Mixing
Name for the class of functions of the form $f:A^Nto A$
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Let $A$ be some space and $N$ be an integer. Does a function of the form $f:A^Nto A$ have a particular name? I'm looking for a keyword to see if there are any general properties of functions of this type.
reference-request
asked Jan 8 at 16:45
jonemjonem
378415
$endgroup$
add a comment |
$begingroup$
Let $A$ be some space and $N$ be an integer. Does a function of the form $f:A^Nto A$ have a particular name? I'm looking for a keyword to see if there are any general properties of functions of this type.
reference-request
asked Jan 8 at 16:45
jonemjonem
378415
$endgroup$
$begingroup$
That depends, is $A$ a space of scalars, or any space?
$endgroup$
– R. Burton
Jan 8 at 16:48
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Usually it's $f:Nrightarrow A^N rightarrow A$ and it's called evaluation, i.e. normal function call in any programming language.
$endgroup$
– tp1
Jan 8 at 16:50
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@R.Burton Any space will do
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– jonem
Jan 8 at 16:50
1
$begingroup$
$N$-ary operator? (See en.wikipedia.org/wiki/Arity.)
$endgroup$
– user614671
Jan 8 at 17:03
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@jonem I'm not sure then. If $A$ is a scalar space, then $f$ belongs to the class of scalar functions (mathworld.wolfram.com/ScalarFunction.html). But if $f$ is tensor-valued, then it could be almost anything. You could consider $f$ to be the class of all possible index/dimension reductions (of which tensor contraction would be a special case). But you would have to allow the Cartesian product of two $n$-dimensional objects to yield an $n^2$-dimensional object (i.e. the vector $((x_1,x_2),(y_1,y_2))$ is equivalent to the matrix $left(begin{matrix}x_1&x_2\y_1&y_2end{matrix}right)$).
$endgroup$
– R. Burton
Jan 8 at 17:03
add a comment |
$begingroup$
Let $A$ be some space and $N$ be an integer. Does a function of the form $f:A^Nto A$ have a particular name? I'm looking for a keyword to see if there are any general properties of functions of this type.
reference-request
asked Jan 8 at 16:45
jonemjonem
378415
$endgroup$
Let $A$ be some space and $N$ be an integer. Does a function of the form $f:A^Nto A$ have a particular name? I'm looking for a keyword to see if there are any general properties of functions of this type.
reference-request
reference-request
asked Jan 8 at 16:45
jonemjonem
378415
asked Jan 8 at 16:45
jonemjonem
378415
asked Jan 8 at 16:45
jonemjonem
378415
asked Jan 8 at 16:45
jonemjonem
378415
asked Jan 8 at 16:45
jonemjonem
378415
378415
$begingroup$
That depends, is $A$ a space of scalars, or any space?
$endgroup$
– R. Burton
Jan 8 at 16:48
$begingroup$
Usually it's $f:Nrightarrow A^N rightarrow A$ and it's called evaluation, i.e. normal function call in any programming language.
$endgroup$
– tp1
Jan 8 at 16:50
$begingroup$
@R.Burton Any space will do
$endgroup$
– jonem
Jan 8 at 16:50
1
$begingroup$
$N$-ary operator? (See en.wikipedia.org/wiki/Arity.)
$endgroup$
– user614671
Jan 8 at 17:03
$begingroup$
@jonem I'm not sure then. If $A$ is a scalar space, then $f$ belongs to the class of scalar functions (mathworld.wolfram.com/ScalarFunction.html). But if $f$ is tensor-valued, then it could be almost anything. You could consider $f$ to be the class of all possible index/dimension reductions (of which tensor contraction would be a special case). But you would have to allow the Cartesian product of two $n$-dimensional objects to yield an $n^2$-dimensional object (i.e. the vector $((x_1,x_2),(y_1,y_2))$ is equivalent to the matrix $left(begin{matrix}x_1&x_2\y_1&y_2end{matrix}right)$).
$endgroup$
– R. Burton
Jan 8 at 17:03
add a comment |
$begingroup$
That depends, is $A$ a space of scalars, or any space?
$endgroup$
– R. Burton
Jan 8 at 16:48
$begingroup$
Usually it's $f:Nrightarrow A^N rightarrow A$ and it's called evaluation, i.e. normal function call in any programming language.
$endgroup$
– tp1
Jan 8 at 16:50
$begingroup$
@R.Burton Any space will do
$endgroup$
– jonem
Jan 8 at 16:50
1
$begingroup$
$N$-ary operator? (See en.wikipedia.org/wiki/Arity.)
$endgroup$
– user614671
Jan 8 at 17:03
$begingroup$
@jonem I'm not sure then. If $A$ is a scalar space, then $f$ belongs to the class of scalar functions (mathworld.wolfram.com/ScalarFunction.html). But if $f$ is tensor-valued, then it could be almost anything. You could consider $f$ to be the class of all possible index/dimension reductions (of which tensor contraction would be a special case). But you would have to allow the Cartesian product of two $n$-dimensional objects to yield an $n^2$-dimensional object (i.e. the vector $((x_1,x_2),(y_1,y_2))$ is equivalent to the matrix $left(begin{matrix}x_1&x_2\y_1&y_2end{matrix}right)$).
$endgroup$
– R. Burton
Jan 8 at 17:03
$begingroup$
That depends, is $A$ a space of scalars, or any space?
$endgroup$
– R. Burton
Jan 8 at 16:48
$begingroup$
That depends, is $A$ a space of scalars, or any space?
$endgroup$
– R. Burton
Jan 8 at 16:48
$begingroup$
Usually it's $f:Nrightarrow A^N rightarrow A$ and it's called evaluation, i.e. normal function call in any programming language.
