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What characterizes smooth functions equal in a sufficiently small neighborhood?
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Let $X$ be the set of all functions $f:mathbbRrightarrowmathbbR$ which are infinitely differentiable at $0$. Let us define an equivalence relation $sim$ on $X$ by saying that $fsim g$ if there exists a $delta>0$ such that $f(x)=g(x)$ for all points in the interval $(-delta,delta)$. And let $Y$ be the set of equivalence classes of elements of $X$ under $sim$.
My question is, what characterizes a given equivalence class in $Y$? The values of $f(0)$ and $f^(n)(0)$ for all $n$ aren't enough, because for any given analytic function $f$ there exists an non-analytic infinitely differentiable function $g$ such that $f(0)=g(0)$ and $f^(n)=g^(n)$ for all $n$ but where it's not the case that $fsim g$.
So what is the minimum information needed to unambiguously specify an element of $Y$?
EDIT: It turns out my concept is an existing mathematical concept, known as a germ. So my question reduces to, what uniquely characterizes the germ of a smooth function?
calculus real-analysis taylor-expansion analyticity analytic-functions
asked 2 days ago
Keshav Srinivasan
1,86311337
This question has an open bounty worth +150
reputation from Keshav Srinivasan ending ending at 2018-08-12 04:42:54Z">in 6 days.
This question has not received enough attention.
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up vote
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Let $X$ be the set of all functions $f:mathbbRrightarrowmathbbR$ which are infinitely differentiable at $0$. Let us define an equivalence relation $sim$ on $X$ by saying that $fsim g$ if there exists a $delta>0$ such that $f(x)=g(x)$ for all points in the interval $(-delta,delta)$. And let $Y$ be the set of equivalence classes of elements of $X$ under $sim$.
My question is, what characterizes a given equivalence class in $Y$? The values of $f(0)$ and $f^(n)(0)$ for all $n$ aren't enough, because for any given analytic function $f$ there exists an non-analytic infinitely differentiable function $g$ such that $f(0)=g(0)$ and $f^(n)=g^(n)$ for all $n$ but where it's not the case that $fsim g$.
So what is the minimum information needed to unambiguously specify an element of $Y$?
EDIT: It turns out my concept is an existing mathematical concept, known as a germ. So my question reduces to, what uniquely characterizes the germ of a smooth function?
calculus real-analysis taylor-expansion analyticity analytic-functions
asked 2 days ago
Keshav Srinivasan
1,86311337
This question has an open bounty worth +150
reputation from Keshav Srinivasan ending ending at 2018-08-12 04:42:54Z">in 6 days.
This question has not received enough attention.
5
Maybe you want to look at this, but it does not contain an answer to your question: en.wikipedia.org/wiki/Germ_(mathematics)
– Christian Blatter
2 days ago
@ChristianBlatter Wow, my concept is an existing mathematical concept!
– Keshav Srinivasan
2 days ago
What do you mean "characterizes"? What you said in the first paragraph is the definition of germ, so one way to characterize it would be just use this definition. Do you want an alternative equivalent definition, or do you want the motivation behind this definition.
– Xiao
6 hours ago
@Xiao No, I mean, what is the minimum information needed to unambiguously specify one germ as opposed to all the other germs? The value of the function at zero along with the values of nth derivative of the function for all n isn’t enough to unambiguously specify a particular germ, because there are multiple germs which share that information. So what is the minimum information needed to specify a single germ?
– Keshav Srinivasan
5 hours ago
@Xiao I’ll tell you the motivation for my question. The higher-order derivative test is inconclusive if $f^(n)(0)=0$ for all $n$. So I’m wondering whether there’s some way to tell if a point is a local maximum or minimum in the case where the higher-order derivative fails, in a way that only depends on the germ of $f$. See my question here: math.stackexchange.com/q/2871719/71829 But in order to figure that out, I need to know what information about a function other than the values of its nth derivatives at the point is captured by the germ of the function.
– Keshav Srinivasan
5 hours ago
add a comment |Â
up vote
6
down vote
favorite
up vote
6
down vote
favorite
Let $X$ be the set of all functions $f:mathbbRrightarrowmathbbR$ which are infinitely differentiable at $0$. Let us define an equivalence relation $sim$ on $X$ by saying that $fsim g$ if there exists a $delta>0$ such that $f(x)=g(x)$ for all points in the interval $(-delta,delta)$. And let $Y$ be the set of equivalence classes of elements of $X$ under $sim$.
