Positive semidefinite propagation time

Let G be a simple, undirected graph. Positive semidefinite (PSD) zero forcing on G is based on the following color-change rule: Let W1,W2, ..., Wk be the sets of vertices of the k connected components in G - B (where B is a set of blue vertices). If w is in Wi and is the only white neighbor of some blue vertex b in the graph G[Bi] (where Bi is the set of blue vertices of B and the white vertices of Wi), then we change w to blue. A positive semidefinite zero forcing set (PSDZFS) is a set of blue vertices that colors the entire graph blue. The positive semidefinite zero forcing number, denoted Z+(G), is the minimum cardinality of a positive semidefinite zero forcing set. The PSD propagation time of a PSDZFS B of graph G is the minimum number of iterations that it takes to color the entire graph blue, starting with B blue, such that at each iteration as many vertices are colored blue as allowed by the color-change rule. The minimum and maximum PSD propagation times are taken over all minimum PSD zero forcing sets of the graph. The PSD propagation time interval of a graph G is the set of integers [pt+(G),pt+(G) +1,...,PT+(G)]. It is believed that every integer in the interval is achievable by some minimum PSDZFS. This thesis develops tools to analyze the minimum and maximum PSD propagation time, tools for analyzing the PSD propagation time interval and applies these tools to study the PSD propagation time of many graph families

Direct determination of 1-aminocyclopropane-1-carboxylic acid in plant tissues by using a gas chromatograph with flame-ionization detection

1–aminocyclopropane–1–carboxylic acid is the immediate precursor for ethylene, a phytohormone that is produced in many plant tissues, which is effective in trace amounts (≤ 1 yl/l). Quantitative determination of ACC in biological tissues is paramount to understanding the regulation of this metabolic process. The most widely accepted method for quantifying 1–aminocyclopropane–1–carboxylic acid requires its oxidation to ethylene, which then is measured by gas chromatography. Our objective was to develop a method for the rapid, direct quantification of 1–aminocyclopropane–1–carboxylic acid by using a gas chromatograph with a flame–ionization detector. Ethylene production was measured in tissues from fruits, leaves, seeds, florets, and flower petals. The remaining tissue samples were ground to a fine powder in liquid nitrogen, and 1–aminocyclopropane–1–carboxylic acid was extracted from the powder with a methanol:chloroform:water mixture (5:12:3 v/v/v). 1– aminocyclopropane–1–carboxylic acid was quantified by using an EZ:faastTM gas chromatography mass-spectrometry free amino acid analysis kit, was identified using gas chromatography mass–spectrometry, and was confirmed with authentic 1–aminocyclopropane–1–carboxylic acid. Additional benefits include a short run time of six minutes and the absence of a potentially hazardous tracer, such as 14C. We show that this method is effective for accurate, direct measurement of 1–aminocyclopropane–1–carboxylic acid from both reproductive and vegetative plant tissues

Anti-van der Waerden Numbers of Graph Products with Trees

Given a graph $G$, an exact $r$-coloring of $G$ is a surjective function $c:V(G) \to [1,\dots,r]$. An arithmetic progression in $G$ of length $j$ with common difference $d$ is a set of vertices $\{v_1,\dots, v_j\}$ such that $dist(v_i,v_{i+1}) = d$ for $1\le i < j$. An arithmetic progression is rainbow if all of the vertices are colored distinctly. The fewest number of colors that guarantees a rainbow arithmetic progression of length three is called the anti-van der Waerden number of $G$ and is denoted $aw(G,3)$. It is known that $3 \le aw(G\square H,3) \le 4$. Here we determine exact values $aw(T\square T',3)$ for some trees $T$ and $T'$, determine $aw(G\square T,3)$ for some trees $T$, and determine $aw(G\square H,3)$ for some graphs $G$ and $H$.Comment: 20 pages, 3 figure

Closing Youth Prisons: Lessons from Agency Administrators

Communities across the United States are fundamentally transforming their approach to juvenile justice services, moving away from outdated youth prisons and investing in community-based solutions. Juvenile justice system administrators are uniquely positioned to lead facility closure efforts as part of broader system reform and many are leading by example. In the summer of 2018, the Urban Institute convened a small group of current and former administrators who had successfully led closure efforts to discuss lessons learned and share their advice with others considering reform. Participants suggested four key strategies: maximize windows of opportunity; strategically partner with advocates; collaborate with youth, families, and other stakeholders; and use data and research to make the case for closure and combat counterproductive narratives. This brief describes the recommendations they offered and illustrates each with real-world examples from across the country