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can we (re)construct some functions?


ahmet
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e.g. defining them by specific (but no more than) several criteria. 

for instance can we say that if we have several specific points and that implied function is passing over these  points, then that would be just one specific function. 

Or are there such specified functions?

 

Thanks 

 

Edited by ahmet
one wrong sentence causing by usage of wrong word was changed
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42 minutes ago, ahmet said:

e.g. defining them by specific (but no more than) several criteria. 

for instance can we say that if we have several specific points and that implied function is passing over these  points, then that would be just one specific function. 

Or are there such specified functions?

 

Thanks 

 

You seem to be talking about curve fitting.

This is a very important subject in Numerical Methods and Finite Element Analysis.

First, taking your example,

The points are called collocating point (Which is the technical term for co locating).

The approximating function matches the desired function at the given points.

However there is, in general, an infinite count of such approximating functions.

Let us say we have a third degree approximating polynomial     ax3 + bx2 + cx +d

Then we can can form a maximum of 4 different simulataneous equations with four points to obtain the coefficients a, b c and d.

But conditions may also be supplied in the form of derivatives at specific points.

Also I have only mentioned polynomials.

Other combinations of elementary functions may be more appropriate.

More complicated functions such as Bessel functions may only be approximated in this way. There are no combinations of elementary functions to determine them.

Edited by studiot
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37 minutes ago, studiot said:

You seem to be talking about curve fitting.

This is a very important subject in Numerical Methods and Finite Element Analysis.

First, taking your example,

The points are called collocating point (Which is the technical term for co locating).

The approximating function matches the desired function at the given points.

However there is, in general, an infinite count of such approximating functions.

Let us say we have a third degree approximating polynomial     ax3 + bx2 + cx +d

Then we can can form a maximum of 4 different simulataneous equations with four points to obtain the coefficients a, b c and d.

But conditions may also be supplied in the form of derivatives at specific points.

Also I have only mentioned polynomials.

Other combinations of elementary functions may be more appropriate.

More complicated functions such as Bessel functions may only be approximated in this way. There are no combinations of elementary functions to determine them.

thank you very much for your share. I see some specific terms (or keywords) in this post. 

Could you share some sources please that would provide comprehensive/detailed explanations?

A specific preference: I need examples in use. The graphs and their sensitivity is highly important for me (We tried to express that such contexts/points would be very important to us (find the therad relevant to R program) or more succintly, the reality of graph (its appearance) is highly important.

And one more reminding: the more example provided in the source,the more it is valuable for me.

....

 

 

 

 

Edited by ahmet
missing words, spelling errors
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mmm, unfortunately the thing I was thinking seems like "impossible"

I was thinking something like this (maybe,more clearly)

---->>whether we could allege it would define specific/unique function or a function which is a member of defined group of functions (E.g. regular functions) under the condition we give  a bunch of points or more than several points on the xy plane or xyz space. 

But that is ,I think clear that not. Because I think we can randomly cut off some parts of that implied /alleged function and compound a new part to it.

such types of function also known as "specifically defined functions"

for instance [math]y=x^{2}[/math] is a specific function but we can cut off the part of this function where x<0 and redefine it as

 

[math] y=x^{3}+5+sgn(x^{2}-4)[/math] where [math]x<0 [/math],and [math] y= x^{2}[/math] where [math] x>0 [/math]

this is also usual...

Edited by ahmet
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1 hour ago, ahmet said:

mmm, unfortunately the thing I was thinking seems like "impossible"

I was thinking something like this (maybe,more clearly)

---->>whether we could allege it would define specific/unique function or a function which is a member of defined group of functions (E.g. regular functions) under the condition we give  a bunch of points or more than several points on the xy plane or xyz space. 

But that is ,I think clear that not. Because I think we can randomly cut off some parts of that implied /alleged function and compound a new part to it.

such types of function also known as "specifically defined functions"

for instance y=x2 is a specific function but we can cut off the part of this function where x<0 and redefine it as

 

y=x3+5+sgn(x24) where x<0 ,and y=x2 where x>0

this is also usual...

 

I'm sorry I can't make head nor tail of what you want to say here.

Google doesn't produce anything relevent for specifically defined functions.

https://www.google.co.uk/search?source=hp&ei=Y3CYX5qXH8ubjLsPnfG3yAI&q=+"specifically+defined+functions"+&oq=+"specifically+defined+functions"+&gs_lcp=CgZwc3ktYWIQA1DaDljaDmDkGGgAcAB4AIAB7gGIAe4BkgEDMi0xmAEAoAECoAEBqgEHZ3dzLXdpeg&sclient=psy-ab&ved=0ahUKEwja1KnHvNXsAhXLDWMBHZ34DSkQ4dUDCAg&uact=5

You seem to be talking about restricting the domain  a function.

 

Please re-organise your thoughts in your own language and have another go at saying what you mean in English.

 

Do you realise that there are functions that cannot be written in terms of 'elementary' functions?

Functions such as elliptic integrals.

 

 

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16 hours ago, studiot said:

 

I'm sorry I can't make head nor tail of what you want to say here.

Google doesn't produce anything relevent for specifically defined functions.

https://www.google.co.uk/search?source=hp&ei=Y3CYX5qXH8ubjLsPnfG3yAI&q=+"specifically+defined+functions"+&oq=+"specifically+defined+functions"+&gs_lcp=CgZwc3ktYWIQA1DaDljaDmDkGGgAcAB4AIAB7gGIAe4BkgEDMi0xmAEAoAECoAEBqgEHZ3dzLXdpeg&sclient=psy-ab&ved=0ahUKEwja1KnHvNXsAhXLDWMBHZ34DSkQ4dUDCAg&uact=5

You seem to be talking about restricting the domain  a function.

