Feb. 5, 2018

C6.txt ,

Homework for Feb. 5, 2018 (due Sunday Feb. 11, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#6

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hw6FirstNameLastName.txt

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#YourName, Math 640 (Spring 2018) homework assignment 6

  1. [typo corrected, Feb. 9, 2018, thanks to Edna Jones] Get procedure CheckZ5(N) to work by first creating a list of floating-point numbers for (θ(x)-x)/x^2 from 2 to N and then adding them, i.e. replacing integration by summation

  2. Read, and completely understand the proof in Don Zagier's nice exposition of Donald Newman's short proof

  3. Write a procedure

    gT(f,t,z,T)

    tha inputs an expression f in a variable t, a complex number z, and a real number T, and outputs the integral of f(t)*exp(-z*t) for t rom 0 to T. Experiment with several f and see for which ones the limit of gT(0), at T goes to infinity, seems to converge, and whether the limit seems to go to g(0).

  4. [Optional Challenge] Empiriclly (or better still, rigorously) conjecture (or determine) answers to the limits in problem 12026 in Feb. 2018 Monthly problems.

    [Note added Feb. 8, 2018: the convergence is extremely slow, but Professor Kauers figured out the answer to the first limit, using ingenuity and symbolic computation. Neither of us knows the answer to the second limit. Update Feb. 9, 2018: Prof. Kauers also figured out the answer to the second limit, suprisingly, the first limit is very simple, and the second limit is rather complicated. Can you find them?]

Added Feb. 12, 2018: Read the nice solutions   .

Programs done on Feb. 8, 2018

Today we were fortunate to have as guest lecturer, Professor Manuel Kauers, here are his great notes, and here the Mathematica source code.

Homework for Feb. 8, 2018 (due Sunday Feb. 11, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#7

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hw7FirstNameLastName.txt

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#YourName, Math 640 (Spring 2018) homework assignment 7

  1. Inspired by Professor Kauers' guest lecture, but ising Maple, conjecture a linear recurrence operator annihilating the sequece
    sum(binomial(2*n,n+k)*binomial(n,k)^2,k=0..n)
    (where N is the shift operator in n (for example the fiboancci numbers are annihilated by N^2-N-1 and n! by N-(n+1)). You may use procedure Findrec in Dr. Z.'s Maple package FindRec.txt
    (Type ezra(Findrecc) for instructions how to use it).

  2. Having found the operator, try to use Dr. Z.'s Maple package AsyRec.txt to figure out the asympotic behavior of the above sequence (Use procedure Asy, or better still, AsyC).

Added Feb. 12, 2018: Read the nice solutions   .


Programs done on Feb. 12, 2018

C8.txt ,

Special Homework for Feb. 12, 2018 (due Wed., Feb. 14, 2018, 6:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#8special

and an attachment

hw8specialFirstNameLastName.jpeg , or hw8specialFirstNameLastName.png , or hw8specialFirstNameLastName.gif

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Added Feb. 14, 2018: Admire the beautiful valentines designed by the students, using Maple.

Usual Homework for Feb. 12, 2018 (due Sunday Feb. 18, 2018, 10:00pm)

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Subject: hw#8

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#YourName, Math 640 (Spring 2018) homework assignment 8

  1. Using the plots package, write a procedure,Contour(R, delta) that inputs a (large) positive real number R, and a small positive integer delta, and draws the the curve C in the proof of the analytical theorem in Don Zagier's paper (p. 707)

  2. Write a procedure, Check5D(s,N), that inputs a complex number s and a positive integer N less than 177, and checks, numerically Fact D in H. Iwaniec's book (p. 21) where the product is taken over the first N roots, by taking the ratio of the left side to the right side, and hopefully getting something close to 1.

    What are: Check5D(3+I,10), Check5D(13+I,100) ?

  3. Write a procedure, Check5E(x,N), that inputs a large real number, x and a positive integer N less than 177, and checks, numerically, fact E, where the sum is over the first N roots of Zeta(s). (See (10.1), p. 37 for the meaning of the "sigma flat"), by taking the difference between the left side and the right side, and hopefully getting something close to 0

  4. Read and understand Chapter 6 (pp. 23-24)

Added Feb. 19, 2018: Read the nice solutions   .


