1,720,958 research outputs found
The Categories of Graphs
Full text linkIn traditional studies of graph theory, the graphs allow only one edge to be incident to any two vertices, not necessarily distinct, and the graph morphisms must map edges to edges and vertices to vertices while preserving incidence. We refer to these restricted morphisms as strict morphisms. We relax the conditions on the graphs by allowing any number of edges to be incident to any two vertices, as well as relaxing the condition on graph morphisms by allowing edges to be mapped to vertices, provided that incidence is still preserved. We call the broader category of these graphs and these morphisms the Category of Conceptual Graphs and Graph Morphisms, denoted Grphs. We then define four other concrete categories of graphs created by combinations of restrictions of the graph morphisms as well as restrictions on the allowed graphs.
We determine the categorial structure of these six categories of graphs by characterizing common categorially defined structures and properties and by characterizing six special types of monomorphisms, and dually six special types of epimorphisms. We also establish the Fundamental Morphism Theorem in two of the categories of graphs.
We then provide an Elementary Theory for five categories of graphs, producing a list of first-order axioms that, when taken with the higher-order axiom of the existence of small products and coproducts, characterizes these five categories of graphs. We also provide a result toward Hedetniemi\u27s conjecture that arose from the study of the categories of graphs
A special Case of Hedetniemi\u27s Conjecture
No full textIn 1966 S. Hedetniemi conjectured that the chromatic number of the categorial product of two graphs with finite chromatic number would be the minimum of the chromatic number of the two graphs. In 1985 A. Hajnal found two graphs with an uncountable chromatic number that when taken together in the categorial product, produce a graph with a countable chromatic number. However Hedetniemi\u27s conjecture remains open today.
In this talk, we will explore a portion of the results and techniques research into this conjecture has created as well as my own research into a new technique that provides a special case of the conjecture
The Fundamental Morphism Theorem in the Categories of Graphs
No full textIn the usual Category of Graphs, the graphs allow only one edge to be incident to any two vertices, not necessarily distinct. Also the usual graph morphisms must map edges to edges and vertices to vertices while preserving incidence. We refer to these morphisms as strict morphisms. We relax the condition on the graphs allowing any number of edges to be incident to any two vertices, as well as relaxing the condition on graph morphisms by allowing edges to be mapped to vertices, provided that incidence is still preserved. We call this broader graph category The Category of Conceptual Graphs, and define four other graph categories created by combinations of restrictions of the graph morphisms as well as restrictions on the allowed graphs.
In 1927 Emmy Noether proved three important theorems concerning the behavior of morphisms between groups. These three theorems are called Noether Isomorphism Theorems, and they have been found to apply to morphisms for many mathematical structures. The first of the Noether Isomorphism Theorems when generalized is called the Fundamental Morphism Theorem, and the Noether Isomorphism Theorems follow as corollaries to the Fundamental Morphism Theorem. We investigate how the Fundamental Morphism Theorem applies to these categories of graphs
Topos-like Properties in Two Categories of Graphs and Graph-like Features in an Abstract Category
No full textOften in the study of graph theory, the usual notion of a graph is that of a simple graph with at most one edge between vertices, and at most one loop on any vertex (some say no loops). The usual notion of a graph homomorphism is a mapping of graphs that sends vertices to vertices, edges to edges, and preserves incidence of the mapped vertices and edges. A more general view is to create a category of objects and morphisms that allows the graphs (i.e. the objects) to have multiple edges between two vertices and multiple loops at a vertex, coupled with more general graph homomorphism (i.e. the morphisms) that allows edges to be mapped to vertices as long as that map still preserves incidence, and of course, still maps vertices to vertices. Here this more general category of graphs is named theCategory of Conceptual Graphs.
We investigate three topos defining properties of two subcategories of the Category of Conceptual Graphs. The first subcategory is the Category of Simple Loopless Graphs with Strict Morphisms in which the graphs are simple and loopless and the morphisms are restricted to only sending edges to edges (i.e. strictly), and the second subcategory is the Category of Simple Graphs with Strict Morphisms where at most one loop is allowed on a vertex. We find that these two graph categories have only a few topos-like properties. We also define some small graph-like objects in an abstract category that are their graph counterparts when viewed in any of the concrete categories of graphs. We also study these graph-like objects in some other familiar (concrete) categories, e.g. the Category of Abelian Groups and Homomorphisms and the Category of Sets and Functions
The Geography of Health: Rethinking Value-Based Payments
No full textGeographic location—specifically ZIP code—profoundly influences health outcomes and healthcare costs, with research suggesting up to 60% of a person\u27s health status is determined by social and environmental conditions associated with where they live. This paper examines the significant disparities between communities, revealing how some areas experience nearly three times the emergency department utilization rates of others, resulting in cost differentials exceeding 50%. The research proposes a comprehensive framework for integrating Social Determinants of Health (SDOH) into value-based payment contracts through five key mechanisms: SDOH-adjusted risk models, informed cost benchmarking, provider incentive restructuring, alternative payment model enhancements, and refined attribution methodologies. While valuable data sources exist to support geographic analysis, the challenge lies in effectively integrating diverse sources into actionable insights—a process requiring substantial investment and specialized expertise. As healthcare transitions to value-based models, organizations that successfully leverage geographic insights will be better positioned to address social needs proactively, ultimately building a more equitable and effective healthcare system that delivers value across all communities. This geographic perspective represents not just an analytical approach but a fundamental shift in how healthcare organizations understand and address population health needs
Topos-like Properties in Two Categories of Graphs and Graph-like Features in an Abstract Category
Full text linkIn the study of the Category of Graphs, the usual notion of a graph is that of a simple graph with at most one loop on any vertex, and the usual notion of a graph homomorphism is a mapping of graphs that sends vertices to vertices, edges to edges, and preserves incidence of the mapped vertices and edges. A more general view is to create a category of graphs that allows graphs to have multiple edges between two vertices and multiple loops at a vertex, coupled with a more general graph homomorphism that allows edges to be mapped to vertices as long as that map still preserves incidence. This more general category of graphs is named the Category of Conceptual Graphs. We investigate topos and topos-like properties of two subcategories of the Category of Conceptual Graphs. The first subcategory is the Category of Simple Loopless Graphs with Strict Morphisms in which the graphs are simple and loopless and the incidence preserving morphisms are restricted to sending edges to edges, and the second subcategory is the Category of Simple Graphs with Strict Morphisms where at most one loop is allowed on a vertex. We also define graph objects that are their graph equivalents when viewed in any of the graph categories, and mimic their graph equivalents when they are in other categories. We conclude by investigating the possible reflective and corefective aspects of our two subcategories of graphs
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
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