Notation Question

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Pathway
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Notation Question

Postby Pathway » Mon Aug 11, 2008 8:55 am UTC

In this Wikipedia article http://en.wikipedia.org/wiki/Cauchy%27s ... l_equation, the notation f(.) is used. Would someone be kind enough to tell me why?
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Re: Notation Question

Postby tkb » Mon Aug 11, 2008 9:46 am UTC

I have seen that [imath]f(\cdot)[/imath] notation used simply to indicate that f is a function which takes a single argument.

Similarly, [imath]f(\cdot,\cdot)[/imath] can denote a function which takes two arguments. I've also seen [imath](\cdot)[/imath] used as formal argument which can appear in other (simple) expressions: for example, "[imath]f(\cdot)\equiv g(\cdot, h(\cdot))[/imath]" would be equivalent to "[imath]f(t)\equiv g(t, h(t))[/imath] for all [imath]t[/imath]".

EDIT: I note I didn't answer the "why" question: I'd consider that it's simply to emphasise/remind us that f is a function (taking a single argument), rather than a constant value or anything else which a letter might denote.

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Re: Notation Question

Postby Doraki » Mon Aug 11, 2008 2:28 pm UTC

the dot represents a hole, its there to show that there is a variable without having to give it a name.
It allows to quickly talk about partially applied functions, like [imath]f(2,\cdot,17)[/imath] where [imath]f[/imath] is initially a function of 3 variables.

tkb, your [imath]g(\cdot, h(\cdot))[/imath] irks me, it looks like a function with 2 variables to me.
What would you make out of "[imath]f(\cdot,\cdot)\equiv g(\cdot, h(\cdot,\cdot))[/imath]" ?

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Re: Notation Question

Postby skeptical scientist » Mon Aug 11, 2008 5:26 pm UTC

I have no idea. The notation f(.) can certainly mean what people here have been saying, but it's use on that Wikipedia page seems bizarre. Looking at the history, it was originally introduced to replace f(x), which of course is a single value of a function rather than a function; however, it would be much more natural to say, simply, "f is continuous," etc.

I more commonly see the period notation for functions which have an unusual notation, such as norms. If you have a norm ||.||, you don't want to write it with out the dot ("||||"), as that would be hard to read and interpret, but you also don't want to put in a variable, as that would give you the norm of that variable, not the norm as a function.
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Re: Notation Question

Postby tkb » Mon Aug 11, 2008 10:01 pm UTC

Doraki wrote:tkb, your [imath]g(\cdot, h(\cdot))[/imath] irks me, it looks like a function with 2 variables to me.
What would you make out of "[imath]f(\cdot,\cdot)\equiv g(\cdot, h(\cdot,\cdot))[/imath]" ?


I'd agree that [imath]g(\cdot, h(\cdot))[/imath] on its own may look like a function of two variables, but the left-hand side of [imath]f(\cdot) \equiv g(\cdot, h(\cdot))[/imath] is intended to provide the context that it is supposed to be a function of only one.

Faced with "[imath]f(\cdot,\cdot)\equiv g(\cdot, h(\cdot,\cdot))[/imath]", I'd cheerfully concede that the notation had hit its limits: my qualification of "simple" expressions was there for a reason! But notation of the sort I gave (where [imath]\cdot[/imath] refers to the same variable throughout) is seen, so I'd think it's as well to be aware of it. (As an example which I have readily to hand, it's freely used in Yong and Zhou's Stochastic Controls..., and when I was studying that subject, I found it convenient - if used with appropriate care.)

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Pathway
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Re: Notation Question

Postby Pathway » Tue Aug 12, 2008 4:12 am UTC

Okay, cool. Then I'm not missing anything deep here, it's just another way to express concepts I already know. Thank you, everyone.

p.s. does anyone know of books using this notation?
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Re: Notation Question

Postby gorcee » Tue Aug 12, 2008 5:49 pm UTC

Pathway wrote:Okay, cool. Then I'm not missing anything deep here, it's just another way to express concepts I already know. Thank you, everyone.

p.s. does anyone know of books using this notation?



Dozens.

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Pathway
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Re: Notation Question

Postby Pathway » Wed Aug 13, 2008 7:35 am UTC

Thanks, but the answer I was looking for was "yes."
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Re: Notation Question

Postby Hit3k » Wed Aug 13, 2008 8:21 am UTC

Pathway wrote:Thanks, but the answer I was looking for was "yes."

Yes. Can I name them? No.
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Re: Notation Question

Postby gorcee » Wed Aug 13, 2008 5:59 pm UTC

Pathway wrote:Thanks, but the answer I was looking for was "yes."


Just looking at the books and papers open on my desk right now...

Computational Methods for Inverse Problems, Curtis R. Vogel
Adaptive Control Design and Analysis, Gang Tao
"The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations," D. Xiu, G. E. Karniadakis

And that's just the first three things that are sitting on my desk. So yeah, it's a fairly ubiquitous notational convention.


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