Re: Finding the inverse of a function.

Re: Finding the inverse of a function.

[peasthope]

I should have attributed the quotations.

From:	Dusko Pavlovic <dusko@kestrel.edu>
Date:	Wed, 05 Oct 2011 13:52:25 +0100
> ... the function x sin(x), i think, ...

Correct.

> ... as studied by undergraduates in calculus I.

According to the course descriptions here at UBC, there is at least
a mention of series in first year courses.  Fourier and other series
appear in 2nd & 3rd years.

> ... might be worth while to rework widder's book on transform theory
coalgebraically.

Engineer speaking.  Mathematicians, don't be too critical.

What I recall from a brief study decades ago is that each of the
familiar series--Taylor, Laurent, Fourier & etc.--is based upon a
set of orthogonal functions.  So I wondered whether the category of
sets of orthogonal functions has been thoroughly studied.  Such a
study might show the necessity of infinite series to represent the
inverses of some functions.  Wishful thinking?

Thanks,                  ... Peter E.

-- 
Telephone 1 360 450 2132.  bcc: peasthope at shaw.ca
Shop pages http://carnot.yi.org/ accessible as long as the old drives survive.
Personal pages http://members.shaw.ca/peasthope/ .



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Re: Finding the inverse of a function.

[peasthope]

> ... the function x sin(x), i think, ...

Correct.

> ... as studied by undergraduates in calculus I.

According to the course descriptions here at UBC, there is at least
a mention of series in first year courses.  Fourier and other series
appear in 2nd & 3rd years.

> ... might be worth while to rework widder's book on transform theory
coalgebraically.

Engineer speaking.  Mathematicians, don't be too critical.

What I recall from a brief study decades ago is that each of the
familiar series--Taylor, Laurent, Fourier & etc.--is based upon a
set of orthogonal functions.  So I wondered whether the category of
sets of orthogonal functions has been thoroughly studied.  Such a
study should show the necessity of infinite series to represent the
inverses of some functions.  Wishful thinking?

Thanks,                  ... Peter E.

-- 
Telephone 1 360 450 2132.  bcc: peasthope at shaw.ca
Shop pages http://carnot.yi.org/ accessible as long as the old drives survive.
Personal pages http://members.shaw.ca/peasthope/ .



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Finding the inverse of a function.

[peasthope]

Is CT any help in getting an overview of infinite series?

I'm curious to find an inverse of f(\theta) = \theta \sin \theta
and wonder whether there is an approach more insightful than
the traditional course in applied analysis.

Thanks,             ... Peter E.



-- 
Telephone 1 360 450 2132.  bcc: peasthope at shaw.ca
Shop pages http://carnot.yi.org/ accessible as long as the old drives survive.
Personal pages http://members.shaw.ca/peasthope/ .



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Finding the inverse of a function.

[Dusko Pavlovic]

infinite series and analytic functions can be simply and conveniently manipulated in categories of coalgebras. their taylor and laplace transforms turn up as coalgebra isomorphims. the basics of this approach are in my LICS 98 paper with martin escardo,
http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=5684#
or
http://www.isg.rhul.ac.uk/dusko/coalgebra.html
neither martin nor i really pursued this path, which is perhaps a mistake, since it seems that a powerful categorical tool lies there.

2c,
-- dusko

On Sep 16, 2011, at 5:42 PM, peasthope@shaw.ca wrote:

> Is CT any help in getting an overview of infinite series?
> 
> I'm curious to find an inverse of f(\theta) = \theta \sin \theta
> and wonder whether there is an approach more insightful than
> the traditional course in applied analysis.
> 
> Thanks,             ... Peter E.
> 

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Re: Finding the inverse of a function.

[Martin Escardo]

This has been further developed in several papers by Rutten and other
people.

(This is entertaining but is not categorical:
http://www.cs.dartmouth.edu/~doug/music.ps.gz)

Martin


On 20/09/11 18:55, Dusko Pavlovic wrote:
> infinite series and analytic functions can be simply and conveniently manipulated in categories of coalgebras. their taylor and laplace transforms turn up as coalgebra isomorphims. the basics of this approach are in my LICS 98 paper with martin escardo,
> http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=5684#
> or
> http://www.isg.rhul.ac.uk/dusko/coalgebra.html
> neither martin nor i really pursued this path, which is perhaps a mistake, since it seems that a powerful categorical tool lies there.
>
> 2c,
> -- dusko
>
> On Sep 16, 2011, at 5:42 PM, peasthope@shaw.ca wrote:
>
>> Is CT any help in getting an overview of infinite series?
>>
>> I'm curious to find an inverse of f(\theta) = \theta \sin \theta
>> and wonder whether there is an approach more insightful than
>> the traditional course in applied analysis.
>>
>> Thanks,             ... Peter E.

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Re: Finding the inverse of a function.

[Dusko Pavlovic]

hi martin,

On Sep 21, 2011, at 2:02 PM, Martin Escardo wrote:

> This has been further developed in several papers by Rutten and other people.

i know, of course, that jan rutten developed theory of power series to a great length, as used in combinatorics and automata theory. he has a paper about coalgebraic differential calculus --- but of *bitstreams*. the query on the list concerned the function 
x sin(x), i think, as studied by undergraduates in calculus I. did jan really work on such things? i would be really interested in that.

i used to think that it might be worth while to rework widder's book on transform theory coalgebraically. but even the coalgebraic laplace transform in our paper does not seem to have been useful for anything. i thought no one noticed it. it would be good to know that it was further developed.

all the best,
-- dusko

> 
> (This is entertaining but is not categorical: http://www.cs.dartmouth.edu/~doug/music.ps.gz)
> 
> Martin
> 
> 
> On 20/09/11 18:55, Dusko Pavlovic wrote:
>> infinite series and analytic functions can be simply and conveniently manipulated in categories of coalgebras. their taylor and laplace transforms turn up as coalgebra isomorphims. the basics of this approach are in my LICS 98 paper with martin escardo,
>> http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=5684#
>> or
>> http://www.isg.rhul.ac.uk/dusko/coalgebra.html
>> neither martin nor i really pursued this path, which is perhaps a mistake, since it seems that a powerful categorical tool lies there.
>> 
>> 2c,
>> -- dusko
>> 
>> On Sep 16, 2011, at 5:42 PM, peasthope@shaw.ca wrote:
>> 
>>> Is CT any help in getting an overview of infinite series?
>>> 
>>> I'm curious to find an inverse of f(\theta) = \theta \sin \theta
>>> and wonder whether there is an approach more insightful than
>>> the traditional course in applied analysis.
>>> 
>>> Thanks,             ... Peter E.


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