Is there an accepted notation for the $n$th sum/integral of a function?












3












$begingroup$


I'm working on a thesis in image processing at the moment, and wanted to include a portion concerning the methodology for n dimension images (it will include a fully write up and functioning code for 2D and 3D images), and that would involve taking the $n$th integral of a function of $n$ variables, which in this case would mean the nth sum. For the sum portion, it would something like the sum of $x_1$ from $0$ to $M$, then the sum of $x_1$ from $0$ to $M$, and so on with a total of of $N$ summations.



Is there any sort of widely accepted convention for annotating this, or should I just put something like sum1 of sum2 /dots then the last sum?



edit: specific examples (sorry, didn't know you could use LaTeX here):



2D: $$sum_{x_1=0}^M sum_{x_2=0}^M f(x_1,x_2)$$
3D: $$sum_{x_1=0}^M sum_{x_2=0}^M sum_{x_3=0}^M f(x_1,x_2,x_3)$$
Is there a nicer way to express something like:
$$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n)$$






integration functional-analysis summation







share|cite|improve this question











$endgroup$












  • $begingroup$
    Your question is not clear. Please edit it to show us an explicit calculation for $2$ and $3$ dimensional images with $M$ small (say $3$).
    $endgroup$
    – Ethan Bolker
    Feb 8 at 2:34






  • 1




    $begingroup$
    you can also make your own notation, similar to what they do when they have the same function being applied many times e.g. $f^n(x)$ as long as you have mentioned that the function is reaplied
    $endgroup$
    – Arjang
    Feb 8 at 3:22






  • 1




    $begingroup$
    I've seen (and used) multiple summations expressed in the way you have in your final line and I think it's perfectly clear. It would be clear also if you dropped the second summation (over $x_2$) to save space. I find the other answers a bit opaque and I have to think much harder than I do when I read the sums with ellipses in between.
    $endgroup$
    – Jagerber48
    Feb 8 at 7:05










  • $begingroup$
    @EthanBolker how is what they have now unclear? Why do they need to provide an example $f$ and actually compute a value? That's of no relevance here.
    $endgroup$
    – The Great Duck
    Feb 8 at 7:10






  • 2




    $begingroup$
    @TheGreatDuck The question was unclear (to me) before the very useful edit.
    $endgroup$
    – Ethan Bolker
    Feb 8 at 11:25
















3












$begingroup$


I'm working on a thesis in image processing at the moment, and wanted to include a portion concerning the methodology for n dimension images (it will include a fully write up and functioning code for 2D and 3D images), and that would involve taking the $n$th integral of a function of $n$ variables, which in this case would mean the nth sum. For the sum portion, it would something like the sum of $x_1$ from $0$ to $M$, then the sum of $x_1$ from $0$ to $M$, and so on with a total of of $N$ summations.



Is there any sort of widely accepted convention for annotating this, or should I just put something like sum1 of sum2 /dots then the last sum?



edit: specific examples (sorry, didn't know you could use LaTeX here):



2D: $$sum_{x_1=0}^M sum_{x_2=0}^M f(x_1,x_2)$$
3D: $$sum_{x_1=0}^M sum_{x_2=0}^M sum_{x_3=0}^M f(x_1,x_2,x_3)$$
Is there a nicer way to express something like:
$$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n)$$






integration functional-analysis summation







share|cite|improve this question











$endgroup$












  • $begingroup$
    Your question is not clear. Please edit it to show us an explicit calculation for $2$ and $3$ dimensional images with $M$ small (say $3$).
    $endgroup$
    – Ethan Bolker
    Feb 8 at 2:34






  • 1




    $begingroup$
    you can also make your own notation, similar to what they do when they have the same function being applied many times e.g. $f^n(x)$ as long as you have mentioned that the function is reaplied
    $endgroup$
    – Arjang
    Feb 8 at 3:22






  • 1




    $begingroup$
    I've seen (and used) multiple summations expressed in the way you have in your final line and I think it's perfectly clear. It would be clear also if you dropped the second summation (over $x_2$) to save space. I find the other answers a bit opaque and I have to think much harder than I do when I read the sums with ellipses in between.
    $endgroup$
    – Jagerber48
    Feb 8 at 7:05










  • $begingroup$
    @EthanBolker how is what they have now unclear? Why do they need to provide an example $f$ and actually compute a value? That's of no relevance here.
    $endgroup$
    – The Great Duck
    Feb 8 at 7:10






