Publications

• Optimal Numbers and Solutions in the Euclidean Algorithm

  The Pentagon, A Mathematics Magazine for Students

• Optimal Numbers and Solutions in the Euclidean Algorithm

  The Pentagon, A Mathematics Magazine for Students

• Tiling Annular Regions with Skew and T-tetrominoes

  Involve, a Journal of Mathematics

  In this paper, we study tilings of annular regions in the integer lattice by skew and T-tetrominoes. We demonstrate the tileability of most annular regions by the given tile set, enumerate the tilings of width-2 annuli, and determine the tile counting group associated to this tile set and the family of all width-2 annuli.

• Optimal Numbers and Solutions in the Euclidean Algorithm

  The Pentagon, A Mathematics Magazine for Students

• Tiling Annular Regions with Skew and T-tetrominoes

  Involve, a Journal of Mathematics

  In this paper, we study tilings of annular regions in the integer lattice by skew and T-tetrominoes. We demonstrate the tileability of most annular regions by the given tile set, enumerate the tilings of width-2 annuli, and determine the tile counting group associated to this tile set and the family of all width-2 annuli.

• Leading Coefficients and the Multiplicity of Known Roots

  Submitted.

  We show that a monic univariate polynomial over a field of characteristic zero, with k distinct non-zero known roots, is determined by its k proper leading coefficients by providing an explicit algorithm for computing the multiplicities of each root. We provide a version of the result and accompanying algorithm when the field is not algebraically closed by considering the minimal polynomials of the roots. Furthermore, we show how to perform the aforementioned algorithm in a numerically stable manner over ℂ, and then apply it to obtain new characteristic polynomials of hypergraphs.

• Optimal Numbers and Solutions in the Euclidean Algorithm

  The Pentagon, A Mathematics Magazine for Students

• Tiling Annular Regions with Skew and T-tetrominoes

  Involve, a Journal of Mathematics

  In this paper, we study tilings of annular regions in the integer lattice by skew and T-tetrominoes. We demonstrate the tileability of most annular regions by the given tile set, enumerate the tilings of width-2 annuli, and determine the tile counting group associated to this tile set and the family of all width-2 annuli.

• Leading Coefficients and the Multiplicity of Known Roots

  Submitted.

  We show that a monic univariate polynomial over a field of characteristic zero, with k distinct non-zero known roots, is determined by its k proper leading coefficients by providing an explicit algorithm for computing the multiplicities of each root. We provide a version of the result and accompanying algorithm when the field is not algebraically closed by considering the minimal polynomials of the roots. Furthermore, we show how to perform the aforementioned algorithm in a numerically stable manner over ℂ, and then apply it to obtain new characteristic polynomials of hypergraphs.

• Using Block Designs in Crossing Number Bounds

  Journal of Combinatorial Designs

  The crossing number cr(G) of a graph G=(V,E) is the smallest number of edge crossings over all drawings of G in the plane. For any k≥1, the k-planar crossing number of G, cr_k(G), is defined as the minimum of cr(G_1)+cr(G_2)+…+cr(G_k) over all graphs G_1,G_2,…,G_k whose union is G. Pach et al. showed that for every k≥1, we have cr_k(G)≤(2/k^2−1/k^3)cr(G) and that this bound does not remain true if we replace the constant 2/k^2−1/k^3 by any number smaller than 1/k^2. We improve the upper bound to 1/k^2(1+o(1)) as k→∞. For the class of bipartite graphs, we show that the best constant is exactly 1/k^2 for every k. The results extend to the rectilinear variant of the k-planar crossing number. To appear.

• Optimal Numbers and Solutions in the Euclidean Algorithm

  The Pentagon, A Mathematics Magazine for Students

• Tiling Annular Regions with Skew and T-tetrominoes

  Involve, a Journal of Mathematics

  In this paper, we study tilings of annular regions in the integer lattice by skew and T-tetrominoes. We demonstrate the tileability of most annular regions by the given tile set, enumerate the tilings of width-2 annuli, and determine the tile counting group associated to this tile set and the family of all width-2 annuli.

• Leading Coefficients and the Multiplicity of Known Roots

  Submitted.

  We show that a monic univariate polynomial over a field of characteristic zero, with k distinct non-zero known roots, is determined by its k proper leading coefficients by providing an explicit algorithm for computing the multiplicities of each root. We provide a version of the result and accompanying algorithm when the field is not algebraically closed by considering the minimal polynomials of the roots. Furthermore, we show how to perform the aforementioned algorithm in a numerically stable manner over ℂ, and then apply it to obtain new characteristic polynomials of hypergraphs.