Programs done on Feb. 15, 2018

C9.txt ,

Homework for Feb. 15, 2018 (due Sunday Feb. 18, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#9

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hw9FirstNameLastName.txt

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#YourName, Math 640 (Spring 2018) homework assignment 9

  1. [Modified Feb. 18, 2018, thanks to Matthew Charnley, who won a dollar.]
    Write a procedure

    FindN(f,x,k,x0)

    that inputs an explicit function f in the variable x, (for example) that is obviously continuos on the open interval (0,1), and a positive integer k, as well as a number x0 strictly between 0 and 1 (i.e. in the open interval (0,1)) and outputs the smallest N such that abs(FSn(f,x,N,x0)-subs(x=x0,f)) is less than 1/10k, in other words what N should you take to guarantee k decimal digits. What are (take Digits:=30)

    FindN(1/(1+x)^3,x,3,0.5) , FindN(sin(cos(x)),x,3,0.8) , FindN(abs(x-1/2),x,3,0.6), FindN(abs(x-1/2),x,3,0.51), , FindN(abs(x-1/2),x,3,0.501)

  2. Read and understand these two pages, proving the identity that is crucial for the proof of the functional equation for Zeta(s)
    (Note the last paragraph of the proof is in the next page, that is not scanned, but it is the end of the standard proof that int(exp(-x^2),x=-infinity..infinity) is sqrt(2*Pi) that you should know from Calc3. (If I=int(exp(-x^2),x=-infinity..infinity) then I^2=int(exp(-x^2),x=-infinity..infinity)*int(exp(-y^2),x=-infinity..infinity)= int(int(exp(-x^2-y^2),x=-infinity..infinity),y=-infinity..infinity), now convert to polar coordinates)

  3. Read and understand Chapter 7 in H. Iwaniec's book

  4. Write a procedure

    Check82(f,R)

    that checks Lemma 8.2 (p. 28) In H.Iwaniec's book for a holomorphic function f(z) and a positive real number R. Try it out for

    Check82(exp(z)*mul((z-i),i=1..5),R);

    for R=1,R=2,R=3,R=4,R=5, and R=7

  5. Do a first reading of Chapter 8

Added Feb. 19, 2018: Read the nice solutions   .


Programs done on Feb. 19, 2018

C10.txt ,

Homework for Feb. 19, 2018 (due Sunday Feb. 25, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#10

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#YourName, Math 640 (Spring 2018) homework assignment 10

  1. Read and understand Euler's proof that Zeta(2)=pi^2/6 as nicely told in Brendan Sullivan's paper Numerous Proofs that ζ(2)=π2/2. Try to read and understand the proof for Zeta(2n). Also read and understand Apostol's proof for Zeta(2).

  2. Write a procedure Check127(t,x,N) that inputs a real number t and a real number x (not far from sqrt(sqrt(1/4+t^2)/(2*pi)) and checks Eq. 12.7, where the infinite sums are replaced by sums from n=1 to n=N. Test the procedure with some reasonable inputs.

  3. Recall the Newton-Raphson method (that also works for complex functions) that approximately finds a root of an equation P(z)=0 by starting with an initial guess z0, and replaces it by z0-P(z0)/P'(z0). Write a procedure

    NR(f,z,z0,eps)

    that inputs an expression (representing a function) f in the variable z, a complex number z0, and a small positive real number eps, and outputs an approximate root obtained by iterating the above until the difference in absolute value of two consecutive iterations is less than eps.

    Test it with several random polynomials and initial values with eps=1/108.

  4. By using NR(Zeta(s),s, 1/2+I*n, 1/10^7), for integer n (starting with n=10) try to find, from scratch as many zeros of Zeta(s) that you can. Compare it with the Odlyzko table.
    Comment added Feb. 28, 2018: Edna Jones made the interesting comment that the reason NR does not converge to the nearest zero is because (even for a cubic!) the beahvior is "chaotic" and the "basins of attraction" are fractals. See wikipedia article .

Added Feb. 26, 2018: Read the nice solutions   .


Programs done on Feb. 22, 2018

C11.txt ,

Contains

Homework for Feb. 22, 2018 (due Sunday Feb. 25, 2018, 10:00pm)

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Subject: hw#11

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#YourName, Math 640 (Spring 2018) homework assignment 11

  1. Read and understand Dr. Z's gorgeous combinatorial proof of Newton's identitities and find the typo(s) [there is at least one], and correct it (them).

  2. Write a Maple procdeure T(A,j,l) that implements the bijection, make sure that it is an involution. Also write a program Wt(A,j,l,x) that outputs the weight.

  3. Use Procedure PfE() to write a procedure ZetaEven(n) that inputs a positive integer and outputs Zeta(2n)

  4. Try to see whether PfE([seq(1/(2*i)!,i=1..n)]), PfE([seq(1/(3*i+1)!,i=1..n)]) etc produce interesting sequences of rational numbers.

  5. Write a procedure LogDerD(L,n) that inputs the first nops(L) coefficients of a Dirichlet polynomial A(s), and outputs -A'(s)/A(s) (Eq. (13.8) in Iwaniec's book.)

