Incompressible surfaces in handlebodies

AbstractLet F be a compact surface and let I be the unit interval. This paper gives a standard form for all 2-sided incompressible surfaces in the 3-manifold F×I. It also supplies a simple sufficient condition for when 2-sided surfaces in this form are incompressible. Since F×I is a handlebody when F has boundary, the paper applies to incompressible surfaces in handlebodies

A Causal Comparative Study of The Difference in Achievement Scores of At-Risk, Minority Students Based on Learning Styles

The American education system has historically promoted free and equal education for all, but history has shown that ethnic minority students’ academic achievement has been lower than ethnic majority students’ academic achievement. The resulting gap in achievement has been addressed throughout the research, but definitive solutions remain elusive. The study addressed the lack of research surrounding cultural learning patterns of at-risk students. The purpose of this quantitative, casual comparative study is to determine the difference among annual state mandated test scores of at-risk students who have visual, auditory, read/write or kinesthetic (VARK) learning styles. The participants for the study were drawn from a convenience sample of students located in a northeastern state during the spring semester of the 2020-2021 school year. The assessment was administered to students who were in grades six, seven and eight during the 2020-2021 school year to determine preferred learning styles. The researcher also used 2020-2021 Northwest Evaluation Association (NWEA) results to determine the differences among annual, state-mandated test scores of at-risk students who have VARK learning styles. The data were analyzed from each of the visual, auditory, read/write, and kinesthetic learning styles profiles. A one-way analysis of variance (ANOVA) was used to analyze the samples. The research shows that students benefit from increased teacher/student connections, faith-baesd partnerships, and increased awareness of cultural influences on learning. The data supports implementing tiered supports to address absenteeism and student disengagement

A Case Study of the Relationship between Role-taking Skills and Intellectual Development

A series of thirteen role-taking and cognitive development tasks, taken from research done on Piaget\u27s developmental theories, were presented to ascertain the relationship between refined role-taking skills and the level of intellectual development demonstrated by a child in Piaget\u27s pre-operational phase (two to seven years). The results indicated that a child in this age range is not able to remove himself from his egocentric position and utilize the concepts of decentering and reversibility in constructing a meaningful message. This would indicate a lack of role-taking skill refinement. It was concluded that the use of this type of task series would aid a therapist in determining a beginning level of therapy that was meaningful to the client

Student demographic characteristics and how they relate to student achievement

AbstractIn higher education, student performance in many classes still is not at the expected level. Numerous research has been conducted on how test scores and demographic characteristics correlate to academic performance. The purpose of this study is to determine what factors are predictors of success. Therefore the study will look at college COMPASS placement and exit scores, SAT scores, and high school GPA. A demographic inventory will be used to depict the background education of the students. The study will aim to establish correlations between the test scores, the demographic characteristics of the students, and end of semester grades. © 2011 Published by Elsevier Ltd

Polyhedra with hexagonal and triangular faces and three faces around each vertex

We analyze polyhedra composed of hexagons and triangles with three faces around each vertex, and their 3-regular planar graphs of edges and vertices, which we call "trihexes". Trihexes are analogous to fullerenes, which are 3-regular planar graphs whose faces are all hexagons and pentagons. Every trihex can be represented as the quotient of a hexagonal tiling of the plane under a group of isometries generated by $180^\circ$ rotations. Every trihex can also be described with either one or three "signatures": triples of numbers $(s, b, f)$ that describe the arrangement of the rotocenters of these rotations. Simple arithmetic rules relate the three signatures that describe the same trihex. We obtain a bijection between trihexes and equivalence classes of signatures as defined by these rules. Labeling trihexes with signatures allows us to put bounds on the number of trihexes for a given number vertices $v$ in terms of the prime factorization of $v$ and to prove a conjecture concerning trihexes that have no "belts" of hexagons.Comment: 26 pages, 19 figure