Monthly Past, Monthly Present, Monthly Future (With ... · Monthly Past, Monthly Present, Monthly...

Monthly Past,Monthly Present,Monthly Future

(With Apologies to Charles Dickens)

Scott Chapman

Sam Houston State University

October 27, 2016

Chapman (Sam Houston State University) October 27, 2016 1 / 47

Prologue

Some resources if you like this talk:

[1] David H. Bailey and Jonathan Borwein. ”Pi Day is upon us again andwe still do not know if Pi is normal.” American Mathematical Monthly121.3 (2014): 191-206.

[2] Jonathan Borwein and Scott Chapman. ”I Prefer Pi: A Brief Historyand Anthology of Articles in the American Mathematical Monthly.” TheAmerican Mathematical Monthly 122.03 (2015): 195-216.

[3] John Ewing (Ed.), A Century of Mathematics Through the Eyes of theMonthly, MAA, Washington, D.C., 1994.

[4] P. Baginski and S. T. Chapman, Factorizations of Algebraic Integers,Block Monoids and Additive Number Theory, Amer. Math. Monthly118(2011), 901–920.

[5] D. Koukoulopoulos and J. Thiel, Arrangements of Stars on theAmerican Flag, Amer. Math. Monthly 119(2012), 443–450.

Prologue

One of my favorite Monthly papers

Greetings from Huntsville!

Our Office Big Sam

Some Basic Facts

Founded: 1896

Published Since 1915 byThe Mathematical Association ofAmerica

The Monthly is the most widelyRead Mathematics Journalin the WorldSource: JSTOR and Ingenta data

Some Basic Facts

Founded: 1896

Some Basic Facts

Founded: 1896

More Specifically

Of the 16,200 journal titles that Ingenta ranks monthly (by numberof downloads), The American Mathematical Monthly constantlyranks in the top 50 (top 1%) and often ranks in the top 20.

The most downloaded article (almost every month): The ProblemSection.

A Monthly Problem led to Bill Gate’s only mathematicalpublication.

Dweighter, H. (pseudonym for Jacob E. Goodman). 1975. ProblemE2569. American Mathematical Monthly 82:1010.

Gates, W.H., and C.H. Papadimitriou. 1979. Bounds for sorting by prefixreversal. Discrete Mathematics 27:47-57

More Specifically

Monthly Past

Benjamin Finkel (1865–1947)Founder of the American Mathematical Monthly

and Editor 1896–1912

Quote From First Issue

“While realizing that the solution of problems is one of thelowest forms of Mathematical research, and that, in general, ithas no scientific value, yet its educational value cannot be overestimated. . . . . . . The American Mathematical Monthly will,therefore, devote a due portion of its space to the solution ofproblems, whether they be the easy problems in Arithmetic, orthe difficult problems in the Calculus, Mechanics, Probability, orModern Higher Mathematics. ”

L. E. Dickson

L. E. Dickson (1874–1954)Editor of the Monthly 1903–1906

Turning Point 1915

In 1915 the American Mathematical Society, by a 3-2 Committee vote,decided not to take control of The Monthly.

It did decide to lend support to any other organization which took over thejournal.

This led to the formation of the Mathematical Association of America,which still publishes The Monthly.

Turning Point 1915

Some Notable Past Monthly Editors

RobertCarmichael(1918)

Herbert Wilf(1987-1991)

Paul Halmos(1982–1986)

John Ewing(1992–1996)

Some Notable Monthly Authors

E. R. Hedrick D. H. Lehmer R. W. HammingL. R. Ford Peter Sarnak Andre WeilI. Kaplansky N. Bourbaki Martin D. KruskalG. Polya William Feller E. T. BellC. L. Siegel C. Fefferman Walter RudinWalter Feit Michael Atiyah Steve SmaleL. E. Dickson George D. Birkhoff Saunders Mac LaneAndrew Gleason Felix Browder George AndrewsPhillip Griffiths Barry Mazur S. S. ChernStephen Wolfram Herbert Wilf David EisenbudJohn Conway Joseph Silverman Branko Grunbaum

Most Cited Articles

Most Cited Articles according to Google Scholar (as of October 25, 2016)

1 Cited 4398 times: College Admissions and the Stability of Marriage,David Gale and Lloyd Shapley, 69(1962), 9-15.

2 Cited 3996 times: Period Three Implies Chaos, Tien-Yien Li andJames A. Yorke, 82(1975), 985-992.

3 Cited 1942 times: Semi-Open Sets and Semi-Continuity inTopological Spaces, Norman Levine, 70(1963), 36-41.

4 Cited 1668 times: The Mathematics of Sentence Structure, JoachimLambek, 65(1958), 154-170.

5 Cited 1612 times: Can One Hear the Shape of a Drum?, Mark Kac,73(1966), 1-23.

Most Cited Articles

More Monthly Past

In honor of Lloyd Shapley’s Nobel Prize, we reprinted in May 2013 his 1962paper with David Gale, which was cited by the Nobel Awards Committee,in its entirety. The paper is preceded by a Foreword by Ehud Kalai,Distinguished Professor of Game Theory at Northwestern University.

