The B-G News April 9, 1962

The BGSU campus student newspaper April 9, 1962. Volume 46 - Issue 46https://scholarworks.bgsu.edu/bg-news/2655/thumbnail.jp

Variations on a Theme: Graph Homomorphisms

This thesis investigates three areas of the theory of graph homomorphisms: cores of graphs, the homomorphism order, and quantum homomorphisms. A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that, for vertex transitive graphs, the size of the core must divide the size of the graph. This motivates the following question: when can the vertex set of a vertex transitive graph be partitioned into sets which each induce a copy of its core? We show that normal Cayley graphs and vertex transitive graphs with cores half their size always admit such partitions. We also show that the vertex sets of vertex transitive graphs with cores less than half their size do not, in general, have such partitions. Next we examine the restriction of the homomorphism order of graphs to line graphs. Our main focus is in comparing this restriction to the whole order. The primary tool we use in our investigation is that, as a consequence of Vizing's theorem, this partial order can be partitioned into intervals which can then be studied independently. We denote the line graph of X by L(X). We show that for all n ≥ 2, for any line graph Y strictly greater than the complete graph Kₙ, there exists a line graph X sitting strictly between Kₙ and Y. In contrast, we prove that there does not exist any connected line graph which sits strictly between L(Kₙ) and Kₙ, for n odd. We refer to this property as being ``n-maximal", and we show that any such line graph must be a core and the line graph of a regular graph of degree n. Finally, we introduce quantum homomorphisms as a generalization of, and framework for, quantum colorings. Using quantum homomorphisms, we are able to define several other quantum parameters in addition to the previously defined quantum chromatic number. We also define two other parameters, projective rank and projective packing number, which satisfy a reciprocal relationship similar to that of fractional chromatic number and independence number, and are closely related to quantum homomorphisms. Using the projective packing number, we show that there exists a quantum homomorphism from X to Y if and only if the quantum independence number of a certain product graph achieves |V(X)|. This parallels a well known classical result, and allows us to construct examples of graphs whose independence and quantum independence numbers differ. Most importantly, we show that if there exists a quantum homomorphism from a graph X to a graph Y, then ϑ̄(X) ≤ ϑ̄(Y), where ϑ̄ denotes the Lovász theta function of the complement. We prove similar monotonicity results for projective rank and the projective packing number of the complement, as well as for two variants of ϑ̄. These immediately imply that all of these parameters lie between the quantum clique and quantum chromatic numbers, in particular yielding a quantum analog of the well known ``sandwich theorem". We also briefly investigate the quantum homomorphism order of graphs

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Daily Eastern News: November 02, 1966

https://thekeep.eiu.edu/den_1966_nov/1000/thumbnail.jp

Toolkit Volume II: A Case for Dental Academic/Community Partnerships for Leadership and Diversity

The ADEA/W.K. Kellogg Foundation Minority Dental Faculty and Inclusion (MDFDI) grant (2015–2017) built on lessons learned and best practices from dental and allied dental pilots in the previous WKKF grants as the focus continued to be leadership, academic/community partnerships and increased diversity in the dental workforce. The resulting toolkit, available in two volumes, focuses on developing dental and allied dental teams and strengthening the pool of underrepresented minority dental health professionals. The toolkit builds on institutional models, the convenings, lessons learned and best practices that have helped change institutional climates

Using the Clinical Interview as a Complementary Assessment for Minority Elementary Students to Determine Their In-depth Understanding of Mathematical Concepts

While some African American students perform as well as or better than their White peers on standardized tests, African Americans as a group attain lower scores on standardized tests than their White peers. This phenomenon has been addressed extensively in educational research. However, not much empirical research has been conducted to investigate whether a complementary assessment, such as the clinical interview, would provide more information about African American students’ mathematical knowledge than a standardized test. Qualitative clinical interview methodology was used to explore the performance of the student participants on the clinical interviews and on the standardized test as well as teacher feedback about these students’ mathematical knowledge. Data were gathered from three main sources: clinical interviews, standardized test results, and teacher interviews. Data were coded to identify common themes that shed light on the research questions. Most of the students performed better on specific items from the standardized assessment, the New Jersey Proficiency Assessment of State Standards (NJPASS), than on the clinical interview items. The scores on both assessments revealed much disparity among participants’ mathematical competencies. The clinical interviews affirmed or changed the teachers’ opinions of their students. The information from the clinical interviews fostered discussion of pedagogical practices

The application of visualization methods to educational data sets with inspiration from statistical and fluid mechanics

textThis dissertation focuses on the development of visualization methods that enable us to examine longitudinal data in a unique way. We take inspiration from statistical and fluid mechanics to represent our data as a "flow" through time. Our visualizations represent vector fields (or flow plots), streamlines, and trajectories, and they are constructed in a similar manner to how one might analyze the aggregate motion of particles in a fluid. However, the subject of our research extends beyond ordinary fluid mechanics. We will use our visualizations to examine statewide standardized test scores in Texas from 2003 to 2011. The nature of the data makes it a perfect match for our methodology, since students' test scores tend to change over time in a semi-deterministic but nonlinear manner. Furthermore, our methods represent a departure from the standard ways of analyzing educational data. By visualizing the changes in students' test scores over a nine-year period, we discovered that our flow plots were changing with the eventual graduating class of 2012. The change in our visualizations was caused by an educational policy known as the Student Success Initiative, or SSI. The policy forced students to pass their standardized tests in 5th and 8th grade, or risk being held back a grade. To help with this process, students who initially failed were given extra instruction and additional opportunities to take the test. SSI was implemented in such a way that it would affect the class of 2012 and beyond, although we did not know of the program's existence until our plots had been developed. SSI had a successful impact on the educational career of Texas students; a far greater percentage of students were able to pass the 5th and 8th grade standardized tests after SSI was implemented. The striking feature of SSI, however, is that it also significantly improved test scores in 6th, 7th, 9th, and 10th grade. Despite its success at improving test scores over many years and grades, the program was eventually defunded. This was partially due to an inability to construct a lengthy longitudinal analysis of the program's influence. Our methodology would have conclusively shown the effectiveness of the SSI policy. Despite the defunding of the SSI, I am confident our methodology can be extended to illustrate changes in other data systems. These systems may or may not be related to education; our code and techniques are designed to be as universal as possible. We will explore several extensions to other data sets at the end of this dissertation.Physic