Planar graphs : a historical perspective.

The field of graph theory has been indubitably influenced by the study of planar graphs. This thesis, consisting of five chapters, is a historical account of the origins and development of concepts pertaining to planar graphs and their applications. The first chapter serves as an introduction to the history of graph theory, including early studies of graph theory tools such as paths, circuits, and trees. The second chapter pertains to the relationship between polyhedra and planar graphs, specifically the result of Euler concerning the number of vertices, edges, and faces of a polyhedron. Counterexamples and generalizations of Euler\u27s formula are also discussed. Chapter III describes the background in recreational mathematics of the graphs of K5 and K3,3 and their importance to the first characterization of planar graphs by Kuratowski. Further characterizations of planar graphs by Whitney, Wagner, and MacLane are also addressed. The focus of Chapter IV is the history and eventual proof of the four-color theorem, although it also includes a discussion of generalizations involving coloring maps on surfaces of higher genus. The final chapter gives a number of measurements of a graph\u27s closeness to planarity, including the concepts of crossing number, thickness, splitting number, and coarseness. The chapter conclused with a discussion of two other coloring problems - Heawood\u27s empire problem and Ringel\u27s earth-moon problem

Livestock-Influenced Water Quality Risk Assessment Tool

This article describes a livestock-influenced water quality risk assessment tool that was developed to assist livestock producers with conducting a self-assessment of their operation and management relative to a facility\u27s risk of negatively affecting water quality. The tool focuses on factors likely to influence designation of the operation as a Concentrated Animal Feeding Operation by a permitting authority and was also designed to be used in cooperation with a technical service provider to make a site-specific assessment. The tool is available in paper format and an interactive Microsoft ExcelTM spreadsheet version

Ethanol suppression of peripheral blood mononuclear cell trafficking across brain endothelial cells in immunodeficiency virus infection

Earlier studies suggested that the combination of alcohol use and immunodeficiency virus infection resulted in more severe neurologic disease than either condition individually. These deleterious interactions could be due to increased immune cell and virus trafficking or may result from interactions between ethanol and human immunodeficiency virus (HIV)-associated toxicity within the brain. To determine the extent to which increased trafficking played a role, we examined the effect of ethanol on the migration of different peripheral blood mononuclear cell (PBMCs) subsets across a brain endothelial cell monolayer. We utilized combinations of feline brain endothelial cells with astrocytes, and/or microglia with either acute exposure to 0.08 g/dL ethanol, a combination of ethanol and feline immunodeficiency virus (FIV), or FIV alone. Adherence of PBMCs to endothelium was increased in all combinations of cells with the addition of ethanol. Despite increased PBMC adhesion with ethanol treatment, transmigration of B cells, monocytes, CD4 T cells and CD8 T cells was not increased and was actually decreased in the presence of astrocytes. Expression of three common adhesion molecules, intercellular adhesion molecule-1 (ICAM1), ICAM2, and vascular cell adhesion molecule, was unchanged or slightly decreased by ethanol. This indicated that although adherence is increased by ethanol it is not due to an increased expression of adhesion molecules. RANTES, MIP1α, MIP1β, and MCP-1 mRNA expression was also studied in brain endothelial cells, astrocytes and microglia by reverse transcriptase-polymerase chain reaction. Ethanol treatment of astrocytes resulted in modest changes of message while FIV caused 7–92-fold increases. The combination of ethanol and FIV reversed the large increase in RANTES and MIP1α message in astrocytes but increased MIP1β and MCP to 20–38-fold over control cells. Thus, modest concentrations of alcohol do not directly influence immune cell trafficking at the endothelium but may exert more complex effects on chemokine expression from astrocytes when combined with FIV

Secondary Pre-Service Teachers’ Algebraic Reasoning About Linear Equation Solving

This study analyzes the responses of 12 secondary pre-service teachers on two tasks focused on reasoning when solving linear equations. By documenting the choices PSTs made while engaging in these tasks, we gain insight into how new teachers work mathematically, reason algebraically, communicate their thinking, and make pedagogical decisions. We will share qualitative results from our examination of teacher knowledge through pre-service teachers’ explanations, models, language, and conjectures about student thinking

Secondary Pre-Service Teachers’ Algebraic Reasoning About Linear Equation Solving

