Is there any equivalent of calculus in a modular field?
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For example, does $fracdydxequiv cmod n$ hold any meaning? How are the rules of calculus modified to allow for this, if possible.
For a simple example, $axequiv bmod n$ behaves the same as $ax=b$ for $xin[0,n)$, after which there is a discontinuity and then the initial interval output repeats because $axequiv((amod n)(xmod n))mod n$. We could therefore define $fracdydxequiv amod n$ at all points not a multiple of $n$ for lines. It would seem apt to be able to extend this definition to at least higher order polynomials.
One may also need to keep in mind $xmod n=fracxn-nlfloorfracxnrfloor$ as a real number extension for issues of continuity, although this makes derivatives even more confuddling.
calculus number-theory modular-arithmetic
asked Sep 10 at 21:26
Tejas Rao
768
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For example, does $fracdydxequiv cmod n$ hold any meaning? How are the rules of calculus modified to allow for this, if possible.
For a simple example, $axequiv bmod n$ behaves the same as $ax=b$ for $xin[0,n)$, after which there is a discontinuity and then the initial interval output repeats because $axequiv((amod n)(xmod n))mod n$. We could therefore define $fracdydxequiv amod n$ at all points not a multiple of $n$ for lines. It would seem apt to be able to extend this definition to at least higher order polynomials.
One may also need to keep in mind $xmod n=fracxn-nlfloorfracxnrfloor$ as a real number extension for issues of continuity, although this makes derivatives even more confuddling.
calculus number-theory modular-arithmetic
asked Sep 10 at 21:26
Tejas Rao
768
I don‘t know of a concept that satisfies your question entirely, but maybe you should look into $p$-adic analysis, which is kind of a generalisation of what you‘re talking about. Of course you can manipulate polynomials and power series mod $n$, too. Maybe have a look at Teichmüller lifts which are a kind of discrete notion of a limit mod $n$.
– Lukas Kofler
Sep 10 at 22:28
add a comment |Â
up vote
4
down vote
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up vote
4
down vote
favorite
For example, does $fracdydxequiv cmod n$ hold any meaning? How are the rules of calculus modified to allow for this, if possible.
For a simple example, $axequiv bmod n$ behaves the same as $ax=b$ for $xin[0,n)$, after which there is a discontinuity and then the initial interval output repeats because $axequiv((amod n)(xmod n))mod n$. We could therefore define $fracdydxequiv amod n$ at all points not a multiple of $n$ for lines. It would seem apt to be able to extend this definition to at least higher order polynomials.
One may also need to keep in mind $xmod n=fracxn-nlfloorfracxnrfloor$ as a real number extension for issues of continuity, although this makes derivatives even more confuddling.
calculus number-theory modular-arithmetic
asked Sep 10 at 21:26
Tejas Rao
768
For example, does $fracdydxequiv cmod n$ hold any meaning? How are the rules of calculus modified to allow for this, if possible.
For a simple example, $axequiv bmod n$ behaves the same as $ax=b$ for $xin[0,n)$, after which there is a discontinuity and then the initial interval output repeats because $axequiv((amod n)(xmod n))mod n$. We could therefore define $fracdydxequiv amod n$ at all points not a multiple of $n$ for lines. It would seem apt to be able to extend this definition to at least higher order polynomials.
One may also need to keep in mind $xmod n=fracxn-nlfloorfracxnrfloor$ as a real number extension for issues of continuity, although this makes derivatives even more confuddling.
calculus number-theory modular-arithmetic
calculus number-theory modular-arithmetic
asked Sep 10 at 21:26
Tejas Rao
768
asked Sep 10 at 21:26
Tejas Rao
768
edited Sep 11 at 6:17
asked Sep 10 at 21:26
Tejas Rao
768
asked Sep 10 at 21:26
Tejas Rao
768
asked Sep 10 at 21:26
Tejas Rao
768
768
I don‘t know of a concept that satisfies your question entirely, but maybe you should look into $p$-adic analysis, which is kind of a generalisation of what you‘re talking about. Of course you can manipulate polynomials and power series mod $n$, too. Maybe have a look at Teichmüller lifts which are a kind of discrete notion of a limit mod $n$.
– Lukas Kofler
Sep 10 at 22:28
add a comment |Â
I don‘t know of a concept that satisfies your question entirely, but maybe you should look into $p$-adic analysis, which is kind of a generalisation of what you‘re talking about. Of course you can manipulate polynomials and power series mod $n$, too. Maybe have a look at Teichmüller lifts which are a kind of discrete notion of a limit mod $n$.
– Lukas Kofler
Sep 10 at 22:28
I don‘t know of a concept that satisfies your question entirely, but maybe you should look into $p$-adic analysis, which is kind of a generalisation of what you‘re talking about. Of course you can manipulate polynomials and power series mod $n$, too. Maybe have a look at Teichmüller lifts which are a kind of discrete notion of a limit mod $n$.
– Lukas Kofler
Sep 10 at 22:28
I don‘t know of a concept that satisfies your question entirely, but maybe you should look into $p$-adic analysis, which is kind of a generalisation of what you‘re talking about. Of course you can manipulate polynomials and power series mod $n$, too. Maybe have a look at Teichmüller lifts which are a kind of discrete notion of a limit mod $n$.
– Lukas Kofler
Sep 10 at 22:28
add a comment |Â
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I don‘t know of a concept that satisfies your question entirely, but maybe you should look into $p$-adic analysis, which is kind of a generalisation of what you‘re talking about. Of course you can manipulate polynomials and power series mod $n$, too. Maybe have a look at Teichmüller lifts which are a kind of discrete notion of a limit mod $n$.
– Lukas Kofler
Sep 10 at 22:28