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Encyclopedia > Characterization (mathematics)

In the jargon of mathematics, the statement that "Property P characterizes object X" means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as "Property Q characterises Y up to isomorphism". The first type of statement says in different words that the extension of P is a singleton set. The second says that the extension of Q is a single equivalence class (for isomorphism, in the given example — depending on how up to is being used, some other equivalence relation might be involved). Math sucks. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich. ... The extension of an idea or (linguistic) expression consists of the things that it applies to; it contrasts with intension. ... In mathematics, a singleton is a set with exactly one element. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...


Examples

  • "According to Bohr-Mollerup theorem, among all functions f such that f(1) = 1 and x f(x) = f(x + 1) for x > 0, log-convexity characterizes the gamma function." This means that among all such functions, the gamma function is the only one that is log-convex. (A function f is log-convex iff log(f) is a convex. The base of the logarithm does not matter as long as it is more than 1, but conventionally mathematicians take "log" with no subscript to mean the natural logarithm, whose base is e.)

  Results from FactBites:
 
Characterization - Wikipedia, the free encyclopedia (260 words)
Characterization is the process of creating characters in fiction, often those who are different from and have different beliefs than the author.
Thorough characterization makes characters well-rounded and complex even though the writer may not be like the character or share his or her attitudes and beliefs.
Characterization can involve developing a variety of aspects of a character, such as appearance, age, gender, educational level, vocation or occupation, financial status, marital status, hobbies, religious beliefs, ambitions, motivations, etc. Often these can be shown through the actions and language of the character, rather than by telling the reader directly.
Insights Into Algebra 1 . Workshop 6 (2885 words)
In the mathematics classroom, the affective domain is concerned with students' perception of mathematics, their feelings toward solving problems, and their attitudes about school and education in general.
Mathematics students, particularly in the middle grades and high school, can do their part by engaging seriously with the material and striving to make mathematical connections that will support their learning.
No mathematics classroom is free of the question "When are we ever going to use this?" Students ask this question all the time, and unless we are able to provide acceptable answers, students may believe that mathematics has no use in their lives.
  More results at FactBites »

 
 

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