My question is, what characterizes a given equivalence class in $Y$? The values of $f(0)$ and $f^(n)(0)$ for all $n$ aren't enough, because for any given analytic function $f$ there exists an non-analytic infinitely differentiable function $g$ such that $f(0)=g(0)$ and $f^(n)=g^(n)$ for all $n$ but where it's not the case that $fsim g$.
So what is the minimum information needed to unambiguously specify an element of $Y$?
EDIT: It turns out my concept is an existing mathematical concept, known as a germ. So my question reduces to, what uniquely characterizes the germ of a smooth function?
calculus real-analysis taylor-expansion analyticity analytic-functions
asked 2 days ago
Keshav Srinivasan
1,86311337
Let $X$ be the set of all functions $f:mathbbRrightarrowmathbbR$ which are infinitely differentiable at $0$. Let us define an equivalence relation $sim$ on $X$ by saying that $fsim g$ if there exists a $delta>0$ such that $f(x)=g(x)$ for all points in the interval $(-delta,delta)$. And let $Y$ be the set of equivalence classes of elements of $X$ under $sim$.
My question is, what characterizes a given equivalence class in $Y$? The values of $f(0)$ and $f^(n)(0)$ for all $n$ aren't enough, because for any given analytic function $f$ there exists an non-analytic infinitely differentiable function $g$ such that $f(0)=g(0)$ and $f^(n)=g^(n)$ for all $n$ but where it's not the case that $fsim g$.
So what is the minimum information needed to unambiguously specify an element of $Y$?
EDIT: It turns out my concept is an existing mathematical concept, known as a germ. So my question reduces to, what uniquely characterizes the germ of a smooth function?
calculus real-analysis taylor-expansion analyticity analytic-functions
asked 2 days ago
Keshav Srinivasan
1,86311337
edited 6 hours ago
asked 2 days ago
Keshav Srinivasan
1,86311337
asked 2 days ago
Keshav Srinivasan
1,86311337
asked 2 days ago
Keshav Srinivasan
1,86311337
1,86311337
This question has an open bounty worth +150
reputation from Keshav Srinivasan ending ending at 2018-08-12 04:42:54Z">in 6 days.
This question has not received enough attention.
This question has an open bounty worth +150
reputation from Keshav Srinivasan ending ending at 2018-08-12 04:42:54Z">in 6 days.
This question has not received enough attention.
5
Maybe you want to look at this, but it does not contain an answer to your question: en.wikipedia.org/wiki/Germ_(mathematics)
– Christian Blatter
2 days ago
@ChristianBlatter Wow, my concept is an existing mathematical concept!
– Keshav Srinivasan
2 days ago
What do you mean "characterizes"? What you said in the first paragraph is the definition of germ, so one way to characterize it would be just use this definition. Do you want an alternative equivalent definition, or do you want the motivation behind this definition.
– Xiao
6 hours ago
@Xiao No, I mean, what is the minimum information needed to unambiguously specify one germ as opposed to all the other germs? The value of the function at zero along with the values of nth derivative of the function for all n isn’t enough to unambiguously specify a particular germ, because there are multiple germs which share that information. So what is the minimum information needed to specify a single germ?
– Keshav Srinivasan
5 hours ago
@Xiao I’ll tell you the motivation for my question. The higher-order derivative test is inconclusive if $f^(n)(0)=0$ for all $n$. So I’m wondering whether there’s some way to tell if a point is a local maximum or minimum in the case where the higher-order derivative fails, in a way that only depends on the germ of $f$. See my question here: math.stackexchange.com/q/2871719/71829 But in order to figure that out, I need to know what information about a function other than the values of its nth derivatives at the point is captured by the germ of the function.
– Keshav Srinivasan
5 hours ago
add a comment |Â
5
Maybe you want to look at this, but it does not contain an answer to your question: en.wikipedia.org/wiki/Germ_(mathematics)
– Christian Blatter
2 days ago
@ChristianBlatter Wow, my concept is an existing mathematical concept!
– Keshav Srinivasan
2 days ago
What do you mean "characterizes"? What you said in the first paragraph is the definition of germ, so one way to characterize it would be just use this definition. Do you want an alternative equivalent definition, or do you want the motivation behind this definition.
– Xiao
6 hours ago
@Xiao No, I mean, what is the minimum information needed to unambiguously specify one germ as opposed to all the other germs? The value of the function at zero along with the values of nth derivative of the function for all n isn’t enough to unambiguously specify a particular germ, because there are multiple germs which share that information. So what is the minimum information needed to specify a single germ?