  • 2




    $begingroup$
    @TheGreatDuck The question was unclear (to me) before the very useful edit.
    $endgroup$
    – Ethan Bolker
    Feb 8 at 11:25














3












3








3





$begingroup$


I'm working on a thesis in image processing at the moment, and wanted to include a portion concerning the methodology for n dimension images (it will include a fully write up and functioning code for 2D and 3D images), and that would involve taking the $n$th integral of a function of $n$ variables, which in this case would mean the nth sum. For the sum portion, it would something like the sum of $x_1$ from $0$ to $M$, then the sum of $x_1$ from $0$ to $M$, and so on with a total of of $N$ summations.



Is there any sort of widely accepted convention for annotating this, or should I just put something like sum1 of sum2 /dots then the last sum?



edit: specific examples (sorry, didn't know you could use LaTeX here):



2D: $$sum_{x_1=0}^M sum_{x_2=0}^M f(x_1,x_2)$$
3D: $$sum_{x_1=0}^M sum_{x_2=0}^M sum_{x_3=0}^M f(x_1,x_2,x_3)$$
Is there a nicer way to express something like:
$$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n)$$






integration functional-analysis summation







share|cite|improve this question











$endgroup$




I'm working on a thesis in image processing at the moment, and wanted to include a portion concerning the methodology for n dimension images (it will include a fully write up and functioning code for 2D and 3D images), and that would involve taking the $n$th integral of a function of $n$ variables, which in this case would mean the nth sum. For the sum portion, it would something like the sum of $x_1$ from $0$ to $M$, then the sum of $x_1$ from $0$ to $M$, and so on with a total of of $N$ summations.



Is there any sort of widely accepted convention for annotating this, or should I just put something like sum1 of sum2 /dots then the last sum?



edit: specific examples (sorry, didn't know you could use LaTeX here):



2D: $$sum_{x_1=0}^M sum_{x_2=0}^M f(x_1,x_2)$$
3D: $$sum_{x_1=0}^M sum_{x_2=0}^M sum_{x_3=0}^M f(x_1,x_2,x_3)$$
Is there a nicer way to express something like:
$$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n)$$






integration functional-analysis summation




integration functional-analysis summation






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Feb 8 at 7:37









Asaf Karagila♦

307k33440773




307k33440773










asked Feb 8 at 2:30









Peace BlasterPeace Blaster

557




557












  • $begingroup$
    Your question is not clear. Please edit it to show us an explicit calculation for $2$ and $3$ dimensional images with $M$ small (say $3$).
    $endgroup$
    – Ethan Bolker
    Feb 8 at 2:34






  • 1




    $begingroup$
    you can also make your own notation, similar to what they do when they have the same function being applied many times e.g. $f^n(x)$ as long as you have mentioned that the function is reaplied
    $endgroup$
    – Arjang
    Feb 8 at 3:22






  • 1




    $begingroup$
    I've seen (and used) multiple summations expressed in the way you have in your final line and I think it's perfectly clear. It would be clear also if you dropped the second summation (over $x_2$) to save space. I find the other answers a bit opaque and I have to think much harder than I do when I read the sums with ellipses in between.
    $endgroup$
    – Jagerber48
    Feb 8 at 7:05










  • $begingroup$
    @EthanBolker how is what they have now unclear? Why do they need to provide an example $f$ and actually compute a value? That's of no relevance here.
    $endgroup$
    – The Great Duck
    Feb 8 at 7:10






  • 2




    $begingroup$
    @TheGreatDuck The question was unclear (to me) before the very useful edit.
    $endgroup$
    – Ethan Bolker
    Feb 8 at 11:25


















  • $begingroup$
    Your question is not clear. Please edit it to show us an explicit calculation for $2$ and $3$ dimensional images with $M$ small (say $3$).
    $endgroup$
    – Ethan Bolker
    Feb 8 at 2:34






  • 1




    $begingroup$
    you can also make your own notation, similar to what they do when they have the same function being applied many times e.g. $f^n(x)$ as long as you have mentioned that the function is reaplied
    $endgroup$
    – Arjang
    Feb 8 at 3:22






  • 1




    $begingroup$
    I've seen (and used) multiple summations expressed in the way you have in your final line and I think it's perfectly clear. It would be clear also if you dropped the second summation (over $x_2$) to save space. I find the other answers a bit opaque and I have to think much harder than I do when I read the sums with ellipses in between.
    $endgroup$
    – Jagerber48
    Feb 8 at 7:05










