Splines mod m Nealy Bowden Smith College July 24, 2014 Bowden - - PowerPoint PPT Presentation

Splines mod m

Nealy Bowden

Smith College

July 24, 2014

Bowden Splines mod m

1

Spline Basics

2

Special Properties (mod m)

3

Characterizations and the Role of Primes

4

Further Research and New Ideas

Bowden Splines mod m

Spline Basics

Bowden Splines mod m

◮ Here is a graph with edges labeled

with elements of Z/27Z x2 x1 3

◮ Here is a graph with edges labeled

with elements of Z/27Z x2 x1 3

◮ Can you label the vertices with

ring elements x1 and x2 so that their difference is a multiple of 3?

◮ Here is a graph with edges labeled

with elements of Z/27Z x2 x1 3

◮ Can you label the vertices with

ring elements x1 and x2 so that their difference is a multiple of 3?

◮ Of course you can!

Bowden Splines mod m

Here’s one set of vertex labels you might have found: 3 9 . the set of vertex labels 9

• is a spline on the graph.

Bowden Splines mod m

Here are some other splines on the same graph: 3 6 3 3 17 20 3 1 16 3 18 18

Bowden Splines mod m

Minimal generating sets are very helpful when talking about splines mod m: 3 x2 x1 .. Here is an edge labeled graph. B = 3

• ,

1 1

• ..

Here is a minimal generating set for all splines on the edge labeled graph.

Bowden Splines mod m

Definition (Spline mod m) Let G be an edge labeled graph such that the set of edge labels of G is a subset of Z/mZ. A spline mod m is a set of vertex labels in Z/mZ that satisfy the following condition:

◮ if two vertices labeled x1 and x2 are joined by an edge labeled

ℓ1 then |x1 − x2| ∈ ℓ1

Bowden Splines mod m

◮ We can look for splines on any type of graph. ◮ We can find splines on graphs labeled with other rings. ◮ Let’s look at a few examples of some other cool splines.

Bowden Splines mod m

more splines

2 10 20 2 12 22 an integer spline on a 3-cycle 2x 8x2 4x a polynomial spline on one edge 2 10 20 2 4 3 5 7 17 7 a spline on K4 in Z/30Z

Bowden Splines mod m

Special Properties (mod m)

Bowden Splines mod m

Special Properties of Splines mod m

◮ Finite sets to label with ◮ Don’t label with 0 or units ◮ Variability of the modulus ◮ Generating set size

Bowden Splines mod m

2 3 x3 x2 x1

  x3 x2 x1   : xi ∈ Z

Bowden Splines mod m

2 3 x3 x2 x1

  x3 x2 x1   : xi ∈ Z

     1 1 1   ,   3 2   ,   3     

Bowden Splines mod m

2 3 x3 x2 x1

  x3 x2 x1   : xi ∈ Z/6Z

     1 1 1   ,   3 2   ,   3     

Bowden Splines mod m

2 3 x3 x2 x1

  x3 x2 x1   : xi ∈ Z/6Z

     1 1 1   ,   3 2   ,   3      3   3 2   ≡   3  

Bowden Splines mod m

2 3 x3 x2 x1

  x3 x2 x1   : xi ∈ Z/6Z

     1 1 1   ,   3 2   ,   3      3   3 2   ≡   3        1 1 1   ,   3 2     

Bowden Splines mod m

Our minimal generating sets can be very small. Theorem (Tymoczko, Hagen) Let G be an edge labeled graph on n vertices. A minimal generating set for integer splines on G must contain exactly n elements. Theorem (Tymoczko, Bowden) Let G be an edge labeled graph on n vertices. A minimal generating set for splines mod m on G can have anywhere between 1 and n elements.*

Bowden Splines mod m

◮ Generating sets are important and they sometimes behave in

surprising ways.

◮ Linear independence can be tricky! ◮ The value of m matters a lot.

Bowden Splines mod m

Role of Primes

Bowden Splines mod m

Z/p2Z example

5 15 5 10 10 20 x1 x2 x4 x5 x3

      x5 x4 x3 x2 x1       : xi ∈ Z/25Z

Let’s say we want to find a minimal generating set to describe all splines on this graph mod 25...

Bowden Splines mod m

Z/p2Z theorem

Theorem Let p be a prime number. If G is a graph on n vertices in Z/p2Z, then a minimal generating set for all splines on G is: B =                                    1 1 1 . . . 1 1             ,             . . . p             ,             . . . p             ,             . . . p             , ...,             p . . .                                   

Bowden Splines mod m

5 15 5 10 10 20 1 1 1 1 1 5 15 5 10 10 20 5 5 15 5 10 10 20 5 5 15 5 10 10 20 5 5 15 5 10 10 20 5

Bowden Splines mod m

Z/32Z example

4 12 6 2 x1 x2 x3 x4

    x4 x3 x2 x1     : xi ∈ Z/32Z

How about all splines on this graph in Z/32Z?

Bowden Splines mod m

Z/pnZ theorem

Theorem Let p be a prime number. If Cn is a cycle on n vertices in Z/pkZ, then B is a minimal generating set for all splines on G (up to rotation).

B =                                    1 1 1 . . . 1 1             ,             ℓ1 ℓ1 . . ℓ1 ℓ1 ℓ1             ,             ℓ2 ℓ2 . . ℓ2 ℓ2             , ...,             ℓi . . ℓi . .             , ...,             ℓn−2 ℓn−2 . . .             ,             ℓn−1 . . .                                   

ℓ1 ℓ2 ℓ3 ℓn−1 ℓn .. .. .. .. .. ..

Bowden Splines mod m

4 12 6 2 1 1 1 1 4 12 6 2 4 4 4 4 12 6 2 12 12 4 12 6 2 6

Bowden Splines mod m

The Importance of Prime Characterizations

◮ We are working out a structure theorem that uses the prime

factorization of m to understand splines mod m in terms of splines mod pk.

◮ This gives an algorithm to compute minimal generating sets. ◮ In this way Z/pkZ lets us understand more complex modules

• f splines.

Bowden Splines mod m

Future Research

Bowden Splines mod m

Future Research

◮ Investigate the relationship between graphs and subgraphs. ◮ Continue to explore variations in minimal generating set size. ◮ Continue to investigate other moduli. ◮ Explore, in greater detail, the relationship between splines

mod m and splines over other rings.

◮ Describe all splines over Z/pkZ for arbitrary G

Bowden Splines mod m

Thank you!

◮ Thank you to everyone in the math department at Smith for

their continued support and guidance.

◮ Thank you to everyone involved with Math 301 for the

amazing opportunity to do and share research together.

◮ Special thanks to the other members of our wonderful splines

research group: Sarah Hagen, Yue Cao, Melanie King, Stephanie Reinders, Chloe Xie, and Dr. Elizabeth Drellich.

◮ Special thanks to Julianna Tymoczko for introducing many

students to the wonderful world of splines.

Bowden Splines mod m