– Keshav Srinivasan
5 hours ago
@Xiao I’ll tell you the motivation for my question. The higher-order derivative test is inconclusive if $f^(n)(0)=0$ for all $n$. So I’m wondering whether there’s some way to tell if a point is a local maximum or minimum in the case where the higher-order derivative fails, in a way that only depends on the germ of $f$. See my question here: math.stackexchange.com/q/2871719/71829 But in order to figure that out, I need to know what information about a function other than the values of its nth derivatives at the point is captured by the germ of the function.
– Keshav Srinivasan
5 hours ago
5
5
Maybe you want to look at this, but it does not contain an answer to your question: en.wikipedia.org/wiki/Germ_(mathematics)
– Christian Blatter
2 days ago
Maybe you want to look at this, but it does not contain an answer to your question: en.wikipedia.org/wiki/Germ_(mathematics)
– Christian Blatter
2 days ago
@ChristianBlatter Wow, my concept is an existing mathematical concept!
– Keshav Srinivasan
2 days ago
@ChristianBlatter Wow, my concept is an existing mathematical concept!
– Keshav Srinivasan
2 days ago
What do you mean "characterizes"? What you said in the first paragraph is the definition of germ, so one way to characterize it would be just use this definition. Do you want an alternative equivalent definition, or do you want the motivation behind this definition.
– Xiao
6 hours ago
What do you mean "characterizes"? What you said in the first paragraph is the definition of germ, so one way to characterize it would be just use this definition. Do you want an alternative equivalent definition, or do you want the motivation behind this definition.
– Xiao
6 hours ago
@Xiao No, I mean, what is the minimum information needed to unambiguously specify one germ as opposed to all the other germs? The value of the function at zero along with the values of nth derivative of the function for all n isn’t enough to unambiguously specify a particular germ, because there are multiple germs which share that information. So what is the minimum information needed to specify a single germ?
– Keshav Srinivasan
5 hours ago
@Xiao No, I mean, what is the minimum information needed to unambiguously specify one germ as opposed to all the other germs? The value of the function at zero along with the values of nth derivative of the function for all n isn’t enough to unambiguously specify a particular germ, because there are multiple germs which share that information. So what is the minimum information needed to specify a single germ?
– Keshav Srinivasan
5 hours ago
@Xiao I’ll tell you the motivation for my question. The higher-order derivative test is inconclusive if $f^(n)(0)=0$ for all $n$. So I’m wondering whether there’s some way to tell if a point is a local maximum or minimum in the case where the higher-order derivative fails, in a way that only depends on the germ of $f$. See my question here: math.stackexchange.com/q/2871719/71829 But in order to figure that out, I need to know what information about a function other than the values of its nth derivatives at the point is captured by the germ of the function.
– Keshav Srinivasan
5 hours ago
@Xiao I’ll tell you the motivation for my question. The higher-order derivative test is inconclusive if $f^(n)(0)=0$ for all $n$. So I’m wondering whether there’s some way to tell if a point is a local maximum or minimum in the case where the higher-order derivative fails, in a way that only depends on the germ of $f$. See my question here: math.stackexchange.com/q/2871719/71829 But in order to figure that out, I need to know what information about a function other than the values of its nth derivatives at the point is captured by the germ of the function.
– Keshav Srinivasan
5 hours ago
add a comment |Â
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5
Maybe you want to look at this, but it does not contain an answer to your question: en.wikipedia.org/wiki/Germ_(mathematics)
– Christian Blatter
2 days ago
@ChristianBlatter Wow, my concept is an existing mathematical concept!
– Keshav Srinivasan
2 days ago
What do you mean "characterizes"? What you said in the first paragraph is the definition of germ, so one way to characterize it would be just use this definition. Do you want an alternative equivalent definition, or do you want the motivation behind this definition.
– Xiao
6 hours ago
@Xiao No, I mean, what is the minimum information needed to unambiguously specify one germ as opposed to all the other germs? The value of the function at zero along with the values of nth derivative of the function for all n isn’t enough to unambiguously specify a particular germ, because there are multiple germs which share that information. So what is the minimum information needed to specify a single germ?
– Keshav Srinivasan
5 hours ago
@Xiao I’ll tell you the motivation for my question. The higher-order derivative test is inconclusive if $f^(n)(0)=0$ for all $n$. So I’m wondering whether there’s some way to tell if a point is a local maximum or minimum in the case where the higher-order derivative fails, in a way that only depends on the germ of $f$. See my question here: math.stackexchange.com/q/2871719/71829 But in order to figure that out, I need to know what information about a function other than the values of its nth derivatives at the point is captured by the germ of the function.
– Keshav Srinivasan
5 hours ago