$endgroup$
– tp1
Jan 8 at 16:50
$begingroup$
Usually it's $f:Nrightarrow A^N rightarrow A$ and it's called evaluation, i.e. normal function call in any programming language.
$endgroup$
– tp1
Jan 8 at 16:50
$begingroup$
@R.Burton Any space will do
$endgroup$
– jonem
Jan 8 at 16:50
$begingroup$
@R.Burton Any space will do
$endgroup$
– jonem
Jan 8 at 16:50
1
1
$begingroup$
$N$-ary operator? (See en.wikipedia.org/wiki/Arity.)
$endgroup$
– user614671
Jan 8 at 17:03
$begingroup$
$N$-ary operator? (See en.wikipedia.org/wiki/Arity.)
$endgroup$
– user614671
Jan 8 at 17:03
$begingroup$
@jonem I'm not sure then. If $A$ is a scalar space, then $f$ belongs to the class of scalar functions (mathworld.wolfram.com/ScalarFunction.html). But if $f$ is tensor-valued, then it could be almost anything. You could consider $f$ to be the class of all possible index/dimension reductions (of which tensor contraction would be a special case). But you would have to allow the Cartesian product of two $n$-dimensional objects to yield an $n^2$-dimensional object (i.e. the vector $((x_1,x_2),(y_1,y_2))$ is equivalent to the matrix $left(begin{matrix}x_1&x_2\y_1&y_2end{matrix}right)$).
$endgroup$
– R. Burton
Jan 8 at 17:03
$begingroup$
@jonem I'm not sure then. If $A$ is a scalar space, then $f$ belongs to the class of scalar functions (mathworld.wolfram.com/ScalarFunction.html). But if $f$ is tensor-valued, then it could be almost anything. You could consider $f$ to be the class of all possible index/dimension reductions (of which tensor contraction would be a special case). But you would have to allow the Cartesian product of two $n$-dimensional objects to yield an $n^2$-dimensional object (i.e. the vector $((x_1,x_2),(y_1,y_2))$ is equivalent to the matrix $left(begin{matrix}x_1&x_2\y_1&y_2end{matrix}right)$).
$endgroup$
– R. Burton
Jan 8 at 17:03
add a comment |
1 Answer
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$begingroup$
$f$ is an operation on the set $A$ of arity $N.$ More specifically, a finitary operation.
Of course "operation" has many meanings, but this meaning is certainly common particularly when discussing algebraic theories in general. For example it is used in Grätzer's Universal Algebra, a standard reference.
answered Jan 9 at 20:02
DapDap
16.4k738
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add a comment |
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1 Answer
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$begingroup$
$f$ is an operation on the set $A$ of arity $N.$ More specifically, a finitary operation.
Of course "operation" has many meanings, but this meaning is certainly common particularly when discussing algebraic theories in general. For example it is used in Grätzer's Universal Algebra, a standard reference.
answered Jan 9 at 20:02
DapDap
16.4k738
$endgroup$
add a comment |
$begingroup$
$f$ is an operation on the set $A$ of arity $N.$ More specifically, a finitary operation.
Of course "operation" has many meanings, but this meaning is certainly common particularly when discussing algebraic theories in general. For example it is used in Grätzer's Universal Algebra, a standard reference.
answered Jan 9 at 20:02
DapDap
16.4k738
$endgroup$
add a comment |
$begingroup$
$f$ is an operation on the set $A$ of arity $N.$ More specifically, a finitary operation.
Of course "operation" has many meanings, but this meaning is certainly common particularly when discussing algebraic theories in general. For example it is used in Grätzer's Universal Algebra, a standard reference.
answered Jan 9 at 20:02
DapDap
16.4k738
$endgroup$
$f$ is an operation on the set $A$ of arity $N.$ More specifically, a finitary operation.
Of course "operation" has many meanings, but this meaning is certainly common particularly when discussing algebraic theories in general. For example it is used in Grätzer's Universal Algebra, a standard reference.
answered Jan 9 at 20:02
DapDap
16.4k738
answered Jan 9 at 20:02
DapDap
16.4k738
answered Jan 9 at 20:02
DapDap
16.4k738
answered Jan 9 at 20:02
DapDap
16.4k738
16.4k738
add a comment |
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$begingroup$
That depends, is $A$ a space of scalars, or any space?
$endgroup$
– R. Burton
Jan 8 at 16:48
$begingroup$
Usually it's $f:Nrightarrow A^N rightarrow A$ and it's called evaluation, i.e. normal function call in any programming language.
$endgroup$
– tp1
Jan 8 at 16:50
$begingroup$
@R.Burton Any space will do
$endgroup$
– jonem
Jan 8 at 16:50
1
$begingroup$
$N$-ary operator? (See en.wikipedia.org/wiki/Arity.)
$endgroup$
– user614671
Jan 8 at 17:03
$begingroup$
@jonem I'm not sure then. If $A$ is a scalar space, then $f$ belongs to the class of scalar functions (mathworld.wolfram.com/ScalarFunction.html). But if $f$ is tensor-valued, then it could be almost anything. You could consider $f$ to be the class of all possible index/dimension reductions (of which tensor contraction would be a special case). But you would have to allow the Cartesian product of two $n$-dimensional objects to yield an $n^2$-dimensional object (i.e. the vector $((x_1,x_2),(y_1,y_2))$ is equivalent to the matrix $left(begin{matrix}x_1&x_2\y_1&y_2end{matrix}right)$).
$endgroup$
– R. Burton
Jan 8 at 17:03