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Full-Text Articles in Algebra
Slₖ-Tilings And Paths In ℤᵏ, Zachery T. Peterson
Slₖ-Tilings And Paths In ℤᵏ, Zachery T. Peterson
Theses and Dissertations--Mathematics
An SLₖ-frieze is a bi-infinite array of integers where adjacent entries satisfy a certain diamond rule. SL₂-friezes were introduced and studied by Conway and Coxeter. Later, these were generalized to infinite matrix-like structures called tilings as well as higher values of k. A recent paper by Short showed a bijection between bi-infinite paths of reduced rationals in the Farey graph and SL₂-tilings. We extend this result to higher kby constructing a bijection between SLₖ-tilings and certain pairs of bi-infinite strips of vectors in ℤᵏ called paths. The key ingredient in the proof is the relation to Plucker friezes and Grassmannian …
Pairs Of Quadratic Forms Over P-Adic Fields, John Hall
Pairs Of Quadratic Forms Over P-Adic Fields, John Hall
Theses and Dissertations--Mathematics
Given two quadratic forms $Q_1, Q_2$ over a $p$-adic field $K$ in $n$ variables, we consider the pencil $\mathcal{P}_K(Q_1, Q_2)$, which contains all nontrivial $K$-linear combinations of $Q_1$ and $Q_2$. We define $D$ to be the maximal dimension of a subspace in $K^n$ on which $Q_1$ and $Q_2$ both vanish. We define $H$ to be the maximal number of hyperbolic planes that a form in $\mathcal{P}_K(Q_1, Q_2)$ splits off over $K$. We will determine which values for $(D, H)$ are possible for a nonsingular pair of quadratic forms over a $p$-adic field $K$.