  • $begingroup$
    @EthanBolker how is what they have now unclear? Why do they need to provide an example $f$ and actually compute a value? That's of no relevance here.
    $endgroup$
    – The Great Duck
    Feb 8 at 7:10






  • 2




    $begingroup$
    @TheGreatDuck The question was unclear (to me) before the very useful edit.
    $endgroup$
    – Ethan Bolker
    Feb 8 at 11:25
















$begingroup$
Your question is not clear. Please edit it to show us an explicit calculation for $2$ and $3$ dimensional images with $M$ small (say $3$).
$endgroup$
– Ethan Bolker
Feb 8 at 2:34




$begingroup$
Your question is not clear. Please edit it to show us an explicit calculation for $2$ and $3$ dimensional images with $M$ small (say $3$).
$endgroup$
– Ethan Bolker
Feb 8 at 2:34




1




1




$begingroup$
you can also make your own notation, similar to what they do when they have the same function being applied many times e.g. $f^n(x)$ as long as you have mentioned that the function is reaplied
$endgroup$
– Arjang
Feb 8 at 3:22




$begingroup$
you can also make your own notation, similar to what they do when they have the same function being applied many times e.g. $f^n(x)$ as long as you have mentioned that the function is reaplied
$endgroup$
– Arjang
Feb 8 at 3:22




1




1




$begingroup$
I've seen (and used) multiple summations expressed in the way you have in your final line and I think it's perfectly clear. It would be clear also if you dropped the second summation (over $x_2$) to save space. I find the other answers a bit opaque and I have to think much harder than I do when I read the sums with ellipses in between.
$endgroup$
– Jagerber48
Feb 8 at 7:05




$begingroup$
I've seen (and used) multiple summations expressed in the way you have in your final line and I think it's perfectly clear. It would be clear also if you dropped the second summation (over $x_2$) to save space. I find the other answers a bit opaque and I have to think much harder than I do when I read the sums with ellipses in between.
$endgroup$
– Jagerber48
Feb 8 at 7:05












$begingroup$
@EthanBolker how is what they have now unclear? Why do they need to provide an example $f$ and actually compute a value? That's of no relevance here.
$endgroup$
– The Great Duck
Feb 8 at 7:10




$begingroup$
@EthanBolker how is what they have now unclear? Why do they need to provide an example $f$ and actually compute a value? That's of no relevance here.
$endgroup$
– The Great Duck
Feb 8 at 7:10




2




2




$begingroup$
@TheGreatDuck The question was unclear (to me) before the very useful edit.
$endgroup$
– Ethan Bolker
Feb 8 at 11:25




$begingroup$
@TheGreatDuck The question was unclear (to me) before the very useful edit.
$endgroup$
– Ethan Bolker
Feb 8 at 11:25










2 Answers
2






active

oldest

votes


















5












$begingroup$

I don't know of a convention. Your last version is perfectly clear. If you need to use it more than once then define a short symbol the first time. That could be as simple as $sum_0^M f$.



Here are two suggestions each using just a single $sum$.



$$
sum_{x in [1, 2, ldots, M]^n} f(x)
$$



$$
sum_{x_i = 0, (i = 1, ldots n)}^M f(x_i, x_2, ldots, x_n)
$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks! I just wanted to make sure I wasn't missing something glaringly obvious, though that second example looks pretty nice for when I put it in beamer and have limited space.
    $endgroup$
    – Peace Blaster
    Feb 8 at 3:15










  • $begingroup$
    I think that the interval brackets should be exchanged for set braces. An improvement would be setting the dimensions ${1,2,dots,M}$ to just $D$
    $endgroup$
    – Chase Ryan Taylor
    Feb 8 at 14:49



















2












$begingroup$

How about something like :
$$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n) = large]_{k=x_0}^{k=x_n}sum_{x_k=0}^{M_n}f(x_1,x_2,dots,x_n)$$



Once you use the above as definition then you can reuse it all over the place, also writing repeated sums was something that caused the the tensor notation to be invented. You might want to look up tensors instead.






share|cite|improve this answer









$endgroup$














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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    5












    $begingroup$

    I don't know of a convention. Your last version is perfectly clear. If you need to use it more than once then define a short symbol the first time. That could be as simple as $sum_0^M f$.



