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Homerigid

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# rigid

Suppose $C$ is a collection of mathematical objects
(for instance, sets, or functions).
Then we say that $C$ is *rigid* if every $c\in C$
is uniquely determined by less information about $c$ than
one would expect.

It should be emphasized that the above “definition” does not
define a *mathematical object*. Instead, it describes in what sense
the adjective rigid is typically used in mathematics,
by mathematicians.

Let us illustrate this by some examples:

1. Harmonic functions on the unit disk are rigid in the sense that they are uniquely determined by their boundary values.

2. By the fundamental theorem of algebra, polynomials in $\mathbb{C}$ are rigid in the sense that any polynomial is completely determined by its values on any countably infinite set, say $\mathbb{N}$, or the unit disk.

3. Linear maps $\mathscr{L}(X,Y)$ between vector spaces $X,Y$ are rigid in the sense that any $L\in\mathscr{L}(X,Y)$ is completely determined by its values on any set of basis vectors of $X$.

4. Mostow’s rigidity theorem

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## Corrections

re: examples by rspuzio ✓

re: examples by rspuzio ✓

emphasized by Derk ✓

quotes by alozano ✓

## Comments

## Rigid

I don't quite understand this definition. To me, it is not clear what a "degree of freedom" is, or what does it mean to be determined by a 'limited' number of them. Limited in what sense? (assuming we know what a degree of freedom is)

## Re: Rigid

I think the intent of the entry is to give a feel for what the notion of rigidity should be, and how it's used in mathematical conversation, rather than a definition of the term itself.

While somewhat nonstandard, I do think entries like this have a place in the encyclopedia...being able to communicate ideas of imprecisely-defined notions is something that mathematicians do. This notion or "rigidity" is yet another piece of vocabulary at their disposal.

So my suggestion for matte is that he make it more clear that this is not a formal definition (actually, I think the first sentence of Version 1 of this entry was pretty good)...how about something like

"The term 'rigid' is used in mathematics to describe a collection of objects in which each object is completely determined by fewer parameters than expected."

Hope this helps,

Cam