  6. Write a procedure Check1312(L,T) that checks Theorem 13.1 (Eq. 13.12) for a Dirichlet polynomial, of degree nops(L), given by the list L. Try it out for several random L and (large) T.

  7. Write a procedure Check1318(T) that checks Eq. (13.18) (p. 50) of H. Iwaniec's book, by bounding the "implied constant". How big is it?

  8. Write a procedure Check1320(T) that checks Eq. (13.20) (p. 51) of H. Iwaniec's book, by bounding the "implied constant". How big is it?

  9. [Optional challenge, I have no clue, 5 dollars to be divided] Explain the inequality sign going from line 4 of p. 50 to line 5 of p.50 in Professor Henryk Inwaniec's book "Lectures on the Riemann Zeta Function" .
    Added Feb. 26, 2018: Professor Iwaniec confirmed that it was an error, see the corrected page . [Thanks to Edna Jones]

Added Feb. 26, 2018: Read the nice solutions   .


Programs done on Feb. 26, 2018

C12.txt ,

Contains

Homework for Feb. 26, 2018 (due Sunday March 4, 2018, 10:00pm)

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Subject: hw#12

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hw12FirstNameLastName.txt

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#YourName, Math 640 (Spring 2018) homework assignment 12

  1. Write procedures TrapInt(f,x,a,b,N) and SimpsInt(f,x,a,b,N) for numerical integration

  2. [Rephrased Feb. 28, 2018 (thanks to Edna Jones)]
    Write procedures a(k,eps) that approximates a(k) by truncating the infinite product and sums (stopping them, at say the N-th prime and then the 2N-th prime and making sure that their difference is less than eps) For the definition of a(k) see Chris Houghes' nice slides on the moments of the Riemann Zeta function and Random Matrix Theory,

  3. Using SimpsInt(f,x,a,b,N) for sufficiently large N, and the above a(k,eps), verify empirically the The Keating-Snaith conjecture given on p.13 of Chris Houges' slides, by writing procedure

    KS(T,k)

    that computes the ratio of the left side to the right side.
    [Note: G(k) (k ≥ 2) is nothing but "iterated-factorial": 0!*1!*...*(k-2)! and G(1)=1, see wikipedia, thanks to Edna Jones for correcting a previous error]

    What are (approximations of) KS(100,3), KS(1000,3); KS(100,3), KS(100,4) ?

  4. [corrected March 4, 2018, thanks to Edna Jones]
    Write a procedure

    VerifyHoughesP18(T,k) that computes the ratio of the left side to the right side in the conjecture on top of p.18 of Chris Houges' slides. What is the values of VerifyHoughesP18(10000,2)?

  5. [Rephrased Feb. 28, 2018 (thanks to Edna Jones)]
    Write a procedure

    VerifyGonekConj(T)   ,

    that computes the ratio of the left side to the right side in Gonek's conjecture given on the bottom of p.18 of Chris Houges' slides. are the values of (keep trying until it takes too long) VerifyGonek(T) for

    T=100, T=1000, T=10000, T=100000, etc.


Added March 5 2018: Read the nice solutions   .


Programs done on March 1, 2018

C13.txt ,

Contains

Homework for March 1, 2018 (due Sunday March 4, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#13

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hw13FirstNameLastName.txt

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#YourName, Math 640 (Spring 2018) homework assignment 13

  1. Read and understand the bottom of p. 58 and top of p. 59. Explain why he set S contains a sequence {t1, ..., tR} of Δ-spaced points with R ≥ T/(2Δ)

  2. Read and try to understand the proof of Lemma 15.2, by explaining every line. (I know it is tough going!)

  3. Write a program

    CheckP59L23()   ,

    that checks Line 23, p. 59 of Iwaniec's book by estimating (i.e. finding an upper bound) , for large T and t, and any u, the implied constant in the O((u^2+1)/T)

  4. Do a first reading of Chapters 16 and 17 of H. Iwaniec's book .

Added March 5 2018: Read the nice solutions   .


Programs done on March 5, 2018

C14.txt ,

Contains

Special Homework for March 5, 2018 (due March 7, 2018, 10:00pm, In Early celebration of Pi Day)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#14Pi

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hw14PiFirstNameLastName.txt

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#YourName, Math 640 (Spring 2018) homework assignment 14Pi

  1. Using Maple, simulate a program, Buffon(K), that approximates Pi by throwing a Buffon needle K times. What did you get for Buffon(1000)? Buffon(10000)?

