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In rural, low socioeconomic public schools, the diversity of the student population challenges teachers beyond their means to meet the wide-ranging needs of students, and therefore creating opportunity gaps. The purpose of this study was to explore one district's leadership support of teachers to create an equitable learning environment for all students. The present study built on existing literature by discovering leadership practices and perceived necessary supports for closing achievement gaps in K-12 classrooms. Through interviews with district and school administrators, classroom teachers, and observations of leadership meetings, the research for this case study included data collection and analysis. From the interview questions and observations, six themes emerged. The themes revealed from the data collected and analysis for this qualitative study aligned with the components of the conceptual framework, a sequence of initiative requirements for equitable learning environments. Developing a sequence of initiatives among all levels of educators could benefit sustainment of an equitable learning environment in Rural ISD. Recommended for further research is professional development to both leadership and teachers about professional learning communities (PLCs). In addition, study of parent education programs and how relationships achieve more parental involvement requires further research. A final recommendation for collegiate education preparation programs should include parental involvement, differentiation, and small group instruction as a requirement for teacher development when planning instruction for equitable lessons resulting in student achievement.

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Food availability is a serious problem for some low-income neighborhoods. This study examines food access in Eatonville, Florida, a small town in Orlando, Florida. Eatonville was one of the first African American towns incorporated into the United States after emancipation. It is a low-income community with 25% of the overall population and 30% of children living below the poverty line. This study will examine the state of food availability through food store and resident surveys in hopes of diagnosing need in order to alleviate it. There are serious implications for residents of cities with inadequate access to nutritious, affordable food. Children living with unequal access will face many future disadvantages in education, employment, and health. These compounding problems lead to a cycle of poverty that can be alleviated with appropriate public policy measures and other neighborhood changes that address food access in low-income neighborhoods.

The purpose of this qualitative, single case study was to identify leadership practices and processes that contributed to the success of a single High Needs, Title 1 elementary school that has experienced acknowledged superior academic success, particularly for its minoritized and low socioeconomically disadvantaged students, as indicated by the school's success on the State of Texas Assessment of Academic Readiness (STAAR) tests. The study sought to explore the principal's and teachers' conception of leadership for success and to examine what effective practices and processes were employed at a high-needs, high-performing campus that accounted for its success. The study applied the culturally responsive school leadership (CRSL) framework to analyze the practices employed at the school. Results of the study, from interviews with the principal and a focus group of three teachers, indicated that not only did they apply the principles of the CRSL framework, which employed the critical self awareness, culturally responsive curricula and teacher preparation in a culturally responsive and inclusive school environment which engaged parents and students in community contexts, but they also enhanced their effectiveness through the provision of additional resources for teachers and students, with a heavy reliance on data to guide academic decisions.

To properly design the microwave components such as transmission lines, filters, capacitors, inductors, and many others, it is important to know the characteristics of the construction materials at microwave frequencies. One of the most reliable techniques in material characterization at microwave frequencies is the coplaner coupled stripline resonator technique. This technique is an enhancement to the classical stripline resonator technique. In this technique, the measured resonance frequency and quality factor of the resonator are used to determine the complex permitivity. One of the main problems in this technique is the proper modeling of the coupling gaps. In this dissertation we will introduce an accurate model of the coupling gap, which will shows that the capacitive behavior of the gap is not pure capacitive as known before, but it turns into more complex one at higher frequencies depending on the dimensions of the gap primarily. The second main problem is the limitation in the frequency range for accurate measurements. At higher frequencies, the coupling reaches its peak value for a given stripline resulting in excessive loading to the resonator and thus a lowered Q value. In this frequency range, measurement of the dielectric properties looses its accuracy because the lowered Q values which means inaccuracies in determining the resonant frequencies as well as great error in determining the Qc and Qd terms. In this dissertation, attempts to remedy this problem by introducing two different approaches to get an improved design for the coplaner coupled stripline resonator are presented. The first approach to optimize the design of the coplaner coupled stripline resonator is based on optimizing the dimensions of the coplaner coupled stripline resonator three sections (coplaner, transition region, and the center stripline). In the second approach, a reactive stub (via) is introduced in the coupling gap between the coplaner line and the center stripline. The added stub is designed to improve the Q values of the structure resonances. Simulations of different designs of the coplaner coupled stripline resonator using different stub dimensions are presented. Advantages and disadvantages of these designs as well as the solution to their resonance frequency shift problems are discussed as well. / Ph. D.

