Harmonic morphisms to P_3 – examples

This post expands on a previous post and gives more examples of harmonic morphisms to the path graph \Gamma_2=P_3.

The path graph P_3

If \Gamma_1 = (V_1, E_1) and \Gamma_2 = (V_2, E_2) are graphs then a map \phi:\Gamma_1\to \Gamma_2 (that is, \phi: V_1\cup E_1\to V_2\cup E_2) is a morphism provided

  1. if \phi sends an edge to an edge then the edges vertices must also map to each other: e=(v,w)\in E_1 and \phi(e)\in E_2 then \phi(e) is an edge in \Gamma_2 having vertices \phi(v)\in V_2 and \phi(w)\in V_2, where \phi(v)\not= \phi(w), and
  2. if \phi sends an edge to a vertex then the edges vertices must also map to that vertex: if e=(v,w)\in E_1 and \phi(e)\in V_2 then \phi(e) = \phi(v) = \phi(w).

As a non-example, if \Gamma_1 is a planar graph, if \Gamma_2 is its dual graph, and if \phi:\Gamma_1\to\Gamma_2 is the dual map V_1\to E_2 and E_1\to V_2, then \phi is not a morphism.

Given a map \phi_E : E_1 \rightarrow E_2 \cup V_2, an edge e_1 is called horizontal if \phi_E(e_1) \in E_2 and is called vertical if \phi_E(e_1) \in V_2. We say that a graph morphism \phi: \Gamma_1 \rightarrow \Gamma_2 is a graph homomorphism if \phi_E (E_1) \subset E_2. Thus, a graph morphism is a homomorphism if it has no vertical edges.

Suppose that \Gamma_2 has at least one edge. Let Star_{\Gamma_1}(v) denote the star subgraph centered at the vertex v. A graph morphism \phi : \Gamma_1 \to \Gamma_2 is called harmonic if for all vertices v \in V(\Gamma_1), the quantity
\mu_\phi(v,f)= |\phi^{-1}(f) \cap Star_{\Gamma_1}(v)|
(the number of edges in \Gamma_1 adjacent to v and mapping to the edge f in \Gamma_2) is independent of the choice of edge f in Star_{\Gamma_2}(\phi(v)).

An example of a harmonic morphism can be described in the plot below as follows: \phi:\Gamma_1\to \Gamma_2=P_3 sends the red vertices in \Gamma_1 to the red vertex of \Gamma_2=P_3, the green vertices in \Gamma_1 to the green vertex of \Gamma_2=P_3, and the white vertices in \Gamma_1 to the white vertex of \Gamma_2=P_3.

Example 1:

P3-C3-V

Example 2:
D3-2110

Example 3:
cyclic4-2101

Integral Calculus and SageMath

Long ago, using LaTeX I assembled a book on Calculus II (integral calculus), based on notes of mine, Dale Hoffman (which was written in word), and William Stein. I ran out of energy to finish it and the source files mostly disappeared from my HD. Recently, Samuel Lelièvre found a copy of the pdf of this book on the internet (you can download it here). It’s licensed under a Creative Commons Attribution 3.0 United States License. You are free to print, use, mix or modify these materials as long as you credit the original authors.

Table of Contents

0 Preface

1 The Integral
1.1 Area
1.2 Some applications of area
1.2.1 Total accumulation as “area”
1.2.2 Problems
1.3 Sigma notation and Riemann sums
1.3.1 Sums of areas of rectangles
1.3.2 Area under a curve: Riemann sums
1.3.3 Two special Riemann sums: lower and upper sums
1.3.4 Problems
1.3.5 The trapezoidal rule
1.3.6 Simpson’s rule and Sage
1.3.7 Trapezoidal vs. Simpson: Which Method Is Best?
1.4 The definite integral
1.4.1 The Fundamental Theorem of Calculus
1.4.2 Problems
1.4.3 Properties of the definite integral
1.4.4 Problems
1.5 Areas, integrals, and antiderivatives
1.5.1 Integrals, Antiderivatives, and Applications
1.5.2 Indefinite Integrals and net change
1.5.3 Physical Intuition
1.5.4 Problems
1.6 Substitution and Symmetry
1.6.1 The Substitution Rule
1.6.2 Substitution and definite integrals
1.6.3 Symmetry
1.6.4 Problems

2 Applications
2.1 Applications of the integral to area
2.1.1 Using integration to determine areas
2.2 Computing Volumes of Surfaces of Revolution
2.2.1 Disc method
2.2.2 Shell method
2.2.3 Problems
2.3 Average Values
2.3.1 Problems
2.4 Moments and centers of mass
2.4.1 Point Masses
2.4.2 Center of mass of a region in the plane
2.4.3 x-bar For A Region
2.4.4 y-bar For a Region
2.4.5 Theorems of Pappus
2.5 Arc lengths
2.5.1 2-D Arc length
2.5.2 3-D Arc length

3 Polar coordinates and trigonometric integrals
3.1 Polar Coordinates
3.2 Areas in Polar Coordinates
3.3 Complex Numbers
3.3.1 Polar Form
3.4 Complex Exponentials and Trigonometric Identities
3.4.1 Trigonometry and Complex Exponentials
3.5 Integrals of Trigonometric Functions
3.5.1 Some Remarks on Using Complex-Valued Functions

4 Integration techniques
4.1 Trigonometric Substitutions
4.2 Integration by Parts
4.2.1 More General Uses of Integration By Parts
4.3 Factoring Polynomials
4.4 Partial Fractions
4.5 Integration of Rational Functions Using Partial Fractions
4.6 Improper Integrals
4.6.1 Convergence, Divergence, and Comparison

5 Sequences and Series
5.1 Sequences
5.2 Series
5.3 The Integral and Comparison Tests
5.3.1 Estimating the Sum of a Series
5.4 Tests for Convergence
5.4.1 The Comparison Test
5.4.2 Absolute and Conditional Convergence
5.4.3 The Ratio Test
5.4.4 The Root Test
5.5 Power Series
5.5.1 Shift the Origin
5.5.2 Convergence of Power Series
5.6 Taylor Series
5.7 Applications of Taylor Series
5.7.1 Estimation of Taylor Series

6 Some Differential Equations
6.1 Separable Equations
6.2 Logistic Equation

7 Appendix: Integral tables

Problem of the Week, #121

A former colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his students, giving a prize of a cookie if they could solve it. Here is one of them.