 

Please re-organise your thoughts in your own language and have another go at saying what you mean in English.

 

Do you realise that there are functions that cannot be written in terms of 'elementary' functions?

Functions such as elliptic integrals.

 

 

hi, :) by specifically defined ,I mean that those functions were generally partial (partially defined with their critic points )

such functions generally contain one of these or the mixture:

1) sgn(f(x))

2) [|f(x)|]

3) |f(x)|

sorry for being unfamiliar with the mathematical terms in english language. 

But here,we can specify any function or a group of functions 

e.g.: continuous functions (broad), regular continuous functions , derivatable functions , integrable functions, etc.

 

 

Edited by ahmet
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On 10/22/2020 at 8:21 PM, ahmet said:

for instance can we say that if we have several specific points and that implied function is passing over these  points, then that would be just one specific function. 

Curve which goes through the all control points is called e.g. Catmull-Rom spline.

https://en.wikipedia.org/wiki/Centripetal_Catmull–Rom_spline

https://en.wikipedia.org/wiki/Catmull-Rom_spline

https://en.wikipedia.org/wiki/Cubic_Hermite_spline

When you know what are positions of control points, you can interpolate and receive position between them, still preserving original curvature (with some level of precision).

 

 

Edited by Sensei
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1 hour ago, ahmet said:

sorry for being unfamiliar with the mathematical terms in english language. 

Nothing to be sorry about.

So as far as I can see we are not talking about 'control' points in curve fitting, but please correct me if I am mistaken.

If you have x then your domain (do you understand what a domains and co-domains for functions ?) is a line we call the x axis.

If you have x and f(x) then you have a plane.

So sgn(x) divides the a axis line into 3 regions thus


[math]\left\{ {{\mathop{\rm sgn}} \left( x \right) = \begin{array}{*{20}{c}}
   {1:x > 0}  \\
   {0:x = 0}  \\
   { - 1:x < 0}  \\
\end{array}} \right\}[/math]


and for the modulus of sgn(x) we have


[math]\left\{ {\left| {{\mathop{\rm sgn}} \left( x \right)} \right| = \begin{array}{*{20}{c}}
   {1:x \ne 0}  \\
   {0:x = 0}  \\
\end{array}} \right\}[/math]


However the function    [math]f\left( x \right) = \sqrt x [/math]

is only valid for [math]x \ge 0[/math]

 

There are also more modern functions called floor and ceiling functions we could discuss.

Am I on the right lines and what did you want to say about such situations ?

 

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hiiiiiii :) :) 

I have to understand or interpret what sensei says. 

(As far as I know that should be relevant to approach methods (i.e. e.g. via integral , polynomial or curvatures methods. )

all the types of functions you wrote are known from my high school level knowledge :) 

but not sure whether we understand the same thing with "control"  point ,keyword. 

 

and are you mentioning about monotone functions or something in "Real Analysis" with ceiling or ground functions? 

or maybe some forms of sande-wich theorem ...not sure about the scope of those keywords. 

 

Edited by ahmet
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1 hour ago, ahmet said:

hiiiiiii :) :) 

I have to understand or interpret what sensei says. 

(As far as I know that should be relevant to approach methods (i.e. e.g. via integral , polynomial or curvatures methods. )

all the types of functions you wrote are known from my high school level knowledge :) 

but not sure whether we understand the same thing with "control"  point ,keyword. 

 

and are you mentioning about monotone functions or something in "Real Analysis" with ceiling or ground functions? 

or maybe some forms of sande-wich theorem ...not sure about the scope of those keywords. 

 

 

Sensei and I are both trying to guess what you want to talk about.

The good news is that your subject (whatever it is) is Scientific (Mathematical), not another nonsense thread  - we already get too many of those.

I'm glad you have seen most of my functions before. That makes it easier to talk about.

Yes they are from Real Analysis, except the ceiling and floor functions which are from Discrete or Concrete Analysis.

 

Sensei is talking about Numerical Analysis, which is indeed a very important point.
His 'control points' are those where some replacement function exactly matches the 'correct' function in value, curvature or some other property.
We variously call the replacement function an approximating function, a test function, a collocating function, an interpolating function and so on.
Numerical Analysis is about what happens between these points ie the difference between the test function and the correct or actual function.
Some of this can be done with Real or Complex Analysis, some needs to be done with Discrete Analysis.

 

But you haven't answered my questions about domains and co domains

Or what you want to discuss about any of these functions, here in this thread.

 

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1 hour ago, studiot said:

Yes they are from Real Analysis, except the ceiling and floor functions which are from Discrete or Concrete Analysis.

well, I have a small amount of knowledge in real analysis ,but not too much. 

generally many mathematicians tend to do pure mathematics in such context. 

and 

1 hour ago, studiot said:

Sensei is talking about Numerical Analysis, which is indeed a very important point.

yes, this indeed was just the thing I was anticipating. But I think I have forgotten some of them, maybe it would be very good if you suggest/provide some numerical analysis sources (but examples are too much important for me (always))

but..

1 hour ago, studiot said:

Complex Analysis

could not be sure how to correlate with complex analysis. Because we were mostly studied basically 

laurent series,

calculating complex integrals and 

properties of complex functions 

and others (like conform transforms etc)

and I could not make a well constructed connection between these and our issue. 

1 hour ago, studiot said:

Sensei and I are both trying to guess what you want to talk about.

oh very good. Keep it,

eventually, as long as we are thinking, we are capable to do something well/better. 

thanks 

:) 

meanwhile, have you tried to mention or refer to a specific paper that you were author of its by,

1 hour ago, studiot said:

I'm glad you have seen most of my functions before. That makes it easier to talk about.

this?

Edited by ahmet
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