This study analyzes the responses of 12 secondary pre-service teachers on two tasks focused on reasoning when solving linear equations. By documenting the choices PSTs made while engaging in these tasks, we gain insight into how new teachers work mathematically, reason algebraically, communicate their thinking, and make pedagogical decisions. We will share qualitative results from our examination of teacher knowledge through pre-service teachers’ explanations, models, language, and conjectures about student thinking

Endothelial cell suppression of peripheral blood mononuclear cell trafficking in vitro during acute exposure to feline immunodeficiency virus

Trafficking of peripheral blood mononuclear cells (PBMCs) into the brain is a critical step in the initiation of human immunodeficiency virus (HIV)-associated central nervous system disease. To examine potential factors that control trafficking during the earliest stages of infection, PBMC transmigration across a cultured feline brain endothelial cell (BECs) monolayer was measured after selective exposure of various cell types to feline immunodeficiency virus (FIV). Infection of the PBMCs with FIV increased the trafficking of monocytes and CD4 and CD8 T cells. Additional exposure of the BECs to FIV suppressed mean monocyte, CD4 T cell, and CD8 T cell trafficking. B cell trafficking was unaltered by these changing conditions. Subsequent exposure of astrocytes or microglia to FIV altered transmigration of different PBMC subsets in different ways. Treated microglia compared with treated astrocytes decreased monocyte transmigration, whereas B cell transmigration was increased significantly. When both astrocytes and microglia were exposed to FIV, an increase in CD8 T cell transmigration relative to BECs alone, to BECs plus astrocytes, or to BECs plus microglia was demonstrated. Thus, initial exposure of PBMCs to FIV is sufficient to induce a general increase in trafficking, whereas initial exposure of endothelial cells to FIV tends to down-regulate this effect. Selectivity of trafficking of specific PBMC subsets is apparent only after exposure of cells of the central nervous system to FIV in co-culture with the endothelium

Transmigration of macrophages across the choroid plexus epithelium in response to the feline immunodeficiency virus

Although lentiviruses such as human, feline and simian immunodeficiency viruses (HIV, FIV, SIV) rapidly gain access to cerebrospinal fluid (CSF), the mechanisms that control this entry are not well understood. One possibility is that the virus may be carried into the brain by immune cells that traffic across the blood–CSF barrier in the choroid plexus. Since few studies have directly examined macrophage trafficking across the blood–CSF barrier, we established transwell and explant cultures of feline choroid plexus epithelium and measured trafficking in the presence or absence of FIV. Macrophages in co-culture with the epithelium showed significant proliferation and robust trafficking that was dependent on the presence of epithelium. Macrophage migration to the apical surface of the epithelium was particularly robust in the choroid plexus explants where 3-fold increases were seen over the first 24 h. Addition of FIV to the cultures greatly increased the number of surface macrophages without influencing replication. The epithelium in the transwell cultures was also permissive to PBMC trafficking, which increased from 17 to 26% of total cells after exposure to FIV. Thus, the choroid plexus epithelium supports trafficking of both macrophages and PBMCs. FIV significantly enhanced translocation of macrophages and T cells indicating that the choroid plexus epithelium is likely to be an active site of immune cell trafficking in response to infection

Compartmentalization and evolution of feline immunodeficiency virus between the central nervous system and periphery following intracerebroventricular or systemic inoculation

The emergence of distinct neuropathogenic strains resulting from the adaptation and the unique evolution of human immunodeficiency virus (HIV) in the brain may contribute to the development of HIV-induced neurological diseases. In this study, the authors tracked early changes in virus evolution and compartmentalization between peripheral tissues and the central nervous system (CNS) after intracerebroventricular (i.c.v.) or intraperitoneal (i.p.) inoculation of animals with cell-free feline immunodeficiency virus (FIV). Using the FIV-NCSU1 envelope V3–V4 heteroduplex tracking assay (HTA), the authors observed a rapid compartmentalization of envelope variants between the CNS and periphery. Animals receiving the i.c.v. inoculation showed two peaks of viral RNA in the cerebrospinal fluid (CSF) with very different HTA patterns. Compared to the initial viral peak in CSF, the second peak showed an increased compartmentalization from plasma, reduced viral diversity, and more divergence from the proviral DNA in peripheral blood mononuclear cells (PBMCs) and the choroid plexus. In contrast, changes in plasma over the same time period were small. Different animals harbored different FIV DNA genotypes with varied regional compartmentalization within the brain. These results demonstrated that the virus within the CNS experienced a relatively independent but variable evolution from the periphery. Initial penetration of virus into the CSF facilitated the development of brain-specific reservoirs and viral diversification within the CNS