Understanding Sign Convention and Its Application in Vectors

[Cinitiator]

Homework Statement

What is sign convention, and how is it applied to vectors? I was reading an economics paper when I came across this term. There's a ":" sign between the vector in question and an entry, which is all supposed to output a real number.

What exactly is sign convention, and how is it applied?

The phrase which was used "Under the sign convention on the input–output vector".

Homework Equations

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The Attempt at a Solution

I've tried searching for it but I found very little information on the topic.

 

[Simon Bridge]

Note: the "sign" associated with a number or a vector is a + or a - ... the colon is not usually considered a "sign" so your question is about "notation".

People use whatever notation takes their fancy in papers - there are lots of conventions surrounding colons and other punctuation marks in notation. Without the specific context in which it is used we can only guess.

What do you mean by "entry"? Is the entry into a spreadsheet and the vector is the cell location? Perhaps you mean that the "vector" is an operator and the "entry" is an operand?

Can you write out the use for us in more detail?
Anyway - it looks like this involves the input-output model in economics...
http://en.wikipedia.org/wiki/Input-output_model
... is that correct?

 

[Cinitiator]

Note: the "sign" associated with a number or a vector is a + or a - ... the colon is not usually considered a "sign" so your question is about "notation".

People use whatever notation takes their fancy in papers - there are lots of conventions surrounding colons and other punctuation marks in notation. Without the specific context in which it is used we can only guess.

What do you mean by "entry"? Is the entry into a spreadsheet and the vector is the cell location? Perhaps you mean that the "vector" is an operator and the "entry" is an operand?

Can you write out the use for us in more detail?
Anyway - it looks like this involves the input-output model in economics...
http://en.wikipedia.org/wiki/Input-output_model
... is that correct?

In the text, it said "Under the sign convention on the input–output vector" as well as "We adopt the usual sign convention for the vector b", and it has a vector, let's name it b, which has its members labeled x1, x2, xn, etc (the subscript notation is used for 1, 2, n, etc).

It said that 'we adopt the usual sign convention for the vector b', and the ":" notation was used as follows:
b:xn < 0


From what I understand, the notation serves to select a given entry (as in, member) from the vector, since it mentioned that if the value is negative, then the firm is a net user of the product in question, and since b is a production vector (which has net quantities of each product a firm uses/produces).

After assuming that this was the case, everything made sense. However, I have no idea what "sign convention" means. Removing the term doesn't seem to change the meaning, though.

Anyway - it looks like this involves the input-output model in economics...
http://en.wikipedia.org/wiki/Input-output_model
... is that correct?

And it's not exactly the Leontief input-output model. The vector is a part of the First Fundamental Theorem of Welfare Economics. This vector is simply a net production vector - negative values imply that the firm inputs more of the product in question than it outputs. It doesn't exactly specify the inputs needed for each output, or for the aggregate output.

 

[Simon Bridge]

Oh - the colon means "such that" (common in math) and there should be a space either side of it ... they are saying that all the elements of b are less than zero ... making the entire vector negative ... which would be the "sign convention" part.

 

[Cinitiator]

Oh - the colon means "such that" (common in math) and there should be a space either side of it ... they are saying that all the elements of b are less than zero ... making the entire vector negative ... which would be the "sign convention" part.

Thanks for your input.

The n is said to be a "given n", though. That is, we select some member from the vector in question.

I assume it just means "a vector b such that x1 (for example) is under 0".

 

[Simon Bridge]

That's why I needed the statement exactly as written - it could mean there is a least one element of b is negative or that all elements of b are negative. The answer to your question though, is that the colon means "such that".

I suspect the passage just has the n doing double duty ... it could be the size of the vector, or a representative integer between 1 and the size of the vector. Both meaning seem to have been used... otherwise x_n would be the last element of b.

But books will have all kinds of sloppy writing in them and you just have to watch for the changes in context. Check to see which interpretation matches the rest of the passage.