Mixing
Saturated Addition Over Tri-state Pulser Input: What kind of algebraic structure is this?
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I'm currently working with a "tri-state pulser" in an engineering context. To excite a response from this device, one provides a voltage sequence consisting of elements from $-1,0,1$. For example, a valid input looks something like this: 
One can write out the shape of this input as a sequence: $(-1,1,0,0,-1,0,1)$. In general if we restrict valid inputs to be of length $L$, then a valid input is a tuple of the form $-1,0,1^L$.
I've been self-learning some algebra on the side, and was realizing that these valid inputs start to look like a vector space. To illustrate in the case of $L=3$:
- we can sometimes add these inputs nicely element-wise: $(0,1,-1) + (1,0,1) = (1,1,0)$
- we have an additive neutral element $(0,0,0)$
- we can multiply inputs element-wise by scalars from $(-1,1,0)$
- under element-wise addition each element has an "inverse" in the sense that another sequence exists that sends it to the neutral element (but elements are not invertible)
But we have a problem here! This is because we can't specify an input bigger than $1$ in absolute value to the pulser. We can model the pulser input as saturating so that the input $(1,-1) + (1,-1) = (1,-1)$. This implies that we do not have have associativity, as $-1+(1+1) = -1 + 1 = 0$ but $(-1+1)+1 = 0+1 = 1$.
Upon realizing this, I began to wonder if there was some other acceptable way of defining the group operation to both faithfully model the tri-state pulser but also get an actual group structure. Inputs are really a sort of "direct sum" of individual inputs, and so we can consider the problem at the individual input level. Unfortunately, there is only one unique group (up to isomorphism) with three elements, and this is the cyclic group $mathbbZ/3mathbbZ$.
Unfortunately, it makes no physical sense to say that providing a very high input voltage level (ex. $6$) to the pulser should be the same thing as providing zero voltage to the pulser. So the cyclic group doesn't seem to provide a good model.
So we have a structure that superficially looks like a vector space, where:
- "addition" is commutative, but not associative
- "addition" has an additive neutral, and "inverses" exist
- multiplication by a scalar is defined, has identity, and is distributive
What algebraic object is this?
(Parenthetical follow-up: Is group theory really the right tool to try and understand this structure? Any introductory book or course note recommendations on the fields of mathematics that study these weird structures are appreciated).
abstract-algebra reference-request mathematical-modeling universal-algebra
asked 58 mins ago
David Egolf
526
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up vote
2
down vote
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I'm currently working with a "tri-state pulser" in an engineering context. To excite a response from this device, one provides a voltage sequence consisting of elements from $-1,0,1$. For example, a valid input looks something like this:
One can write out the shape of this input as a sequence: $(-1,1,0,0,-1,0,1)$. In general if we restrict valid inputs to be of length $L$, then a valid input is a tuple of the form $-1,0,1^L$.
I've been self-learning some algebra on the side, and was realizing that these valid inputs start to look like a vector space. To illustrate in the case of $L=3$:
- we can sometimes add these inputs nicely element-wise: $(0,1,-1) + (1,0,1) = (1,1,0)$
- we have an additive neutral element $(0,0,0)$
- we can multiply inputs element-wise by scalars from $(-1,1,0)$
- under element-wise addition each element has an "inverse" in the sense that another sequence exists that sends it to the neutral element (but elements are not invertible)
But we have a problem here! This is because we can't specify an input bigger than $1$ in absolute value to the pulser. We can model the pulser input as saturating so that the input $(1,-1) + (1,-1) = (1,-1)$. This implies that we do not have have associativity, as $-1+(1+1) = -1 + 1 = 0$ but $(-1+1)+1 = 0+1 = 1$.
Upon realizing this, I began to wonder if there was some other acceptable way of defining the group operation to both faithfully model the tri-state pulser but also get an actual group structure. Inputs are really a sort of "direct sum" of individual inputs, and so we can consider the problem at the individual input level. Unfortunately, there is only one unique group (up to isomorphism) with three elements, and this is the cyclic group $mathbbZ/3mathbbZ$.
Unfortunately, it makes no physical sense to say that providing a very high input voltage level (ex. $6$) to the pulser should be the same thing as providing zero voltage to the pulser. So the cyclic group doesn't seem to provide a good model.
So we have a structure that superficially looks like a vector space, where:
- "addition" is commutative, but not associative
- "addition" has an additive neutral, and "inverses" exist
- multiplication by a scalar is defined, has identity, and is distributive
What algebraic object is this?