    Hint: in order to generate a random location for the (x-coordinate of the) bottom of the needle use
    evalf(RandomTools[Generate](positive)):
    and in order to generate a random orientation (i.e. the angle it makes with the positive x-axis) for the needle, use
    evalf(RandomTools[Generate](positive)*2*Pi):
    now test whether the x-coordinate of the top of the needle is either negative or larger than 1. Now repeat this many times and find the ratio of successes, let's call it r, that should be an approximation for 2/Pi, so to estimate Pi, à la Buffon, compute 2/r.

  2. Using the fact that binomial(2n,n)/2^(2*n) is approximately (for large n) , 1/(sqrt(n*Pi), write a Maple program

    CoinTossing(n,K)

    where each round consists of throwing a fair coin 2*n times, and the round is considered a susccess if there were exactly n Heads. Let S be the number of successful rounds. Use the fact that for large n and K, S/K is approximately 1/sqrt(n*Pi), to estimate Pi. What did you get for CoinTossing(100,10000)? Do it several times.

  3. [Contest, prize is a Pi-shaped chocolate piece] Write a simulation program that is even slower than the above, i.e. find something whose probability can be expressed in terms of Pi, simlate it, using Maple's rand(), and use it to approximate Pi, as slowly and as poorly as possible, BUT, if you had all of the time in the world, and believe in the law of large numbers, you can get it to any desired accuracy.

Added March 8 2018: Read the nice solutions   . The co-winners of the prize (a Pi-shaped chocolate piece) are Yukun Yao and George Hauser.

Regular Homework for March 5, 2018 (due Sunday March 18, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#14

and an attachment

hw14FirstNameLastName.txt

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#YourName, Math 640 (Spring 2018) homework assignment 14

[Rephrased March 7, 2018 (thanks to Edna Jones)]

  1. Write a procedure

    Q(x,T,gL)

    that inputs a real numbre x a (usually) large positive real number T and a finite list of real numberss gL and outputs the expression, Q(x), defined in Eq. (18.12), in Professor Iwaniec's great book , where g_1=gL[1], g_3=gL[2], ...,

  2. Write procedure del(s,T) defined by (18.11) and G(s,T,gL) defined by 18.10

  3. Read Chapter 19.

  4. Use Maple to plot a figure similar to Fig. 19.1 on p. 71.

  5. Browse Atle Selberg's seminal paper on the zeros of Zeta(s) on the Critical Line.

Added March 19 2018: Read the students' nice solutions   .


Programs done on March 8, 2018

C15.txt ,

Contains

Homework for March 8, 2018 (due Sunday March 18, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#15

and an attachment

hw15FirstNameLastName.txt

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#YourName, Math 640 (Spring 2018) homework assignment 15

    Read and understand the proof that if c/d is rational then tan(c/d) is irrational, and hence Pi/4 (and hence Pi) are irrational, as proved here .

  1. Read and understand Oswald Veblen's beautiful proof that Pi and e are transcendental

  2. Use the two formulas in Jesus Guillera's great website to write programs JG1(N), JG2(N) that input a positive integer N and compute Pi to N decimal digits.

  3. Use procedure AZd in the Maple package EKHAD.txt to compute

    int( x^n*(1-x)^n/(1+x^2),x=0..1):

    for n from 2000 to 2010.

    Hint: AZd(x^n*(1-x)^n/(1+x^2),x,n,N)[1] will give you a linear recurrence operator that implies a second-order recurrence. Program it with "option remember" and get the 2000-th through 2010-th terms.

  4. Look up Machin-type formulas and implement them to compute Pi.

Added March 19 2018: Read the students' nice solutions   .


Programs done on March 19, 2018

C16.txt ,

Contains

Homework for March 19, 2018 (due Sunday March 25, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#16

and an attachment

hw16FirstNameLastName.txt

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#YourName, Math 640 (Spring 2018) homework assignment 16

  1. Wrie a procedure

    CIpara(f,z,P,t,alpha,beta): that inputs a function f of a complex variable z, a pair P=[P1,P2] where P1 and P2 are functions of the real variable t, and real numbers alpha,beta and outputs the contour integral

    Int(f(z) dz, z in C from start to finish),

    where C is the parametric curve x=P1(t),y=P2(t) from t=alpha to t=beta (Hint: dz=dx+I*dy, dx=P1'(t)dt, dy=P2'(t))

    Make sure that CIpara(f,z,[c0+r*cos(t),c1+r*sin(t)] ,t,0,2*Pi) gives the same output as CIc(f,z,c0+I*c1,r) for various functions f of z

  2. By writing a short procedure LineSegment(A,B,t) that outputs the parametric representation of a line segment joining point A=[A1,A2] and point B=[B1,B2] from t=0 to t=1, write a procedure

    CIpolygon(f,z,Polygon): that inputs a function f of a complex variable z, and a list of vertices denoting in order, the vertices of a polygon, and outputs

    Int(f(z) dz, z in C from start to finish), where C is the polygon. Make sure that

    CIpolygon(f,z,[[C0,C1], [C0+a,C1],[C0+a,C1+b],[C0,C1+b]] )

    outputs the same as CIrN(f,z,[C0,C1], a,b) for various functions f of z.