Esta tese objetiva apresentar duas abordagens inovadoras acerca de dois assuntos clássicos da teoria dos números: bases aditivas e prime gaps. O primeiro tópico é estudado de um ponto de vista topológico, com o intuito de oferecer uma visão abrangente e resultados gerais sobre o conjunto de todas as bases aditivas, não versando sobre uma base aditiva específica, como é de costume. Por meio da introdução de uma métrica, são apresentadas várias ferramentas topológicas que permitem tratar problemas de difícil ataque direto sobre bases aditivas por meio de argumentos indiretos sobre bases melhor comportadas e suficientemente próximas das originalmente consideradas. Já a contribuição sobre prime gaps é realizada utilizando ferramentas algébricas, no lugar das analíticas, como habitual. Por meio de técnicas oriundas tanto da teoria de Galois quanto da teoria algébrica dos números, é apresentado um estudo da conjectura de Firoozbakht, juntamente com algumas de suas consequências, caso ela venha a ser provada / This thesis aims to present two innovative approaches about two classical subjects on number theory: additive bases and prime gaps. The first topic is studied from a topological point of view, in order to offer a comprehensive treatment and general results on the set of all additive bases, not dealing with one specific additive basis, as usual. By the introduction of a metric, it\'s presented a variety of topological tools that allows to treat problems of difficult direct attack on additive bases through indirect arguments on bases better behaved and sufficiently close to the originally considered. The contribution on the prime gaps subject is performed by the use of algebraic tools instead of analytical methods, as usual. Utilizing techniques from Galois theory and algebraic number theory, it\'s presented a study of the Firoozbakht\'s conjecture, along with some of its consequences if the conjecture is proved

Esta tese objetiva apresentar duas abordagens inovadoras acerca de dois assuntos clássicos da teoria dos números: bases aditivas e prime gaps. O primeiro tópico é estudado de um ponto de vista topológico, com o intuito de oferecer uma visão abrangente e resultados gerais sobre o conjunto de todas as bases aditivas, não versando sobre uma base aditiva específica, como é de costume. Por meio da introdução de uma métrica, são apresentadas várias ferramentas topológicas que permitem tratar problemas de difícil ataque direto sobre bases aditivas por meio de argumentos indiretos sobre bases melhor comportadas e suficientemente próximas das originalmente consideradas. Já a contribuição sobre prime gaps é realizada utilizando ferramentas algébricas, no lugar das analíticas, como habitual. Por meio de técnicas oriundas tanto da teoria de Galois quanto da teoria algébrica dos números, é apresentado um estudo da conjectura de Firoozbakht, juntamente com algumas de suas consequências, caso ela venha a ser provada / This thesis aims to present two innovative approaches about two classical subjects on number theory: additive bases and prime gaps. The first topic is studied from a topological point of view, in order to offer a comprehensive treatment and general results on the set of all additive bases, not dealing with one specific additive basis, as usual. By the introduction of a metric, it\'s presented a variety of topological tools that allows to treat problems of difficult direct attack on additive bases through indirect arguments on bases better behaved and sufficiently close to the originally considered. The contribution on the prime gaps subject is performed by the use of algebraic tools instead of analytical methods, as usual. Utilizing techniques from Galois theory and algebraic number theory, it\'s presented a study of the Firoozbakht\'s conjecture, along with some of its consequences if the conjecture is proved

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The study, Mind the Gap between Grade Three and Four is written within the research fields of Mathematics Education. The research interests in the study concerns how to achieve a smooth transition between grade three and four in compulsory school. As a mathematics teacher I found out that the transition from one grade to another in compulsory school affects both the pupil and the mathematics teacher and not always in a satisfactory way. Formal meetings concerning handing over information about pupils are not sufficient, and there is a definite difference between what is said about the pupils during a formal meeting and what is said in more informal settings. Are their gaps that occur? The aim of this study is to listen to mathematics teachers´ and to illuminate their stories about handing over pupils from grade three and taking over pupils in grade four. I am searching for gaps in their stories. My goal is to safe guard the continuity concerning the changes of grades between grade three and four in compulsory school. The research questions are: What are the important factors for mathematics teacher´s concerning the transition between grade three and four? What themes and gaps concerning the transitions are revealed in their stories? The study is based on a theoretical framework made of; Gees (1999) theory about "language-in-use" and "the seven building blocks of language". Social constructions because the idea is to criticize and change the established order regarding the transition between grade three and grade four. I also made a choice to look at the difficulties concerning the transition as external factors and not as the shortcoming of the mathematics teachers´ therefore I use the critical perspective. Discourse is used because this study is about communication, both spoken and written. In order to investigate important factors for mathematics teachers concerning the transitions I interview six mathematics teachers. They are asked about the existent transition between grade three and four at their schools and what they think are important factors for transitions overall. I also videotape one formal meeting because when different methods are used it helps me verify 2 the outcome. The transcriptions from the interviews and the formal meeting are analyzed with help from the discourse analysis based on Gee (1999). The analysis/result shows themes that are important and signal about gaps that could be filled by further research. Themes revealed concerns: school organization, formal/informal meetings, teaching, assessment and special pedagogy. Gaps that are visible concern the teachers’ different educations, the school-web and the curriculum, time, relations, communication, assessment, feedback, teaching methods, teaching materials, and pupils in special needs. I argue that its´ necessary to create a structure and make guidelines for formal conferences concerning the transition between grade three and four to minimize the discontinuity. It should also be of interest that the gaps mentioned in this study are filled, and one way to begin could be to plan for collegial work between teachers’ in different grades.