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## 2016 Summer Undergraduate Applied Mathematics Institute

### Projects:

• Exploring Origami Generated Structures in ${C}$, Yu Xuan Hong (download paper)
Abstract: The origami generator problem can be described as follows: given a set $U$ of angles, and a set $S$ of points containing 0 and 1, construct lines at angles in $U$ from each point in $S$ and from all possible intersection points of constructed lines, the set $R(U)$ is the closure of all possible points (including initial points) generated by such action. Previous research has shown many properties regarding the algebraic and geometric structures of $R(U)$, given $U$ and $S$ satisfying certain conditions. In particular, results have been proven for cases where $1 \in U$ and $S=\{0,1\}$. In this paper, we venture beyond these restrictions to explore results for more general cases of $U$ and $S$. Our main results hold for cases where $U$ does not contain 1, and when $|S| \geq 2$. We will state and prove the conditions in those cases for $R(U)$ to be a lattice or a ring.

• Fractal Behavior of the Fibonomial Triangle Modulo Prime $p$, where the Rank of Apparition of $p$ is $p+1$, Michael DeBellevue, Ekaterina Kryuchkova (download paper)
Abstract:  Pascal's triangle is known to exhibit fractal behavior modulo prime numbers. We tackle the analogous notion in the fibonomial triangle modulo prime $p$ with the rank of apparition $p^*=p+1$, proving that these objects form a structure similar to the Sierpinski Gasket. Within a large triangle of $p^* p^{m+1}$ many rows, in the $i^{th}$ triangle from the top and the $j^{th}$ triangle from the left, $\binom{n+ip^*p^m}{k+jp^*p^m}_F$ is divisible by $p$ if and only if $\binom{n}{k}_F$ is divisible by $p$. This proves the existence of the recurring triangles of zeroes which are the principal component of the Sierpinski Gasket. The exact congruence classes follow the relationship $\binom{n+ip^*p^m}{k+jp^*p^m}_F\equiv_p(-1)^{ik-nj}\binom{i}{j}\binom{n}{k}_F$, where $0\leq n,k$ < $p^*p^m$. It is a known result that the Fibonacci sequence modulo an integer is periodic. Denote the period modulo $p$ $\pi(p)$.
• Turán numbers of vertex-disjoint cliques in $r$-partite graphs, Adam Kapilow, Anna Schenfisch (download paper)
Abstract: The Turán number of a pair of graphs $G$ and $H$ is denoted $ex(G,H)$, and is the maximum number of edges a subgraph of $G$ may have and still contain no copy of $H$. In this paper, we determine $ex(K_{a_1,a_2,...,a_r}, mK_r)$, where $K_{a_1,a_2,...,a_r}$ denotes a complete $r$-partite graph with part sizes $a_1,...,a_r$ and $mK_r$ denotes $m$ vertex-disjoint copies of $K_r$, the complete graph on $r$ vertices. We prove that for any integers $1 \leq m \leq a_1 \leq a_2 \leq ... \leq a_r$ we have $ex(K_{a_1,a_2, ... , a_r}, mK_r) = \sum \limits_{1\leq i < j \leq r}a_ia_j - a_1a_2 + a_2(m-1).$
• $\tau$-Norm-Perfect and $\tau$-Perfect Eisenstein Integers for $\tau=\omega+2$ and $2$, Carlos Rojas Mena (download paper)
Abstract: Using Robert Spira's [4] definitions of complex Mersenne numbers and the complex sum-of-divisors function, we characterize $(\omega+2)$-norm-perfect and $(\omega+2)$-perfect numbers that are divisble by $\omega+2$ and prove the nonexistence of $2$-norm-perfect numbers that are divisible by $2$ in the Eisenstein integers.