  3. Write a procedure

    IntCauchy(L,x)

    that inputs a list L of positive numbers L and computes

    int(1/mul(x^2+L[i]^2,i=1..nops(L)),x=-infinity..infinity) using Cauchy's residue theorem.

    [Hint: Maple has a command called residue. The integral is the same as the contour integral over the contour consisting of the real axis and the upper infinite semicircle. The answer is 2*Pi*I times the sum of the residues of the poloes of the above function in the upper half-plane. Make sure that you get the same answer for various random lists L.

  4. Read and understand Chapter 19 of Professor Iwaniec's great book ,

  5. Conjecture the answer to March 2018 Monthly Problem 12029, and if possible prove your conjecture.

  6. Check numerically March 2018 Monthly Problem 12031, (both parts) and if possible, prove them.

  7. Conjecture the answer to March 2018 Monthly Problem 12032, and if possible, prove it, and it would be really nice if you can use procedure MultiZeil in this amazing Maple package
    [Hint, in order to use MultiZeil, you need to divide the summand by 2^n, and calling the double-sum of the problem A(n), get a recurrence for A(n)/2^n, that easily translates to a recurrence for A(n). Note that N is the shift operator in n, so, for example, the recurrence for Fibonacci is abbreviated N^2-N-1. Once you get a recurrence, you can either do it by hand (using Discrete Math 101), or use Maple's command rsolve].

Added March 26 2018: Read the students' nice solutions   .


Programs done on March 22, 2018

C17.txt ,

Contains

Homework for March 22, 2018 (due Sunday March 25, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#17

and an attachment

hw17FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 17

  1. Write a program

    Check211(S,C0,a,b)

    that checks the Littlewood lemma (Eq. 21.1, p. 77), where S is a finite set of complex numbers, C0 is a point denoting the bottom-left corner of a rectangle whose horizonal side has length a and vertical side has length b, and taking F to be the function of z that is the product of (z-alpha) over all the members alpha of S. You may use CIr(f,z,C0,a,b) or CIrN(f,z,C0,a,b) of C16.txt . Test it for some examples where some membes of S lie inside the rectangle, and some don't.

  2. Using the amazing Maple package RENE.txt, prove Problem 12027 of March 2018 Monthly.

    [Hint: Use the triangle Te(m,n), and procedures Inradius(m,n), Circumradius(Te(m,n)), Incircle(m,n) (for the equation of the incircle), Pt to find the intersection of two lines, and Le to find the line joining two points, DistSq (for the square of the distance between two points), and SSR(A,n,B), with n=3, A the list of the squares of the two terms on the left, and B the right side (R/r-1/2).

  3. Test empirically Problem 12033 of March 2018 Monthly, and if possible, prove it.

Added March 26 2018: Read the students' nice solutions   .


Programs done on March 26, 2018

C18.txt

Contains

Homework for March 26, 2018 (due Sunday April 1, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#18

and an attachment

hw18FirstNameLastName.txt

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#YourName, Math 640 (Spring 2018) homework assignment 18

  1. Write a program

    FindPeriod1(L,d)

    that inputs a list L and a positive integer d and finds (if they exist), lists L1,L2, such that L is the list L1 followed by the list L2 repeated d times, and the rest is the start of L2. For example

    FindPeriod1([1,2,1,3,2,1,3,2,1,3,2,1,3], 3);

    should output [1,2], [1,3,2] .

  2. Using FindPeriod1(L,d), Write a program

    FindPeriod(L)

    that finds a list L1 and L2 for which there exists a d>=3 such that FindPeriod1(L,d)=[L1,L2]. For example FindPeriod([1,2,1,3,2,1,3,2,1,3,2,1,3]) should return [1],[2,1,3] [I thank Yukun Yao for correcting this, before I wrote [1,2],[1,3,2] that is not mininal in the prefix]

  3. It is a famous theorem in Number Theory that every quadratic irrationality has an ultimately periodic continued fraction. By using (make DIgits large enough) NuToCF(evalf(x),n) followed by FindPeriod(L), write a procedure

    ConjCF(x)

    that inputs a quadratic irrationality, and outputs lists of positive integers, L1 and L2, such that the infinite (simple) continued fraction of x is L1 followed by L2 repeated infinitely many times.

