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Full-Text Articles in Mathematics
Comparing Powers Of Edge Ideals, Mike Janssen, Thomas Kamp, Jason Vander Woude
Comparing Powers Of Edge Ideals, Mike Janssen, Thomas Kamp, Jason Vander Woude
Faculty Work Comprehensive List
Given a nontrivial homogeneous ideal I ⊆ k[x1, x2, . . . ,xd], a problem of great recent interest has been the comparison of the rth ordinary power of I and the mth symbolic power I(m). This comparison has been undertaken directly via an exploration of which exponents m and r guarantee the subset containment I(m) ⊆ Ir and asymptotically via a computation of the resurgence ρ(I), a number for which any m/r > ρ(I) guarantees I(m) ⊆ Ir. Recently, a third quantity, the symbolic defect, was introduced; as It ⊆ I(t), the symbolic defect is the minimal number of generators …
On Robust Colorings Of Hamming-Distance Graphs, Isaiah Harney, Heide Gluesing-Luerssen
On Robust Colorings Of Hamming-Distance Graphs, Isaiah Harney, Heide Gluesing-Luerssen
Mathematics Faculty Publications
Hq(n, d) is defined as the graph with vertex set Znq and where two vertices are adjacent if their Hamming distance is at least d. The chromatic number of these graphs is presented for various sets of parameters (q, n, d). For the 4-colorings of the graphs H2(n, n − 1) a notion of robustness is introduced. It is based on the tolerance of swapping colors along an edge without destroying properness of the coloring. An explicit description of the maximally robust 4-colorings of …
Persistence Equivalence Of Discrete Morse Functions On Trees, Yuqing Liu
Persistence Equivalence Of Discrete Morse Functions On Trees, Yuqing Liu
Mathematics Summer Fellows
We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence. Two functions are considered persistence equivalent if and only if they induce the same persistence diagram. We compare this notion of equivalence to other notions of equivalent discrete Morse functions. We then compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree. We conclude with an example illustrating our construction.