Problem 121

The Maryland “Big Game” lottery is played by selecting 5 different numbers in \{ 1,2,3,\dots, 50\} and then selecting one of the numbers in \{ 1,2,3,\dots, 36\}. The first section is an unordered selection without replacement (so, arrange them in increasing order if you like) but the second selection can repeat one of the 5 numbers initially picked.

How many ways can this be done?

John Cleese on creativity

Creativity is not a talent. It is a way of operating. John Cleese has written and lectured on creativity and listed the 5 factors that you can arrange to be more creative:

  1. Space                                                                                                                                                       You need to escape your usual pressures.
  2. Time                                                                                                                                                         You need your space for a specific block of time.
  3. Time                                                                                                                                                         Giving your mind as long as possible to come up with something  original, be  patient with yourself.
  4. Confidence                                                                                                                                              Nothing will stop you being creative so effectively as the fear of  making a mistake.
  5. Humor                                                                                                                                                      Have fun. Humor gets us from closed mode to the open mind quicker than anything else.

 

Memories of TS Michael, by Thomas Quint

TS Michael passed away on November 22, 2016, from cancer. I will miss him as a colleague and a kind, wise soul. Tom Quint has kindly allowed me to post these reminiscences that he wrote up.


Well, I guess I could start with the reason TS and I met in the first place. I was a postdoc at USNA in about 1991 and pretty impressed with myself. So when USNA offered to continue my postdoc for two more years (rather than give me a tenure track position), I turned it down. Smartest move I ever made, because TS got the position and so we got to know each other.

We started working w each other one day when we both attended a talk on “sphere of influence graphs”. I found the subject moderately interesting, but he came into my office all excited, and I couldn’t get rid of him — wouldn’t leave until we had developed a few research ideas.

Interestingly, his position at USNA turned into a tenure track, while mine didn’t. It wasn’t until 1996 that I got my position at U of Nevada.

Work sessions with him always followed the same pattern. As you may or may not know, T.S. a) refused to fly in airplanes, and b) didn’t drive. Living across the country from each other, the only way we could work together face-to-face was: once each summer I would fly out to the east coast to visit my parents, borrow one of their cars for a week, and drive down to Annapolis. First thing we’d do is go to Whole Foods, where he would load up my car with food and other supplies, enough to last at least a few months. My reward was that he always bought me the biggest package of nigiri sushi we could find — not cheap at Whole Foods!

It was fun, even though I had to suffer through eight million stories about the USNA Water Polo Team.

Oh yes, and he used to encourage me to sneak into one of the USNA gyms to work out. We figured that no one would notice if I wore my Nevada sweats (our color is also dark blue, and the pants also had a big “N” on them). It worked.

Truth be told, TS didn’t really have a family of his own, so I think he considered the mids as his family. He cared deeply about them (with bonus points if you were a math major or a water polo player :-).

One more TS anecdote, complete with photo.  Specifically, TS was especially thrilled to find out that we had named our firstborn son Theodore Saul Quint.  Naturally, TS took to calling him “Little TS”.  At any rate, the photo below is of “Big TS” holding “Little TS”, some time in the Fall of 2000.

tslittlets_photo2000

TS Michael in 2000.

Some favorite quotes on math, science, learning

There are some things which cannot be learned quickly,
and time, which is all we have,
must be paid heavily for their acquiring.
They are the very simplest things,
and because it takes a man’s life to know them
the little new that each man gets from life
is very costly and the only heritage he has to leave.
Ernest Hemingway
(From A. E. Hotchner, Papa Hemingway, Random House, NY, 1966)

I believe that a scientist looking at nonscientific problems is just as dumb as the next guy.
Richard Feynman

The best thing for being sad is to learn something. That is the only thing that never fails. You may grow old and trembling in your anatomies, you may lie awake at night listening to the disorder of your veins, you may miss your only love, you may see the world about you devastated by evil lunatics, or know your honor trampled in the sewers of baser minds. There is only one thing for it then to learn. Learn why the world wags and what wags it. That is the only thing which the mind can never exhaust, never alienate, never be tortured by, never fear or distrust, and never dream of regretting.
T. H. White in The Once and Future King

Truth, like gold, is to be obtained not by its growth, but by washing away from it all that is not gold. Leo Tolstoy

Education is what survives when what has been learnt has been forgotten.
B. F. Skinner

The advantage is that mathematics is a field in which one’s blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one’s best moments that count and not one’s worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician’s reputation.
Norbert Wiener, in Ex-Prodigy: My Childhood and Youth

Science is a differential equation. Religion is a boundary condition.
Alan Turing

Everything is vague to a degree you do not realize till you have tried to make it precise.
Bertrand Russell

For every complicated problem there is a solution that is simple, direct, understandable, and wrong.
H. L. Mencken

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
John Louis von Neumann

To be what we are, and to become what we are capable of becoming, is the only end in life.
Baruch Spinoza