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A map is a function, as in the association of any of the four colored shapes in X to its color in Y. In mathematics, a map or mapping is a function in its general sense. [1] These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper. [2]
Later in the book, but fitting thematically into this part, [1] [4] chapter 9 covers map projections. [3] Moving from geodesy to visualization, [1] chapters 4 and 5 concern the use of color and scale on maps. Chapter 6 concerns the types of data to be visualized, and the types of visualizations that can be made for them.
Cartographic generalization, or map generalization, includes all changes in a map that are made when one derives a smaller-scale map from a larger-scale map or map data. It is a core part of cartographic design .
An animated cobweb diagram of the logistic map = (), showing chaotic behaviour for most values of >. A cobweb plot , known also as Lémeray Diagram or Verhulst diagram is a visual tool used in the dynamical systems field of mathematics to investigate the qualitative behaviour of one-dimensional iterated functions , such as the logistic map .
The data from a study can also be analyzed to consider secondary hypotheses inspired by the initial results, or to suggest new studies. A secondary analysis of the data from a planned study uses tools from data analysis, and the process of doing this is mathematical statistics. Data analysis is divided into:
Mathematical visualization is used throughout mathematics, particularly in the fields of geometry and analysis. Notable examples include plane curves , space curves , polyhedra , ordinary differential equations , partial differential equations (particularly numerical solutions, as in fluid dynamics or minimal surfaces such as soap films ...
It does this by representing data as points in a low-dimensional Euclidean space. The procedure thus appears to be the counterpart of principal component analysis for categorical data. [citation needed] MCA can be viewed as an extension of simple correspondence analysis (CA) in that it is applicable to a large set of categorical variables.
An approach to connections which makes direct use of the notion of transport of "data" (whatever that may be) is the Ehresmann connection. The most abstract approach may be that suggested by Alexander Grothendieck , where a Grothendieck connection is seen as descent data from infinitesimal neighbourhoods of the diagonal ; see ( Osserman 2004 ).