    Here are two suggestions each using just a single $sum$.



    $$
    sum_{x in [1, 2, ldots, M]^n} f(x)
    $$



    $$
    sum_{x_i = 0, (i = 1, ldots n)}^M f(x_i, x_2, ldots, x_n)
    $$






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Thanks! I just wanted to make sure I wasn't missing something glaringly obvious, though that second example looks pretty nice for when I put it in beamer and have limited space.
      $endgroup$
      – Peace Blaster
      Feb 8 at 3:15










    • $begingroup$
      I think that the interval brackets should be exchanged for set braces. An improvement would be setting the dimensions ${1,2,dots,M}$ to just $D$
      $endgroup$
      – Chase Ryan Taylor
      Feb 8 at 14:49
















    5












    $begingroup$

    I don't know of a convention. Your last version is perfectly clear. If you need to use it more than once then define a short symbol the first time. That could be as simple as $sum_0^M f$.



    Here are two suggestions each using just a single $sum$.



    $$
    sum_{x in [1, 2, ldots, M]^n} f(x)
    $$



    $$
    sum_{x_i = 0, (i = 1, ldots n)}^M f(x_i, x_2, ldots, x_n)
    $$






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Thanks! I just wanted to make sure I wasn't missing something glaringly obvious, though that second example looks pretty nice for when I put it in beamer and have limited space.
      $endgroup$
      – Peace Blaster
      Feb 8 at 3:15










    • $begingroup$
      I think that the interval brackets should be exchanged for set braces. An improvement would be setting the dimensions ${1,2,dots,M}$ to just $D$
      $endgroup$
      – Chase Ryan Taylor
      Feb 8 at 14:49














    5












    5








    5





    $begingroup$

    I don't know of a convention. Your last version is perfectly clear. If you need to use it more than once then define a short symbol the first time. That could be as simple as $sum_0^M f$.



    Here are two suggestions each using just a single $sum$.



    $$
    sum_{x in [1, 2, ldots, M]^n} f(x)
    $$



    $$
    sum_{x_i = 0, (i = 1, ldots n)}^M f(x_i, x_2, ldots, x_n)
    $$






    share|cite|improve this answer









    $endgroup$



    I don't know of a convention. Your last version is perfectly clear. If you need to use it more than once then define a short symbol the first time. That could be as simple as $sum_0^M f$.



    Here are two suggestions each using just a single $sum$.



    $$
    sum_{x in [1, 2, ldots, M]^n} f(x)
    $$



    $$
    sum_{x_i = 0, (i = 1, ldots n)}^M f(x_i, x_2, ldots, x_n)
    $$







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Feb 8 at 3:13









    Ethan BolkerEthan Bolker

    45.5k553120




    45.5k553120












    • $begingroup$
      Thanks! I just wanted to make sure I wasn't missing something glaringly obvious, though that second example looks pretty nice for when I put it in beamer and have limited space.
      $endgroup$
      – Peace Blaster
      Feb 8 at 3:15










    • $begingroup$
      I think that the interval brackets should be exchanged for set braces. An improvement would be setting the dimensions ${1,2,dots,M}$ to just $D$
      $endgroup$
      – Chase Ryan Taylor
      Feb 8 at 14:49


















    • $begingroup$
      Thanks! I just wanted to make sure I wasn't missing something glaringly obvious, though that second example looks pretty nice for when I put it in beamer and have limited space.
      $endgroup$
      – Peace Blaster
      Feb 8 at 3:15










    • $begingroup$
      I think that the interval brackets should be exchanged for set braces. An improvement would be setting the dimensions ${1,2,dots,M}$ to just $D$
      $endgroup$
      – Chase Ryan Taylor
      Feb 8 at 14:49
















    $begingroup$
    Thanks! I just wanted to make sure I wasn't missing something glaringly obvious, though that second example looks pretty nice for when I put it in beamer and have limited space.
    $endgroup$
    – Peace Blaster
    Feb 8 at 3:15




    $begingroup$
    Thanks! I just wanted to make sure I wasn't missing something glaringly obvious, though that second example looks pretty nice for when I put it in beamer and have limited space.
    $endgroup$
    – Peace Blaster
    Feb 8 at 3:15












