Comparing Powers of Edge Ideals

Given a nontrivial homogeneous ideal $I\subseteq k[x_1,x_2,\ldots,x_d]$, a problem of great recent interest has been the comparison of the $r$th ordinary power of $I$ and the $m$th symbolic power $I^{(m)}$. This comparison has been undertaken directly via an exploration of which exponents $m$ and $r$ guarantee the subset containment $I^{(m)}\subseteq I^r$ and asymptotically via a computation of the resurgence $\rho(I)$, a number for which any $m/r > \rho(I)$ guarantees $I^{(m)}\subseteq I^r$. Recently, a third quantity, the symbolic defect, was introduced; as $I^t\subseteq I^{(t)}$, the symbolic defect is the minimal number of generators required to add to $I^t$ in order to get $I^{(t)}$. We consider these various means of comparison when $I$ is the edge ideal of certain graphs by describing an ideal $J$ for which $I^{(t)} = I^t + J$. When $I$ is the edge ideal of an odd cycle, our description of the structure of $I^{(t)}$ yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.Comment: Version 2: Revised based on referee suggestions. Lemma 5.12 was added to clarify the proof of Theorem 5.13. To appear in the Journal of Algebra and its Applications. Version 1: 20 pages. This project was supported by Dordt College's undergraduate research program in summer 201

Observational prospects for gravitational waves from hidden or dark chiral phase transitions

We study the gravitational wave (GW) signature of first-order chiral phase transitions ($\chi$PT) in strongly interacting hidden or dark sectors. We do so using several effective models in order to reliably capture the relevant non-perturbative dynamics. This approach allows us to explicitly calculate key quantities characterizing the $\chi$PT without having to resort to rough estimates. Most importantly, we find that the transition's inverse duration $\beta$ normalized to the Hubble parameter $H$ is at least two orders of magnitude larger than typically assumed in comparable scenarios, namely $\beta/H\gtrsim\mathcal{O}(10^4)$. The obtained GW spectra then suggest that signals from hidden $\chi$PTs occurring at around 100 MeV can be in reach of LISA, while DECIGO and BBO may detect a stochastic GW background associated with transitions between roughly 1 GeV and 10 TeV. Signatures of transitions at higher temperatures are found to be outside the range of any currently proposed experiment. Even though predictions from different effective models are qualitatively similar, we find that they may vary considerably from a quantitative point of view, which highlights the need for true first-principle calculations such as lattice simulations.Comment: 35 pages, 9 figures. Extended discussion and updated calculation of gravitational wave spectra, main results unchanged; references added; matches published versio

Splashing and evaporation of nucleons from excited nuclei

The energy spectrum and the emission rate of particles emitted from excited nucleus due to both the evaporation and the splashing (emission from a cold vibrating nucleus) are calculated. We show that the collective motion of the nuclear Fermi liquid is accompanied by direct non-statistical emission of nucleons via the dynamical distortion of the Fermi surface.Comment: Revtex file (12 pages) and 2 figures. Submitted to Nucl. Phys.

Partitions of R^n with Maximal Seclusion and their Applications to Reproducible Computation

We introduce and investigate a natural problem regarding unit cube tilings/partitions of Euclidean space and also consider broad generalizations of this problem. The problem fits well within a historical context of similar problems and also has applications to the study of reproducibility in randomized computation. Given $k\in\mathbb{N}$ and $\epsilon\in(0,\infty)$, we define a $(k,\epsilon)$-secluded unit cube partition of $\mathbb{R}^{d}$ to be a unit cube partition of $\mathbb{R}^{d}$ such that for every point $\vec{p}\in\R^d$, the closed $\ell_{\infty}$ $\epsilon$-ball around $\vec{p}$ intersects at most $k$ cubes. The problem is to construct such partitions for each dimension $d$ with the primary goal of minimizing $k$ and the secondary goal of maximizing $\epsilon$. We prove that for every dimension $d\in\mathbb{N}$, there is an explicit and efficiently computable $(k,\epsilon)$-secluded axis-aligned unit cube partition of $\mathbb{R}^d$ with $k=d+1$ and $\epsilon=\frac{1}{2d}$. We complement this construction by proving that for axis-aligned unit cube partitions, the value of $k=d+1$ is the minimum possible, and when $k$ is minimized at $k=d+1$, the value $\epsilon=\frac{1}{2d}$ is the maximum possible. This demonstrates that our constructions are the best possible. We also consider the much broader class of partitions in which every member has at most unit volume and show that $k=d+1$ is still the minimum possible. We also show that for any reasonable $k$ (i.e. $k\leq 2^{d}$), it must be that $\epsilon\leq\frac{\log_{4}(k)}{d}$. This demonstrates that when $k$ is minimized at $k=d+1$, our unit cube constructions are optimal to within a logarithmic factor even for this broad class of partitions. In fact, they are even optimal in $\epsilon$ up to a logarithmic factor when $k$ is allowed to be polynomial in $d$. We extend the techniques used above to introduce and prove a variant of the KKM lemma, the Lebesgue covering theorem, and Sperner\u27s lemma on the cube which says that for every $\epsilon\in(0,\frac12]$, and every proper coloring of $[0,1]^{d}$, there is a translate of the $\ell_{\infty}$ $\epsilon$-ball which contains points of least $(1+\frac23\epsilon)^{d}$ different colors. Advisers: N. V. Vinodchandran & Jamie Radcliff

Generic canonical form of pairs of matrices with zeros

We consider a family of pairs of m-by-p and m-by-q matrices, in which some entries are required to be zero and the others are arbitrary, with respect to transformations (A,B)--> (SAR,SBL) with nonsingular S, R, L. We prove that almost all of these pairs reduce to the same pair (C, D) from this family, except for pairs whose arbitrary entries are zeros of a certain polynomial. The polynomial and the pair (C D) are constructed by a combinatorial method based on properties of a certain graph.Comment: 13 page