Are there any conventions about the use of $n^*$ as notation of a variable? I have seen it for the first time here.

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    $\begingroup$ typically it is used to show that something is new or different. Yet maintains a lot of the aspects of the old. $\endgroup$ – picakhu Apr 24 '11 at 23:19
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    $\begingroup$ In the post you linked to it's already been defined as the function described in the post. In general it means a lot of things depending on context. $\endgroup$ – Qiaochu Yuan Apr 24 '11 at 23:41
  • $\begingroup$ Huh, the original link broke. Oh well, that's life. $\endgroup$ – Willie Wong Apr 29 '13 at 15:51

There's no general convention. The use in the webpage you cited certainly isn't widely known or used.

Generally, mathematicians like to use adornments to variable names to indicate a particular operation. Mathematicians especially like the use of adornments when the operation is a duality relation. (Perhaps the most prominent counterexample to this rule is the prime-notation for derivatives, where $f'$ is the first derivative of the function $f$ and $f''$ is the second derivative.)

Some examples of adornments that signifies a duality relation include:

  • $p$ and $\bar{p}$ for complex conjugation
  • $A$ and $A^t$ for matrix transposition
  • $A$ and $A^*$ for operator adjoints
  • $f$ and $\hat{f}$ for the Fourier transformation (technically the double Fourier transform is not the identity, so this is not a real symmetric transformation between an object and its dual, but it is close enough)
  • $\omega$ and ${}^*\omega$ for the Hodge dual of forms

A given symbol can however take on different meanings in different contexts. The bar-over-a-symbol and the prime are two very commonly overloaded symbols. The meaning of $*$ for the adjoint operation on operations (or matrices) is sometimes applied to the case of a linear operator on the one-dimensional complex vector space; in this case the linear operators can be identified with the complex numbers and we sometimes see the notation $z$ and $z^*$ for the complex conjugate pairs.

Aside from some of the uses of the star adornment which are already mentioned in other answers, there is one other use that is common in mathematical analysis, especially in conjunction with the study of partial differential equations (which is as close to a convention as I know for $p$ and $p^*$ when $p$ is a real number):

For a fixed integer $n$ (representing the dimension of the space $\mathbb{R}^n$ one is working on), the number $p^*$ is defined to be $$ p^* = \frac{np}{n-p} $$ for any $1 \leq p < n$. This number comes up from scaling considerations on function spaces, and is often called the Sobolev conjugate of $p$ due to its appearance in Sobolev inequalities. Personally I think this notation is lousy, but it is somewhat established in the literature.

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    $\begingroup$ Ohh, I couldn't possibly agree more with "I think this notation is lousy" :) $\endgroup$ – t.b. Apr 24 '11 at 23:41
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    $\begingroup$ "I think this notation is lousy" - yes, ohh yessss... $\endgroup$ – J. M. is a poor mathematician Apr 24 '11 at 23:48
  • $\begingroup$ As an alternative notation for $\frac{np}{n-p}$, I would suggest letting $n^*$ be defined to mean the function $[1,n) \rightarrow \mathbb{R}$ given by $n^*(p) = np/(n-p)$. So instead of $p^*$, we can write $n^*(p)$. $\endgroup$ – goblin Aug 6 '15 at 2:23

It can mean:

the adjoint matrix, if $n$ is a matrix

the set of all powers of $n$ is $n$ is a word or letter in a regular expression

the dual space if $n$ is a vector space

the unit elements if $n$ is a ring


or it can just be used to denote the image of $n$ under a function called * as in your example.


Without context it's difficult to say. One common use of the star is for the optimal value or point in an optimization problem. But in the link you provided it's just something related to n.


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