  4. Write a procedure

    EvalPCF(L1,L2)

    that inputs lists L1 and L2 (L1 may be empty) and finds the quadratic irrationality represented by it. For example

    EvalPCF([1],[2]);

    should output sqrt(2)

    [Hint: First write a procedure for the pure-periodic z=L2infinity , using the fact that z=EvalCF([L2,z]), solve for z, and then do EvalCF([L1,z]).

  5. Using ConjCF(x) and EvalCF(L1,L2), write a procedure

    ConjAndProveCF(x)

    that has the same input and output as ConjCF(x) but now the conjectured infinite (ultimately periodic) (simple) continued fraction is rigorously proved (or returns FAIL, if unable to prove it)

  6. Find the L1 and L2 for all square-roots of positive integers (that are not perfect squares) from 2 to 100 (and if possible even further). Is the sequence of lengths of the periodic part (L2) in the OEIS? What square-root gives the largest period?

Added April 2, 2018: Read the students' nice solutions   .


Programs done on March 29, 2018

C19.txt

Contains

Homework for March 29, 2018 (due Sunday April 1, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#19

and an attachment

hw19FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 19

  1. Write a program

    SofP(n,k,r)

    that inputs a positive integer n and outputs the set of tuples [a1, ..., ak] of non-negative integers, such that a1>=a2>=..>=ak>=0 and

    a1^r+ ...+ak^r=n

    Verify that SofP(n,4,2) is non-empty for all n<=1000, thereby confirming Lagrange's theorem for n <=1000.

    Find the set of primes p less than 1000 for which SofP(p,3,2) is non-empty. Can you characterize it?

  2. In preparation to Henry Smith's proof given a beautiful combinatorial formulation here,write a program

    AllCF(p)

    That inputs a prime p and outputs the set of (simple) continued-fraction representation of the fractions p/a for a from 1 to (p-1)/2. Verify, for all primes less than 1000, that whenver a list shows up, its reverse also shows up.

Added April 2, 2018: Read the students' nice solutions   .


Programs done on April 2, 2018

C20.txt

Contains, in addition to the contents of C18.txt and C19.txt, the following procedures

Homework for April 2, 2018 (due Sunday April 8, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#20

and an attachment

hw20FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 20

  1. Check the corrected version of EvalCFg(A,B) and verify that

    EvalCFg([seq(a[i],i=1..n)],[1$n])

    gives the same thing as

    1/EvalCF([seq(a[i],i=1..n)]);

    for n between 1 and 10. Also prove is in general, by looking at the definition.

  2. Let Qg(A,B) be the denominator of EvalCFg(A,B), find the natural generalization of wt(s,a) from C19.txt, call it wtG(s,a,b), such that Qg(A,B) is the sum of wtG(s,a,b) of all the members of FibS(n)

  3. Let Pg(A,B) be the numerator of EvalCFg(A,B)), express it in terms of Qg(A',B') for some related lists A', B'.

  4. Using the combinatorial interpration, find a second order recurrence satisfied by Qg(A,B)

  5. By using Maple's

    convert(tan(x),confrac,20);

    find the typo in Eq. (15) (p. 522) of Chrystal's book [Or convince me that it is not a typo, but I misunderstood the notation]

  6. [Optional Challenge, 5 dollars to be divided among correct solutions] Define the sequence of polynomials

    Pn:=(n,x)->denom(EvalCFg([seq(2*i-1,i=1..n)],[x,x^2$(n-1)]))

    conjecture a closed form expression for the coefficient of x^k in Pn(n,x), and hence conjecture an explicit expression for Pn(n,x). Then use this to find an expression for Qn(n,x) (the numerator), and finally prove that the limit of Qn(n,x)/Pn(n,x) as n goes to infinity, converges to tanh(x)

  7. Write a program

    Verify(a,b,c,d,e)

    that empirically verifies theorem 1 of Marcin Mazur's paper (Amer. Math. Monthly, March 2018). Assume that the vertices of the tetrahedron are

    A=[0,0,0], B=[1,0,0], C=[a,b,0], D=[c,d,e]

    and hence its volume is b*e/6.

  8. [Optional Challenge, 5 dollars to be divided among correct solutions] Prove this Theorem using Maple.

  9. Using Maple, prove completely Proposition 2 of Marcin Mazur's paper (Amer. Math. Monthly, March 2018).
    [Take A,B,C,D as above, look up the definition of the centroid, and let P be an arbitary point [p1,p2,p3]]

  10. [Optional Challenge] Try to make sense out Attempted proof of RH by Nagy Adel, Mena Asham, and Marian Fady. (received by Dr. Z.'s on April 1, 2018).

