What Are Inverse Transformations and Generating Functions?

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    Inverse Transformation
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Discussion Overview

The discussion revolves around the concepts of transformations, direct and inverse transformations, and generating functions. Participants explore definitions and relationships between these terms, as well as their applications in mathematics, particularly in linear algebra and combinatorics.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Meta-discussion

Main Points Raised

  • One participant defines a transformation as a function that maps x to y, specifically noting that in linear algebra, it often refers to linear transformations.
  • Another participant suggests that the terms "direct" and "inverse" transformations require more specification, as they could be applied to different functions depending on context.
  • A participant mentions that generating functions relate to sequences of numbers and their Taylor series, but questions their connection to transformations.
  • One participant recommends the book "Generatingfunctionology" by Wilf for understanding generating functions, while expressing concerns about certain concepts within it.
  • Another participant suggests that the inquiry about transformations indicates a lack of "mathematical maturity," implying that the original poster may benefit from more foundational texts.
  • There is a suggestion to read textbooks for clearer explanations and to refine the original question for better assistance.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and relationships of transformations and generating functions, with no consensus reached on their connections or the appropriateness of the original inquiry.

Contextual Notes

Some participants note that the terms used may depend on specific mathematical contexts and definitions, and there is an acknowledgment of varying levels of understanding among participants.

aggarwal
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please explain the terms with example:----Transformation,direct transformation,inverse transformation,genreating function
 
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A transformation, in general, is just a function: y= f(x) "transforms" x to y. In linear algebra at least, a transformation usually means a linear transformation: f(ax+ by)= af(x)+ bf(y). I don't you can use the terms "direct" and "inverse" transformation without more specification. If I start with the transformation f, then its inverse function, f-1(x) is its "inverse transformation" and I guess one would call f the "direct transformation". Of course, one could as easily think of g(x)=f-1(x) as the "direct transformation" and its inverse, g-1(x)= f(x), would be its "inverse transformation".

I don't see what "generating function" has to do with a "transformation".

In general, if you have a sequence of numbers (such as the moments of a probability distribution), then the generating function is the function whose Taylor series has those numbers as coefficients.
 
just read your book.
 
thanks a lot....
 
sorry, it seemed appropriate. maybe if you ask a more precise question that is not specifically answerted in every text, i could help more.

did you actually follow my suggestion, i.e.read your book? if so, where did you get stuck?
 
Suggest two books

A good book on generating functions is Wilf, Generatingfunctionology, although I feel the "exponential family" stuff is a terrible idea since it utterly obscures what is really going on with the stuff, wreath product and properties of the Joyal cycle index of a structor (aka combintorial species).

Inquiry about direct and inverse transformations plus failure to cite sources in an PF inquiry suggests a certain lack of "mathematical maturity" (to avoid appearance of sarcasm I should perhaps point out that "mathematical maturity" is a commonly used term in mathematics pedagogy at the university level), so Wilf may be too advanced for the OP. However, a good book written for bright high school students (!) on transformations is Yaglom, Geometric Transformations.
 
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