Lloyd Shapley delivers his Nobel Prize Lecture

More Monthly Past

In honor of Lloyd Shapley’s Nobel Prize, we reprinted in May 2013 his 1962paper with David Gale, which was cited by the Nobel Awards Committee,in its entirety. The paper is preceded by a Foreword by Ehud Kalai,Distinguished Professor of Game Theory at Northwestern University.

Lloyd Shapley delivers his Nobel Prize Lecture

Monthly Past: The Special Issue

The March 2013 issue was the firstSpecial Issue of the Monthly in over20 years

The issue contains papers writtenby the contributors to theInternational Summer School forStudents held at Jacobs Universityin Bremen, Germany in the summerof 2011.

Monthly Past: The Special Issue

The March 2013 issue was the firstSpecial Issue of the Monthly in over20 years

The issue contains papers writtenby the contributors to theInternational Summer School forStudents held at Jacobs Universityin Bremen, Germany in the summerof 2011.

How Tokieda Sees It

The Special Issue includes the Ford-Halmos Award winning paperRoll Models by Tadashi Tokieda.

Tokieda’s paper contains a wonderful explanation of the mathematicsbehind simple rolling motion. In fact, he goes to great lengths to explainhow a golf ball can completely disappear in the cup and then reemergewithout hitting the bottom of the cup.

How Tokieda Sees It

The Inaugural Pi Day Paper

The Math Biology Special Issue

The November 2014 Special Issuewas dedicated to MathematicalBiology

It contains papers from some of theleading people in the field includingMike Steel, Mark Lewis, DavidTerman, Trachette Jackson, andJeffrey Poet

The Math Biology Special Issue

The November 2014 Special Issuewas dedicated to MathematicalBiology

It contains papers from some of theleading people in the field includingMike Steel, Mark Lewis, DavidTerman, Trachette Jackson, andJeffrey Poet

Some Other Papers of Present and FutureInterest

The Second Pi Paper

Monthly Future!

Here are some titles that will round out 2016.

1 Arcology, by Janos Kollar and Jennifer Johnson.

2 The LEGO Counting Problem, by Søren Eilers.

3 The Range of a Rotor Walk, by Laura Florescu, Lionel Levine, andYuval Peres.

4 Erdos, Klamer, and the 3x + 1 Problem, by Jeff Lagarias.

5 Anonymity in Predicting the Future, by D. Bajpai and Dan Velleman.

Monthly Future!

Back to Monthly Past

The Setting

LetZ = . . .− 3,−2,−1, 0, 1, 2, 3, . . .

represent the integersN = 1, 2, 3, . . .

represent the natural numbers and

N0 = 0, 1, 2, . . .

represent the natural numbers adjoin 0.

The Setting

LetZ = . . .− 3,−2,−1, 0, 1, 2, 3, . . .

N0 = 0, 1, 2, . . .

The Setting

LetZ = . . .− 3,−2,−1, 0, 1, 2, 3, . . .

N0 = 0, 1, 2, . . .

Notation

This part of the talk will be based on the simple congruence relation on Zdefined by

a ≡ b (mod n)

if and only if

n | a− b in Z

Notation

a ≡ b (mod n)

if and only if

n | a− b in Z

Notation

a ≡ b (mod n)

if and only if

n | a− b in Z

Notation

a ≡ b (mod n)

if and only if

n | a− b in Z

Why we love a ≡ b (mod n)

One of our basic arithmetic operations work well here:

If a ≡ b (mod n) and c ∈ Z, then

ca ≡ cb (mod n)

ca ≡ cb (mod n) does not imply that a ≡ b (mod n).

ca ≡ cb (mod n)

The Sequences

Let’s consider a very simple arithmetic sequence:

1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49 =

1 + 4k | k ∈ N0 =

1 + 4N0

is known as The Hilbert Monoid.

The Sequences

1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49 =

1 + 4k | k ∈ N0 =

1 + 4N0

The Sequences

1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49 =

1 + 4k | k ∈ N0 =

1 + 4N0

The Sequences

1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49 =

1 + 4k | k ∈ N0 =

1 + 4N0

The Sequences

1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49 =

1 + 4k | k ∈ N0 =

1 + 4N0

What is a Monoid?