Added April 9, 2018: Read the students' nice solutions   .


Programs done on April 5, 2018

C21.txt [courtsey of Edna Jones. I lost the original file]

Contains


Homework for April 5, 2018 (due Sunday April 8, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#21

and an attachment

hw21FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 21

  1. y(x)=tanh(x) happens to satisfy the differential equation y'(x)=y(x)^2-1

    Write a program

    GuessODE(f,x,N,k,y)

    that inputs a function f, or the the start of a Taylor series, in x, a positive integer N, a positive integer k, and a letter y, and outputs a polynomial in y, of degree k, such that f satisfies the ode

    f'(x)=P(f(x)),

    up to the N-th Taylor coefficient (and hopefully for ever). For example

    GuessODE(tanh(x),x,20,2,y) ;

    should return y^2-1 .

  2. Using procedure

    FWBg(n,x,y,p)

    added in C21.txt after class (FWBg(n,x,x^2,j->2*j-1) gives tanh(x)), define the family of polynomials (depending on the integer parameters a and b)

    Pn(x,y) = numer(FWBg(n,x,y,j->a*j+b))   Qn(x,y) = denom(FWBg(n,x,y,j->a*j+b))

    to form the sequence consisting of the first, say, 30, terms of each , and use procedure

    Findrec(Sequence,n,N,C)

    (with C=7) in the Maple package Findrec.txt to find the recurrences satisfied by these for various choices of a, and b. Can you conjecture a form for the recurrence(s) in terms of a and b?

Added April 9, 2018: Read the students' nice solutions   .


Programs done on April 9, 2018

C22.txt

Contains

Homework for April 9, 2018 (due Sunday April 15, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#22

and an attachment

hw22FirstNameLastName.txt

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#YourName, Math 640 (Spring 2018) homework assignment 22

The rest of the homework, for this semester, will be working on the two class projects, where the templates are Truncated Zeta Project and Random Polygon Project. Soon we will divied the labor and get organized, but meanwhile please do all the problems below, that eventualy will be part of the class' papers.

  1. For the truncated Zeta project, finish what we did in class and write a program

    TrunZetaMinima(N,T,err),

    that inputs a positive integer N, a large real number T, and a small error, (positive real number), and outputs the list of all pairs [location, value] where location is a local minimum of ZNtR(N,t), and value is its value there.

    Try to generated as much data as possible.
    [Added April 12, 2018: we did this one in C23.txt, so it is officially optional now (of course, you can make it better)]

  2. For the random polygon project, get a feel for it, by

  3. [Challenge (no peeking in the internet)] Explain why this (limiting) pattern happens.

Added April 16, 2018: Read the students' nice solutions   .


Programs done on April 12, 2018

C23.txt

Contains

Homework for April 12, 2018 (due Sunday April 15, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#23

and an attachment

hw23FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 23

  1. Write a program

    GlobalMinimumZN(N,T)

    that inputs a positive integer N and a large positive integer T, and outputs the location and value of the global minimum of Z(N,t) in 0<=t<=T

    What are GlobalMinimumZN(N,1000) for N from 2 to 10?

  2. Write a program

    GlobalMaximumZN(N,T)

    that inputs a positive integer N and a large positive integer T, and outputs the location and value of the global maximum of Z(N,t) in 0<=t<=T

    What are GlobalMaximumZN(N,1000) for N from 2 to 10?

Added April 16, 2018: Read the students' nice solutions   .


Programs done on April 16, 2018 (virtual class)

C24.txt

Homework for April 16, 2018 (due Wed. April 18, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#24

and an attachment

hw24FirstNameLastName.txt

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#YourName, Math 640 (Spring 2018) homework assignment 24

  1. Finish up EVCm(M) so that it will coincide with EVC(M), in other words, sort the list of eigenvalues into a weakly-decreasing list (by absolute value), and make all eigenvectors unit. Of course the eigenvectors should correspond to the respective eigenvalues.

  2. Write a procedure

    Coeffs(V,M)

    that inputs a vector V (given as a list, of size n, say) and a list of vectors M (of the same size of V), and outputs the list a[1], ..., a[n] such that

    V=a[1]*M[1]+ ...+ a[n]*M[n]

    In other words express V as a linear combination of the vectors M[1], ..., M[n]

    [Hint: you can either use solve, or matrix inversion, using inverse(M) and the procedure MtV(M,V) created today]

  3. Write a procedure, A(n), that inputs a positive integer n, and outputs the n by n matrix (given as a list of lists), corresponding to the "averaging process"

    [x1,x2, ..., xn] ->[(x1+x2)/2, (x2+x3)/2, ...., (xn+x1)/2]

Added April 18, 2018: Read the students' nice solutions   .


