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

Spline Basics 1 Special Properties (mod m ) 2 Characterizations and the Role of Primes 3 Further Research and New Ideas 4 Bowden Splines mod m

Spline Basics Bowden Splines mod m

◮ Here is a graph with edges labeled with elements of Z / 27 Z x 2 3 x 1

◮ Here is a graph with edges labeled with elements of Z / 27 Z ◮ Can you label the vertices with x 2 3 ring elements x 1 and x 2 so that their difference is a multiple of 3? x 1

◮ Here is a graph with edges labeled with elements of Z / 27 Z ◮ Can you label the vertices with x 2 3 ring elements x 1 and x 2 so that their difference is a multiple of 3? x 1 ◮ Of course you can! Bowden Splines mod m

Here’s one set of vertex labels you might have found: 9 3 0 . � 9 � the set of vertex labels is a spline on the graph. 0 Bowden Splines mod m

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

Minimal generating sets are very helpful when talking about splines mod m : �� 3 � 1 � �� x 2 B = , .. 0 1 3 x 1 .. Here is a minimal generating set for all splines on the edge labeled graph. Here is an 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 / m Z . A spline mod m is a set of vertex labels in Z / m Z that satisfy the following condition: ◮ if two vertices labeled x 1 and x 2 are joined by an edge labeled ℓ 1 then | x 1 − x 2 | ∈ � ℓ 1 � 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 22 10 17 20 12 20 10 2 2 3 7 7 4 an integer spline on a 3-cycle 2 2 2 x 8 x 2 5 4 x a spline on K 4 in Z / 30 Z a polynomial spline on one edge 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 x 2 x 1 x 3  x 3   : x i ∈ Z x 2  x 1 Bowden Splines mod m

2 3 x 2 x 1 x 3         1 3 3    ,  , 1 2 0     1 0 0    x 3   : x i ∈ Z x 2  x 1 Bowden Splines mod m

2 3 x 2 x 1 x 3   x 3  : x i ∈ Z / 6 Z x 2  x 1         1 3 3    ,  , 1 2 0     1 0 0   Bowden Splines mod m

2 3 x 2 x 1 x 3     3 3  ≡ 3 2 0      x 3 0 0  : x i ∈ Z / 6 Z x 2  x 1         1 3 3    ,  , 1 2 0     1 0 0   Bowden Splines mod m

2 3 x 2 x 1 x 3     3 3  ≡ 3 2 0      x 3 0 0  : x i ∈ Z / 6 Z x 2  x 1    1   3            1 3 3  , 1 2       ,  , 1 2 0 1 0       1 0 0   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 / p 2 Z example x 3 10 x 4 10 5 15 x 5 x 2 Let’s say we want to find a 20 minimal generating set to 5 x 1 describe all splines on this graph mod 25...  x 5  x 4     x 3 : x i ∈ Z / 25 Z     x 2   x 1 Bowden Splines mod m

Z / p 2 Z theorem Theorem Let p be a prime number. If G is a graph on n vertices in Z / p 2 Z , then a minimal generating set for all splines on G is:   1   0   0   0   p      1 . . . .                              1 . . . .                             . . . . .             B = , , , , ...,           . 0 0 p 0                         . 0 p 0 0                             1 p 0 0 0                    1 0 0 0 0  Bowden Splines mod m

1 0 10 10 1 5 10 5 10 5 15 15 1 1 0 0 5 10 20 20 5 5 0 1 0 10 5 15 0 0 0 0 10 10 20 5 0 0 0 10 5 10 5 15 15 0 5 5 0 20 20 5 5 0 0 Bowden Splines mod m

Z / 32 Z example x 3 12 6 x 4 x 2 2 How about all splines on this 4 x 1 graph in Z / 32 Z ?  x 4  x 3    : x i ∈ Z / 32 Z   x 2  x 1 Bowden Splines mod m

Z / p n Z theorem Theorem Let p be a prime number. If C n is a cycle on n vertices in Z / p k Z , then B is a minimal generating set for all splines on G (up to rotation). ℓ n − 1 ..   1   ℓ 1   ℓ 2   ℓ i   ℓ n − 2   ℓ n − 1   ..    1 ℓ 1 ℓ 2 . ℓ n − 2                                  .. 1 . . . . .                                 . . . ℓ i . .   ℓ n ℓ 3             B = , , , ..., , ..., ,             . ℓ 1 ℓ 2 0 . . ..                             . ℓ 1 ℓ 2 . 0 0   ℓ 2                               ..  1 ℓ 1 0 . 0 0                ..     ℓ 1  1 0 0 0 0 0  Bowden Splines mod m

1 12 12 12 6 6 1 1 12 0 2 2 4 4 1 0 4 0 12 12 6 6 4 4 6 0 2 2 4 4 0 0 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 p k . ◮ This gives an algorithm to compute minimal generating sets. ◮ In this way Z / p k Z lets us understand more complex modules of 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 / p k Z 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