Where does transformation collide?

A rule that assigns to every point of a space another point, such as a rotation, reflection, or the linear map carried by a matrix: applied to a whole figure it moves or reshapes it while following fixed algebraic laws, and the transformations that preserve a chosen property compose into a group. Klein's 1893 A comparative review of recent researches in geometry, the English rendering of his Erlangen program, recast each geometry as the study of the properties left unchanged by a particular group of transformations. Here the referent is a function acting on a set of points, defined by the structure it preserves, not the uptake of foreign DNA by a cell, a rule rewriting a sentence, or an operation rescaling a column of data.[1]

Unit

A single map from a space to itself, given by its rule or matrix

Recovery means

An invertible transformation has an inverse map that returns each point to its origin

Measured by

Specified by a matrix, a formula, or its action on basis points, and classified by the properties and group it preserves

The process by which a cell takes up naked DNA from its surroundings and incorporates it, so that a heritable trait carried by the foreign DNA becomes part of the cell's own genotype. Griffith's 1928 The significance of pneumococcal types reported that a harmless pneumococcus could be permanently changed into a virulent one by something released from dead virulent cells, the observation later shown to be uptake of DNA and named the transforming principle. Here the referent is a physical event inside a living cell that changes what it inherits, not a map acting on points of a space, a syntactic rule, or a mathematical rescaling of measurements.[2]

Unit

One cell's uptake and inheritance of external DNA

Recovery means

Not applicable

Measured by

Quantified as transformation efficiency, the number of transformed cells obtained per unit of DNA added

A formal rule that relates one syntactic structure to another, for example the rule that ties an active sentence to its passive counterpart, so that a grammar can derive many surface forms from shared underlying structure. Harris's 1957 Co-occurrence and transformation in linguistic structure introduced the term for such structure-to-structure operations, and it became the core device of the transformational grammar developed soon after. Here the referent is a rule operating on sentence structure within a grammar, not the uptake of DNA by a cell, a geometric mapping of a space, or an arithmetic reshaping of a variable.[3]

Unit

One rule rewriting one syntactic structure as another

Recovery means

Not applicable

Measured by

Stated as an explicit rule in a grammar and tested against which sentence pairs speakers accept as related

A mathematical operation applied to every value of a variable to change its distribution or scale, such as taking the logarithm or square root of each observation so the data better meet the assumptions of an analysis. Box and Cox's 1964 An analysis of transformations set out a family of power transformations and a way to estimate which one best makes a variable's spread constant and its shape closer to normal. Here the referent is an operation reshaping a set of measured numbers, chosen to fit a statistical model, not a cell taking up DNA, a geometric map of a space, or a rule rewriting a sentence.[4]

Unit

One function applied uniformly to every value of a variable

Recovery means

A monotone transformation can be inverted to return the fitted results to the original scale

Measured by

Chosen by a criterion such as maximum likelihood over a family of transformations and checked by residual diagnostics

In peace and conflict studies, transformation refers to conflict transformation, changing the relationships, structures and cultures that produce a conflict rather than only settling the dispute. This meaning comes from Lederach (1997) in Building peace.[5]

Unit

Recovery means

Measured by