Bounds on the radius of the p-adic Mandelbrot set

Let f be a degree d polynomial defined over the nonarchimedean field C_p, normalized so f is monic and f(0)=0. We say f is post-critically bounded, or PCB, if all of its critical points have bounded orbit under iteration of f. It is known that if p is greater than or equal to d and f is PCB, then all critical points of f have p-adic absolute value less than or equal to 1. We give a similar result for primes between d/2 and d. We also explore a one-parameter family of cubic polynomials over the 2-adic numbers to illustrate that the p-adic Mandelbrot set can be quite complicated when p is less than d, in contrast with the simple and well-understood p > d case

The Public Education Network Study of LEF Leadership: Report on Baseline Survey Findings

Many nonprofit organizations seek to make change. To that end, much needed "capital" -- variously described as social, public, professional, and human -- is being brought to bear upon pressing social issues. Researchers across the country are attempting to understand how these resources are being generated, deployed, and administered, and to what avail. Of particular interest here are local education funds (LEFs) and their leaders. LEFs are a set of voluntary, intermediate, and mission-driven organizations, conceived by the Ford Foundation in 1983, which sit strategically at the nexus of educational and civic capacity building. This report provides the results of the first phase of the Public Education Network (PEN) leadership study, a baseline survey administered to 59 LEF executive directors.The survey had two purposes: to provide a snapshot of leadership characteristics and perceptions of executive directors -- information never before compiled and examined -- and to gather contextual information on the LEFs and the communities they serve. The results will be used as a foundation for subsequent research on LEF leadership. In time, the findings from this research will assist PEN in its efforts to nurture and sustain LEF leadership

Leading Ways: Preliminary Research on LEF Leadership for the Public Education Network

In the early 1980s, public schools, particularly in urban areas, were struggling with changing demographics and a need for greater community commitment to public education. Small independent community-based organizations, called local education funds (LEFs), were first established at this time by community leaders to bridge the gap between communities and their schools. Twenty years later, the Public Education Network (PEN), the national organization of more than 70 of the country's LEFs, is championing LEFs' unique contributions to educational reform and planning new methods to continue to support their work. One important aspect of PEN's efforts is the development of a research agenda around LEF leadership and effectiveness.The Urban Institute has supported the emerging research on LEF leadership in several ways. First, in 2001, researchers administered and analyzed results of a survey of all current LEF executive directors. The results provided a snapshot of leadership characteristics and attitudes. Second, in 2002, researchers reviewed existing literature on LEFs and nonprofit leadership and interviewed several individuals about the founding of LEFs. Third, also in 2002, researchers conducted 60-to 90-minute telephone interviews with 10 founding directors of early LEFs.This report describes efforts by the Urban Institute and PEN to better understand and describe leadership in LEFs. The research conducted to date reveals the following common features of LEF leadership:LEF leaders create a space for sustained collaboration in communities. Founders, in particular, see their work as establishing unique and lasting forums in their communities.LEF leadership fits well in an adaptive leadership framework, in which the director educates him or herself about community values and exposes issues that must be resolved through collaborative action. Adaptive leaders mobilize others around a community consensus rather than dictate a solution. In the case of LEFs, developing key relationships contributes to successful leadership.These relationships involve community members, principals, government officials, local businesses, and other stakeholders. Because LEFs are meant to provide a permanent venue for collaboration, these relationships are not only instrumental but also ends in themselves.Although LEF leaders are extremely invested in their communities in terms of socioeconomic status and race, they are not typically representative of the communities their LEFs serve. This gap in constituency and representation leads to uncertainty about whether LEFs effectively engage all segments of the community in their work. More inclusive leadership, then, is an issue worth exploring.LEF leadership appears to involve making strategic choices in an environment of constraints. With limited human and financial resources, LEFs must attempt to achieve their goals through a complex interaction with their communities and schools. Key stages in LEF development include the shift to greater community engagement and systemic reform. However, not all LEFs follow the same developmental pattern

The Role of the Vertical Leader in Shared Leadership

The purpose of this quantitative, positivistic study is to investigate the unique roles, actions, and behaviors of vertical team leaders that lead to the emergence of shared leadership, effectiveness, and performance in work teams in organizations. The correlational study design evaluated critical leadership functions relative to a 20-item shared leadership inventory. The study included 34 team leaders and 101 team members associated with primarily Midwestern organizations. The findings revealed that all leadership functions can be shared to a certain extent, but the leadership function of providing feedback was notably less shared than other leader functions. In addition, not only can functional leadership participation be a predictor of shared leadership, but also, this research has established new reliability and validity of the he Team Leadership Questionnaire (TLQ). Other findings from this sample indicate that functional leadership can be a predictor of shared leadership and when leadership functions related to planning and initiating (transition phase) are more shared, then the execution functions (action phase) are also more shared. The transition function predicts the outcome of shared leadership more strongly. Finally, there is a strong correlation with perceived leader effectiveness and leadership satisfaction with shared leadership in this study. This research provides new insights for creating an environment that better supports shared leadership and challenges some traditionally held norms of the unique role of the vertical leader

Parents and Preschool Children Interacting with Storybooks: Children’s Early Literacy Achievement

This research reports on one area of a larger study in Western Canada examining the literacy activities of families from culturally diverse backgrounds. The research focused on parents’ interactions with preschool children in storybook sharing and children’s emergent reading development as measured by the Test of Early Reading Ability-2 (TERA-2). The sample consisted of 35 parents and children. Parents’ and children’s interactions in storybook sharing were videotaped and coded using a modified scale by Shapiro, Anderson, and Anderson (1997). Relationships were found between parents’ and children’s interactions in storybook reading and children’s early literacy achievement

Fathers\u27 and Mothers\u27 Book Selection Preferences for Their Four Year Old Children Abstract

Twelve fathers and 12 mothers of four year olds were presented with 14 children\u27s books representing various genres and were asked to select the five books they would read to their children in the coming week and to give reasons for their selections. They were then asked to identify those books they would not select and to provide reasons. There were some differences between mothers\u27/fathers\u27 book selection and some differences between selecting for sons/daughters. Similar differences were noted in terms of those books which parents would not select

Local Arboreal Representations

Let K be a field complete with respect to a discrete valuation v of residue characteristic p. Let f(z)∈K[z] be a separable polynomial of the form z[superscript ℓ]−c. Given a∈K, we examine the Galois groups and ramification groups of the extensions of K generated by the solutions to fn(z)=a. The behavior depends upon v(c), and we find that it shifts dramatically as v(c) crosses a certain value: 0 in the case p∤ℓ, and −p/(p−1) in the case p=ℓ