    $begingroup$
    I think that the interval brackets should be exchanged for set braces. An improvement would be setting the dimensions ${1,2,dots,M}$ to just $D$
    $endgroup$
    – Chase Ryan Taylor
    Feb 8 at 14:49




    $begingroup$
    I think that the interval brackets should be exchanged for set braces. An improvement would be setting the dimensions ${1,2,dots,M}$ to just $D$
    $endgroup$
    – Chase Ryan Taylor
    Feb 8 at 14:49











    2












    $begingroup$

    How about something like :
    $$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n) = large]_{k=x_0}^{k=x_n}sum_{x_k=0}^{M_n}f(x_1,x_2,dots,x_n)$$



    Once you use the above as definition then you can reuse it all over the place, also writing repeated sums was something that caused the the tensor notation to be invented. You might want to look up tensors instead.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      How about something like :
      $$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n) = large]_{k=x_0}^{k=x_n}sum_{x_k=0}^{M_n}f(x_1,x_2,dots,x_n)$$



      Once you use the above as definition then you can reuse it all over the place, also writing repeated sums was something that caused the the tensor notation to be invented. You might want to look up tensors instead.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        How about something like :
        $$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n) = large]_{k=x_0}^{k=x_n}sum_{x_k=0}^{M_n}f(x_1,x_2,dots,x_n)$$



        Once you use the above as definition then you can reuse it all over the place, also writing repeated sums was something that caused the the tensor notation to be invented. You might want to look up tensors instead.






        share|cite|improve this answer









        $endgroup$



        How about something like :
        $$sum_{x_1=0}^M sum_{x_2=0}^M dots sum_{x_n=0}^M f(x_1,x_2,dots,x_n) = large]_{k=x_0}^{k=x_n}sum_{x_k=0}^{M_n}f(x_1,x_2,dots,x_n)$$



        Once you use the above as definition then you can reuse it all over the place, also writing repeated sums was something that caused the the tensor notation to be invented. You might want to look up tensors instead.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Feb 8 at 3:32









        ArjangArjang

        5,65462364




        5,65462364






























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            2 $begingroup$ Consider the function defined in (1) below related to the fundamental prime counting function $pi(x)$ . Note that A143519(n) is not multiplicative. (1) $quad f(x)=sumlimits_{n=1}^{x}A143519(n)$ https://oeis.org/A143519 The following plot illustrates $f(x)$ defined in formula (1) above. Figure (1) : Illustration of $f(x)$ defined in formula (1) The integer zeros of $f(x)$ for $xle 10,000$ are listed in (2) below. (2) $quad$ {1,6,9,12,19,30,79,80,81,116,193,201,287,288,291,668,673,679,680,685,686,1109} The zero crossings of $f(x)$ for $xle 10,000$ where $f(x)$ doesn't settle at zero are listed in (3) below. (3) $quad$ {14,21,33,114,115,118,195,286,290,295,442,445,665,667,670,671,678,682} Question (1) : Does $f(x)$ have a finite number of integer zeros, and if so ...
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            List of scandals in India

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            List of scandals in India From Wikipedia, the free encyclopedia Jump to navigation Jump to search This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by expanding it with reliably sourced entries. This article has multiple issues. Please help improve it or discuss these issues on the talk page . (Learn how and when to remove these template messages) This article has an unclear citation style . The references used may be made clearer with a different or consistent style of citation and footnoting. ( January 2014 ) (Learn how and when to remove this template message) This list is in reverse order. As a consequence it appears to be slanted towards recent events. Please try to keep recent events in historical perspective and add more content related to non-recent events. ( May 2013 ) (Learn how and when to remove this template message) ...
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            Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

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            0 1 What exactly does the error message beginning with E: mean? I assume it has to do with the file structure on Ubuntu-on-Windows, but what exactly? The following package was automatically installed and is no longer required: os-prober Use 'apt-get autoremove' to remove it. The following extra packages will be installed: libxslt1.1 The following NEW packages will be installed: libxslt1.1 xmlstarlet 0 upgraded, 2 newly installed, 0 to remove and 0 not upgraded. Need to get 435 kB of archives. After this operation, 1023 kB of additional disk space will be used. Get:1 http://archive.ubuntu.com/ubuntu/ trusty/main libxslt1.1 amd64 1.1.28-2build1 [145 kB] Get:2 http://archive.ubuntu.com/ubuntu/ trusty/universe xmlstarlet amd64 1.5.0-1 [290 kB] ...
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