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What are the general uses of the hat and star symbol in math? Or could you please point me to a page that discusses this? Thanks.

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@Martino8: actually, when you write star symbol, could you please also provide it in latex? thanks! – Rudy the Reindeer Mar 10 '11 at 17:42
Lots of things. Could you be more precise about what you're asking for? – Qiaochu Yuan Mar 10 '11 at 17:56
Why don't you provide some context? We can probably make a list of 50 different uses of any given symbol, but that doesn't seem very useful. – wildildildlife Mar 10 '11 at 17:57
up vote 5 down vote accepted

Different branches of mathematics may have varying conventional usages of these kind of "decorations". Typically they denote a transformed version of the base variable (e.g. $\hat{f}$ denoting the Fourier transform of $f$ as mentioned in another answer). Or, they may denote a special or specific value of a variable ($x^*$ giving the value of $x$ minimizing $f$ from another answer.) The $*$ symbol is often used for arbitrary associative binary operations. Etc. etc.

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Not to mention Kleene's star! – Raphael Mar 10 '11 at 19:07
Absolutely! I guess the point is that there is no "standard" usage across all of mathematics, though there are "local" conventions. (Set theorists will have different conventions than K-theorists...) – Apollo Mar 10 '11 at 19:57

There is a nice list for $*$ in this article

I guess another (more general) term for "hat" is Circumflex

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I have seen the star used for multiplication, hermitian conjugate of a matrix, special values of a variable (given a function $f(x), x^*$ might be the value of $x$ that minimizes $f$), among others. In Conway's theory of games, * is the game that wins for the first player.

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how about omitted terms $$ \partial\langle x_0,...,x_n\rangle=\sum_{i=0}^n(-1)^i\langle x_0,...,\hat{x_i},...x_n\rangle $$

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$\hat{}$ can also be used to denote the Fourier transform $\hat{f}$ of an integrable function $f$.

$\ast$ can be used to denote the convolution product $f \ast g$ of two functions $f$ and $g$.

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The Hat symbol can be used to denote a vector. And the Star symbol may possibly used to denote a binary operation. For example a non empty set $G$ with a binary operation $\star$ is said to be a Group if.....

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I have also seen it denoting a unit vector. – Ross Millikan Mar 10 '11 at 17:38
@Ross: Unit vector is also a vector :) – anonymous Mar 10 '11 at 17:40
True, but in these cases an overarrow was a vector of any length and the hat was reserved specifically for unit vectors. – Ross Millikan Mar 10 '11 at 17:41
@Ross: Perhaps, but i have used this symbol to denote vector quantities as well. – anonymous Mar 10 '11 at 17:42
I'll second Ross's comment: I have always seen the hat used to denote a unit vector. Sometimes $\vec x$ is a vector of arbitrary length and $\hat x$ a unit vector; sometimes $\mathbf x$ is an arbitrary vector and $\hat{\mathbf x}$ is a unit vector; but never is a hat placed on a vector that is not of unit length. – Rahul Mar 10 '11 at 19:07

According to ISO 31-11: $$\begin{align} \mathbb{N}^* = \mathbb{N}-\{0\} \\ \mathbb{Z}^* = \mathbb{Z}-\{0\} \end{align}$$ The same goes for $\mathbb{Q}, \mathbb{R}, \mathbb{C}$. $$\begin{align} z^* = \text{complex conjugate of } z. \end{align}$$

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I've seen $\mathbb{R}^*,\mathbb{C}^*$ used more to denote the hyperreals and hypercomplex numbers than that, tbh. – YoTengoUnLCD Jun 15 '15 at 23:40

Another possibility: if $V$ is a vector space over a field $\mathbb{F}$, its dual space, $V^*$, is the set of linear maps $V \to \mathbb{F}$. The dual space is also a vector space in its own right. The double dual of $V$ is $V^{**} = (V^*)^*$, and there's a nice correspondence between $V$ and $V^{**}$ such that given an element $v \in V$ we have a special corresponding element often called $\hat{v} \in V^{**}$.

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