Programs done on April 19, 2018 (virtual class)

C25.txt

Contains procedures

Homework for April 19, 2018 (due Sun. April 22, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#25

and an attachment

hw25FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 24

  1. Prove that 1 is the an eignenvalue of A(n) and the all 1 vector is an eigenvector

  2. [Added April 20,2018] plot the eigenvectors corresponding to the second-and-third largest (i.e. the largest ones that complex) eigenvetors of A(n) for n=30. Recall that its components are complex numbers and the complex number x+y*I corresponds to the point [x,y]. What kind of shape emerges? can you conjecture an equation for it?

  3. [Possibly a challenge] Explain why the limiting shape is an elliplse
    [ Added April 21, 2018: previously I claimed that the inclination is 45 degrees, that was wrong. The inclination can be anything. The shape is an ellipse whose center is the origin, of course, but otherwise can be any shape and any orientation]

  4. Write a program that inputs a good polygon and outputs the eccentricity [Added April 21, 2018: and orientation] of the limiting ellipe, by just using the initial points of the polygon (i.e. the vector of x-coordinates and the vector of y-coordiates, both of which add-up to 0).
    [Correction April 21, previously I said "add up to 1", of course that was wrong, by construction of a good polygon the x and y bectors add up to 0]

    Hints: (1) use Coeffs (2) Convince yourself that if A and B are any complex numbers, the curve
    x=Real(A*exp(I*t)), y=Real(B*exp(I*t))
    is an ellipse. By writing A=|A|*exp(I*arg(A)) and B=|B|*exp(I*arg(B)), can you find the eccentricity and orienation of that ellise?

  5. [Challenge, 10 dollars to be divided; Added April 23, 2018: this prize was shared by Edna Jones and Ahsan Khan] Using experimentation, or otherwise, find an explicit expression, in terms of w:=exp(2*Pi*I/n), for the two largest eigenvalues of A(n) (they are complex-conjugates of each other) and if possible for the corresponding eigenvectors (they are also complext conjuates of each other).


Added April 23, 2018: Read the students' nice solutions   .


Programs done on April 23, 2018

Homework for April 23, 2018 (due Sun. April 29, 2018, 10:00pm)

Please email ShaloshBEkhad at gmail dot com an email with

Subject: hw#26

and an attachment

hw26FirstNameLastName.txt

and indicate whether it is OK to post.
#YourName, Math 640 (Spring 2018) homework assignment 24

  1. Find a typo in Bellman's account of George Polya's beautiful elementary proof

  2. [Challenge] figure out if and when the 45 degrees orientation holds, claimed by Van Loan in this chariming article .

  3. [Challenge, I don't know the answer] Try to use Polya's brilliant idea of using an algebraic identity, l-secting it, plugging-in values and taking the limit, may yield other continuous, and who knows?, new, identitites. For example, start with the trinomial theorem (1/z+1+z)^(3*m).

  4. Work as hard as you can on the class projects Truncated Zeta and Iterated polygon that you belong to (but feel free to contribute to the other one, if you will contribute significantly, the leader will add your name). [The leaders (Yukun Yao and Matt Charnley) should copy these pages, and the package to their own website, and at the very end, send them to me for posting]


Programs done on April 26, 2018

C27.txt

Contains, in addition to those of C23.txt, procedurs

Homework for April 26, 2018 (due Tuesday, May 1)

    Have the templates and the very preliminary versions in the respective leaders' websites the following fronts of the two projects Coordinate who will do what. The leaders will assemble all the pieces, and write a first draft of the papers and the Maple packages, that the team members can modify. It would be nice to have something postable by Aug. 31, 2018.


Programs done on April 30, 2018

C28.txt

Today we were getting ready for the field trip to the Princeton Cemetery (May 1, 2018) , by highlighting the work of some of the geniuses that we will visit.

Contains,

Optional Homework for April 30, 2018 (due whenever you feel like it)

[Please send it to DoronZeil@gmail.com rather than ShaloshBEkhad@gmail.com]

  1. Implement, in Maple, the Cooley-Tukey. Compare its speed to the naive way of finding the Discrete Fourier Transform.


Added June 15, 2018: See the amazing paper by Yonah Biers-Ariel, Matthew Charnley, Brooke Logan, Anthony Zaleski and Dr. Z. that came out from one of the two group projects.

Added Jan. 29, 2019: See the amazing Maple package /TruncatedRiemannZeta.txt by Edna Jones and Yukun Yao. For beatiful pictures and data visit web-page of Truncated Riemann Zeta function project.