• Using Block Designs in Crossing Number Bounds

  Journal of Combinatorial Designs

  The crossing number cr(G) of a graph G=(V,E) is the smallest number of edge crossings over all drawings of G in the plane. For any k≥1, the k-planar crossing number of G, cr_k(G), is defined as the minimum of cr(G_1)+cr(G_2)+…+cr(G_k) over all graphs G_1,G_2,…,G_k whose union is G. Pach et al. showed that for every k≥1, we have cr_k(G)≤(2/k^2−1/k^3)cr(G) and that this bound does not remain true if we replace the constant 2/k^2−1/k^3 by any number smaller than 1/k^2. We improve the upper bound to 1/k^2(1+o(1)) as k→∞. For the class of bipartite graphs, we show that the best constant is exactly 1/k^2 for every k. The results extend to the rectilinear variant of the k-planar crossing number. To appear.

• On the Adjacency Spectra of Hypertrees

  The Electronic Journal of Combinatorics

  We extend the results of Zhang et al. to show that λ is an eigenvalue of a k-uniform hypertree (k≥3) if and only if it is a root of a particular matching polynomial for a connected induced subtree. We then use this to provide a spectral characterization for power hypertrees. Notably, the situation is quite different from that of ordinary trees, i.e., 2-uniform trees. We conclude by presenting an example (an 11 vertex, 3-uniform non-power hypertree) illustrating these phenomena.

• Optimal Numbers and Solutions in the Euclidean Algorithm

  The Pentagon, A Mathematics Magazine for Students

• Tiling Annular Regions with Skew and T-tetrominoes

  Involve, a Journal of Mathematics

  In this paper, we study tilings of annular regions in the integer lattice by skew and T-tetrominoes. We demonstrate the tileability of most annular regions by the given tile set, enumerate the tilings of width-2 annuli, and determine the tile counting group associated to this tile set and the family of all width-2 annuli.

• Leading Coefficients and the Multiplicity of Known Roots

  Submitted.

  We show that a monic univariate polynomial over a field of characteristic zero, with k distinct non-zero known roots, is determined by its k proper leading coefficients by providing an explicit algorithm for computing the multiplicities of each root. We provide a version of the result and accompanying algorithm when the field is not algebraically closed by considering the minimal polynomials of the roots. Furthermore, we show how to perform the aforementioned algorithm in a numerically stable manner over ℂ, and then apply it to obtain new characteristic polynomials of hypergraphs.

• Using Block Designs in Crossing Number Bounds

  Journal of Combinatorial Designs

  The crossing number cr(G) of a graph G=(V,E) is the smallest number of edge crossings over all drawings of G in the plane. For any k≥1, the k-planar crossing number of G, cr_k(G), is defined as the minimum of cr(G_1)+cr(G_2)+…+cr(G_k) over all graphs G_1,G_2,…,G_k whose union is G. Pach et al. showed that for every k≥1, we have cr_k(G)≤(2/k^2−1/k^3)cr(G) and that this bound does not remain true if we replace the constant 2/k^2−1/k^3 by any number smaller than 1/k^2. We improve the upper bound to 1/k^2(1+o(1)) as k→∞. For the class of bipartite graphs, we show that the best constant is exactly 1/k^2 for every k. The results extend to the rectilinear variant of the k-planar crossing number. To appear.

• On the Adjacency Spectra of Hypertrees

  The Electronic Journal of Combinatorics

  We extend the results of Zhang et al. to show that λ is an eigenvalue of a k-uniform hypertree (k≥3) if and only if it is a root of a particular matching polynomial for a connected induced subtree. We then use this to provide a spectral characterization for power hypertrees. Notably, the situation is quite different from that of ordinary trees, i.e., 2-uniform trees. We conclude by presenting an example (an 11 vertex, 3-uniform non-power hypertree) illustrating these phenomena.

• New Bounds on the Biplanar Crossing Number of Low-dimensional Hypercubes

  Bulletin of the Institute of Combinatorics and its Applications (BICA)

  In this note we provide an improved upper bound on the biplanar crossing number of the 8-dimensional hypercube. The k-planar crossing number of a graph crk(G) is the number of crossings required when every edge of G must be drawn in one of k distinct planes. It was shown in Czabarka et al. that cr2(Q8)≤256 which we improve to cr2(Q8)≤128. Our approach highlights the relationship between symmetric drawings and the study of k-planar crossing numbers. We conclude with several open questions concerning this relationship.