The graph-theoretic approach to descriptive set theory

AbstractFor many species, multiple maps are available, often constructed independently by different research groups using different sets of markers and different source material. Integration of these maps provides a higher density of markers and greater genome coverage than is possible using a single study. In this article, we describe a novel approach to comparing and integrating maps by using abstract graphs. A map is modeled as a directed graph in which nodes represent mapped markers and edges define the order of adjacent markers. Independently constructed graphs representing corresponding maps from different studies are merged on the basis of their common loci. Absence of a path between two nodes indicates that their order is undetermined. A cycle indicates inconsistency among the mapping studies with regard to the order of the loci involved. The integrated graph thus produced represents a complete picture of all of the mapping studies that comprise it, including all of the ambiguities and inconsistencies among them. The objective of this representation is to guide additional research aimed at interpreting these ambiguities and inconsistencies in locus order rather than presenting a “consensus order” that ignores these problems.

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Study of the Concrete Production Process -A graph theoretic approach

Journal of Physics Conference Series ◽

10.1088/1742-6596/1706/1/012115 ◽

2020 ◽

Vol 1706 ◽

pp. 012115

Author(s):

P Sangeetha ◽

M Shanmugapriya ◽

R Sundareswaran ◽

K Sowmya ◽

S Srinidhi

Keyword(s):

Production Process ◽

Theoretic Approach ◽

Graph Theoretic ◽

Graph Theoretic Approach ◽

Concrete Production

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A graph theoretic approach to ultrafast information distribution: Borel Cayley graph resizing algorithm

Computer Communications ◽

10.1016/j.comcom.2010.07.013 ◽

2010 ◽

Vol 33 (17) ◽

pp. 2093-2104 ◽

Cited By ~ 14

Author(s):

Jaewook Yu ◽

Eric Noel ◽

K. Wendy Tang

Keyword(s):

Cayley Graph ◽

Theoretic Approach ◽

Information Distribution ◽

Graph Theoretic ◽

Graph Theoretic Approach

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Two Applications of Inner Model Theory to the Study of Sets

Bulletin of Symbolic Logic ◽

10.2307/421049 ◽

1996 ◽

Vol 2 (1) ◽

pp. 94-107 ◽

Cited By ~ 3

Author(s):

Greg Hjorth

Keyword(s):

Los Angeles ◽

Set Theory ◽

Model Theory ◽

Large Cardinals ◽

Descriptive Set Theory ◽

Inner Model Theory ◽

Regular Cardinals ◽

Inner Model ◽

Definable Sets ◽

Descriptive Set

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely descriptive set theoretic arguments, may at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL(ℝ) are measurable. John Steel's analysis also settled a number of structural questions regarding HODL(ℝ), such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals.

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