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The success of representation theory has led to numerous generalizations. One of the most general is in category theory. [15] The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces. [4]
Hasty generalization is the fallacy of examining just one or very few examples or studying a single case and generalizing that to be representative of the whole class of objects or phenomena. The opposite, slothful induction , is the fallacy of denying the logical conclusion of an inductive argument, dismissing an effect as "just a coincidence ...
Inductive reasoning is any of various methods of reasoning in which broad generalizations or principles are derived from a body of observations. [1] [2] This article is concerned with the inductive reasoning other than deductive reasoning (such as mathematical induction), where the conclusion of a deductive argument is certain given the premises are correct; in contrast, the truth of the ...
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. [1] Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model ).
It is also called a representative function. [1] They arise naturally from finite-dimensional representations of G as the matrix -entry functions of the corresponding matrix representations. The Peter–Weyl theorem says that the matrix coefficients on G are dense in the Hilbert space of square-integrable functions on G .
The general theory for Lie groups deals with semidirect products of the two types, by means of general results called Mackey theory, which is a generalization of Wigner's classification methods. Representation theory also depends heavily on the type of vector space on which the group acts. One distinguishes between finite-dimensional ...
Therefore, generalization is a valuable and integral part of learning and everyday life. Generalization is shown to have implications on the use of the spacing effect in educational settings. [13] In the past, it was thought that the information forgotten between periods of learning when implementing spaced presentation inhibited generalization ...
When such an element is chosen, it is called a representative of the class. The choice of a representative in each class defines an injection from X / R {\displaystyle X/R} to X . Since its composition with the canonical surjection is the identity of X / R , {\displaystyle X/R,} such an injection is called a section , when using the terminology ...