What is a monoid?

A set S with a binary operation ∗ (like + or × on the real numbers) whichsatisfies the following.

1 ∗ is closed on S .

2 ∗ is associative on S ((a ∗ b) ∗ c = a ∗ (b ∗ c))

3 ∗ has an identity element e (a ∗ e = e ∗ a)

Examples: Z, R or Q under +.

What is a Monoid?

What is a monoid?

What is a Monoid?

What is a monoid?

What is a Monoid?

What is a monoid?

What is a Monoid?

What is a monoid?

What is a Monoid?

What is a monoid?

Motivation

David Hilbert(1912)

“Legend” has it that Hilbert used thisset when he taught courses inElementary Number Theory toconvince students of the necessity ofproving the unique factorizationproperty of the integers.

Motivation

David Hilbert(1912)

“Legend” has it that Hilbert used thisset when he taught courses inElementary Number Theory toconvince students of the necessity ofproving the unique factorizationproperty of the integers.

Hilbert’s Argument

In 1 + 4N0 we have

21 · 33 = 9 · 77

(3 · 7) · (3 · 11) = (3 · 3) · (7 · 11)

and clearly 9, 21, 33 and 77 cannot be factored in 1 + 4N0. But notice that9, 21, 33 and 77 are not prime in the usual sense of the definition in Z.

In 1 + 4N0 we have

21 · 33 = 9 · 77

(3 · 7) · (3 · 11) = (3 · 3) · (7 · 11)

In 1 + 4N0 we have

21 · 33 = 9 · 77

(3 · 7) · (3 · 11) = (3 · 3) · (7 · 11)

In 1 + 4N0 we have

21 · 33 = 9 · 77

(3 · 7) · (3 · 11) = (3 · 3) · (7 · 11)

An Example in Almost Every Basic AbstractAlgebra Textbook

Let D = Z[√−5] = a + b

√−5 | a, b ∈ Z.

In D6 = 2 · 3 = (1 +

√−5)(1−

√−5)

represents a nonunique factorization into products of irreducibles in D.

To fully understand this, a student must understand units and norms in D.

Let D = Z[√−5] = a + b

√−5 | a, b ∈ Z.

In D6 = 2 · 3 = (1 +

√−5)(1−

√−5)

Let D = Z[√−5] = a + b

√−5 | a, b ∈ Z.

In D6 = 2 · 3 = (1 +

√−5)(1−

√−5)

Basic Notation

Let M be a monoid. Call x ∈ M

(1) prime if whenever x | yz for x , y , and z in M, then either x | y orx | z .

(2) irreducible (or an atom) if whenever x = yz for x , y , and z in M,then either y ∈ M× or z ∈ M×.

As usual,x prime in M =⇒ x irreducible in M

but not conversely.

Basic Notation

but not conversely.

Basic Notation

but not conversely.

Basic Notation

but not conversely.

Basic Notation

but not conversely.

Basic Notation

but not conversely.

How Things Factor in 1 + 4N0

The element x is irreducible in 1 + 4N0 if and only if x is either

1 p where p is a prime and p ≡ 1 (mod 4), or

2 p1p2 where p1 and p2 are primes congruent to 3 modulo 4.

Moreover, x is prime if and only if it is of type 1.

Corollary

Let x ∈ 1 + 4N0. Ifx = α1 · · ·αs = β1 · · ·βt

for αi and βj in A(1 + 4N0), then s = t.

How Things Factor in 1 + 4N0

The element x is irreducible in 1 + 4N0 if and only if x is either

1 p where p is a prime and p ≡ 1 (mod 4), or

2 p1p2 where p1 and p2 are primes congruent to 3 modulo 4.

Moreover, x is prime if and only if it is of type 1.

Corollary

Let x ∈ 1 + 4N0. Ifx = α1 · · ·αs = β1 · · ·βt

for αi and βj in A(1 + 4N0), then s = t.