(Parenthetical follow-up: Is group theory really the right tool to try and understand this structure? Any introductory book or course note recommendations on the fields of mathematics that study these weird structures are appreciated).
abstract-algebra reference-request mathematical-modeling universal-algebra
asked 58 mins ago
David Egolf
526
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I'm currently working with a "tri-state pulser" in an engineering context. To excite a response from this device, one provides a voltage sequence consisting of elements from $-1,0,1$. For example, a valid input looks something like this:
One can write out the shape of this input as a sequence: $(-1,1,0,0,-1,0,1)$. In general if we restrict valid inputs to be of length $L$, then a valid input is a tuple of the form $-1,0,1^L$.
I've been self-learning some algebra on the side, and was realizing that these valid inputs start to look like a vector space. To illustrate in the case of $L=3$:
- we can sometimes add these inputs nicely element-wise: $(0,1,-1) + (1,0,1) = (1,1,0)$
- we have an additive neutral element $(0,0,0)$
- we can multiply inputs element-wise by scalars from $(-1,1,0)$
- under element-wise addition each element has an "inverse" in the sense that another sequence exists that sends it to the neutral element (but elements are not invertible)
But we have a problem here! This is because we can't specify an input bigger than $1$ in absolute value to the pulser. We can model the pulser input as saturating so that the input $(1,-1) + (1,-1) = (1,-1)$. This implies that we do not have have associativity, as $-1+(1+1) = -1 + 1 = 0$ but $(-1+1)+1 = 0+1 = 1$.
Upon realizing this, I began to wonder if there was some other acceptable way of defining the group operation to both faithfully model the tri-state pulser but also get an actual group structure. Inputs are really a sort of "direct sum" of individual inputs, and so we can consider the problem at the individual input level. Unfortunately, there is only one unique group (up to isomorphism) with three elements, and this is the cyclic group $mathbbZ/3mathbbZ$.
Unfortunately, it makes no physical sense to say that providing a very high input voltage level (ex. $6$) to the pulser should be the same thing as providing zero voltage to the pulser. So the cyclic group doesn't seem to provide a good model.
So we have a structure that superficially looks like a vector space, where:
- "addition" is commutative, but not associative
- "addition" has an additive neutral, and "inverses" exist
- multiplication by a scalar is defined, has identity, and is distributive
What algebraic object is this?
(Parenthetical follow-up: Is group theory really the right tool to try and understand this structure? Any introductory book or course note recommendations on the fields of mathematics that study these weird structures are appreciated).
abstract-algebra reference-request mathematical-modeling universal-algebra
asked 58 mins ago
David Egolf
526
I'm currently working with a "tri-state pulser" in an engineering context. To excite a response from this device, one provides a voltage sequence consisting of elements from $-1,0,1$. For example, a valid input looks something like this:
One can write out the shape of this input as a sequence: $(-1,1,0,0,-1,0,1)$. In general if we restrict valid inputs to be of length $L$, then a valid input is a tuple of the form $-1,0,1^L$.
I've been self-learning some algebra on the side, and was realizing that these valid inputs start to look like a vector space. To illustrate in the case of $L=3$:
- we can sometimes add these inputs nicely element-wise: $(0,1,-1) + (1,0,1) = (1,1,0)$
- we have an additive neutral element $(0,0,0)$
- we can multiply inputs element-wise by scalars from $(-1,1,0)$
- under element-wise addition each element has an "inverse" in the sense that another sequence exists that sends it to the neutral element (but elements are not invertible)
But we have a problem here! This is because we can't specify an input bigger than $1$ in absolute value to the pulser. We can model the pulser input as saturating so that the input $(1,-1) + (1,-1) = (1,-1)$. This implies that we do not have have associativity, as $-1+(1+1) = -1 + 1 = 0$ but $(-1+1)+1 = 0+1 = 1$.
Upon realizing this, I began to wonder if there was some other acceptable way of defining the group operation to both faithfully model the tri-state pulser but also get an actual group structure. Inputs are really a sort of "direct sum" of individual inputs, and so we can consider the problem at the individual input level. Unfortunately, there is only one unique group (up to isomorphism) with three elements, and this is the cyclic group $mathbbZ/3mathbbZ$.
Unfortunately, it makes no physical sense to say that providing a very high input voltage level (ex. $6$) to the pulser should be the same thing as providing zero voltage to the pulser. So the cyclic group doesn't seem to provide a good model.
So we have a structure that superficially looks like a vector space, where:
- "addition" is commutative, but not associative
- "addition" has an additive neutral, and "inverses" exist
- multiplication by a scalar is defined, has identity, and is distributive
What algebraic object is this?
(Parenthetical follow-up: Is group theory really the right tool to try and understand this structure? Any introductory book or course note recommendations on the fields of mathematics that study these weird structures are appreciated).
abstract-algebra reference-request mathematical-modeling universal-algebra
asked 58 mins ago
David Egolf
526
asked 58 mins ago
David Egolf
526
asked 58 mins ago
David Egolf
526
asked 58 mins ago
David Egolf
526
526
add a comment |Â
add a comment |Â
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