An Example to Illustrate the Last Two Results

Let’s Factor

141, 851, 281 = 4× (35, 462, 820) + 1 ∈ 1 + 4N0

Now141, 851, 281 = 13 × 17 × 11 × 23 × 43 × 59

Let’s Factor

141, 851, 281 = 4× (35, 462, 820) + 1 ∈ 1 + 4N0

Now141, 851, 281 = 13 × 17 × 11 × 23 × 43 × 59

141, 851, 281 = 13 × 17︸︷︷︸p≡1 (mod 4)

× 11 × 23 × 43 × 59︸︷︷︸p≡3 (mod 4)

= 13× 17× (11× 23)× (43× 59) = 13× 17× 253× 2537

= 13× 17× (11× 43)× (23× 59) = 13× 17× 473× 1357

= 13× 17× (11× 59)× (43× 23) = 13× 17× 649× 989

141, 851, 281 = 13 × 17︸︷︷︸p≡1 (mod 4)

× 11 × 23 × 43 × 59︸︷︷︸p≡3 (mod 4)

= 13× 17× (11× 23)× (43× 59) = 13× 17× 253× 2537

= 13× 17× (11× 43)× (23× 59) = 13× 17× 473× 1357

= 13× 17× (11× 59)× (43× 23) = 13× 17× 649× 989

141, 851, 281 = 13 × 17︸︷︷︸p≡1 (mod 4)

× 11 × 23 × 43 × 59︸︷︷︸p≡3 (mod 4)

= 13× 17× (11× 23)× (43× 59) = 13× 17× 253× 2537

= 13× 17× (11× 43)× (23× 59) = 13× 17× 473× 1357

= 13× 17× (11× 59)× (43× 23) = 13× 17× 649× 989

141, 851, 281 = 13 × 17︸︷︷︸p≡1 (mod 4)

× 11 × 23 × 43 × 59︸︷︷︸p≡3 (mod 4)

= 13× 17× (11× 23)× (43× 59) = 13× 17× 253× 2537

= 13× 17× (11× 43)× (23× 59) = 13× 17× 473× 1357

= 13× 17× (11× 59)× (43× 23) = 13× 17× 649× 989

Conclusion of 1 + 4N0

In general, a monoid with this property, i.e.,

x = α1 · · ·αs = β1 · · ·βt

for αi and βj in A(M), then s = t, is called half-factorial.

Theorem

There is a mapϕ : Z[

√−5]→ 1 + 4N0

which preserves lengths of factorizations into products of irreducibles.Hence, Z[

√−5] is half-factorial.

x = α1 · · ·αs = β1 · · ·βt

Theorem

√−5]→ 1 + 4N0

x = α1 · · ·αs = β1 · · ·βt

Theorem

√−5]→ 1 + 4N0

More Monthly Past

How the Flag Should Look?

(a) Long - This pattern is made by alternating long and short rows ofstars where the long rows contain one more star than the short rows.In this pattern the first and last row are both long.

(a) Long (50 stars, 1960)

Configurations of Stars on the Flag

(b) Short - This pattern is defined in the same way as the long pattern,except that the first and the last row are both short.

(b) Short

(c) Alternate - This pattern is defined in the same way as the longpattern, except that the first row is long and the last row is short, orvice versa.

(c) Alternate (45 stars,1896)

(d) Wyoming - In this pattern, the first and the last row are long, whileall other rows are short.

(d) Wyoming (32 stars,1858)

(e) Equal - All rows in this pattern have the same number of stars.

(e) Equal (48 stars,1912)

(f) Oregon - Similar to the equal pattern, all rows in this pattern are ofthe same length with the exception of the middle row, which is twostars shorter. This pattern requires an odd number of rows.

(f) Oregon (33 stars,1859)

(g) Long (50 stars, 1960) (h) Short (i) Alternate (45 stars,1896)

(j) Wyoming (32 stars,1858)

(k) Equal (48 stars,1912)

(l) Oregon (33 stars,1859)

The Odd Person Out

Figure: 29 star Union Jack

Definition

A nice arrangement of stars on the Union Jack is one that uses one of thesix patterns defined above with a column to row ratio in the interval [1, 2].

Fact: All integers from 1 to 100, except for 29, 69 and 87, have at leastone nice arrangement.

The Odd Person Out

Definition

The Odd Person Out

Definition

A Necessary Deep Theorem

Ω(n) = the number of prime factors of n counted with multiplicity.

Theorem (Hardy, Ramanujan)

For any fixed ε > 0 we have that

limN→∞

N#n ≤ N : (1− ε) log logN ≤ Ω(n) ≤ (1 + ε) log logN = 1.

A Necessary Deep Theorem

Ω(n) = the number of prime factors of n counted with multiplicity.

Theorem (Hardy, Ramanujan)

For any fixed ε > 0 we have that

limN→∞

N#n ≤ N : (1− ε) log logN ≤ Ω(n) ≤ (1 + ε) log logN = 1.

The Surprising Conclusion

S(N) = #n ≤ N : there is a nice arrangement of

n stars on the Union Jack.

By previous computation, S(100) = 97.

Theorem

limN→∞

Theorem

limN→∞

Theorem

limN→∞

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