Collections of Reals with Small Additive PowerSets
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I'm looking for vectors of $n$ numbers that carry the property that the unique sums of elements of the components of the vector is small.
For example for any $n$ the consider the vector $(1,1,...1)$ consisting of $n$ contiguous $1's$. Interpreting the vector as a set of elements the set of possible sums of these elements is $0,1,2,3,4...n$. I'm looking for other examples of classes of vectors (up to a multiplicative factor of a positive real number) where the possible sums of elements is at most $n+1$ but the vectors contain $n$ non-zero, REAL components.
Any vector of the form $(pm 1, pm 1, pm 1, ... pm 1)$ (where each component is a positive or negative 1 selected independently) is of the this form.
And it's not clear to me if any other vectors exist that carry my desired property and are not a positive multiple of one of the $2^n$ vectors in the aforementioned set.
linear-algebra combinatorics
asked Dec 15 '18 at 21:13
frogeyedpeasfrogeyedpeas
7,56072053
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add a comment |
$begingroup$
I'm looking for vectors of $n$ numbers that carry the property that the unique sums of elements of the components of the vector is small.
For example for any $n$ the consider the vector $(1,1,...1)$ consisting of $n$ contiguous $1's$. Interpreting the vector as a set of elements the set of possible sums of these elements is $0,1,2,3,4...n$. I'm looking for other examples of classes of vectors (up to a multiplicative factor of a positive real number) where the possible sums of elements is at most $n+1$ but the vectors contain $n$ non-zero, REAL components.
Any vector of the form $(pm 1, pm 1, pm 1, ... pm 1)$ (where each component is a positive or negative 1 selected independently) is of the this form.
And it's not clear to me if any other vectors exist that carry my desired property and are not a positive multiple of one of the $2^n$ vectors in the aforementioned set.
linear-algebra combinatorics
asked Dec 15 '18 at 21:13
frogeyedpeasfrogeyedpeas
7,56072053
$endgroup$
add a comment |
$begingroup$
I'm looking for vectors of $n$ numbers that carry the property that the unique sums of elements of the components of the vector is small.
For example for any $n$ the consider the vector $(1,1,...1)$ consisting of $n$ contiguous $1's$. Interpreting the vector as a set of elements the set of possible sums of these elements is $0,1,2,3,4...n$. I'm looking for other examples of classes of vectors (up to a multiplicative factor of a positive real number) where the possible sums of elements is at most $n+1$ but the vectors contain $n$ non-zero, REAL components.
Any vector of the form $(pm 1, pm 1, pm 1, ... pm 1)$ (where each component is a positive or negative 1 selected independently) is of the this form.
And it's not clear to me if any other vectors exist that carry my desired property and are not a positive multiple of one of the $2^n$ vectors in the aforementioned set.
linear-algebra combinatorics
asked Dec 15 '18 at 21:13
frogeyedpeasfrogeyedpeas
7,56072053
$endgroup$
I'm looking for vectors of $n$ numbers that carry the property that the unique sums of elements of the components of the vector is small.
For example for any $n$ the consider the vector $(1,1,...1)$ consisting of $n$ contiguous $1's$. Interpreting the vector as a set of elements the set of possible sums of these elements is $0,1,2,3,4...n$. I'm looking for other examples of classes of vectors (up to a multiplicative factor of a positive real number) where the possible sums of elements is at most $n+1$ but the vectors contain $n$ non-zero, REAL components.
Any vector of the form $(pm 1, pm 1, pm 1, ... pm 1)$ (where each component is a positive or negative 1 selected independently) is of the this form.
And it's not clear to me if any other vectors exist that carry my desired property and are not a positive multiple of one of the $2^n$ vectors in the aforementioned set.
linear-algebra combinatorics
linear-algebra combinatorics
asked Dec 15 '18 at 21:13
frogeyedpeasfrogeyedpeas
7,56072053
asked Dec 15 '18 at 21:13
frogeyedpeasfrogeyedpeas
7,56072053
asked Dec 15 '18 at 21:13
frogeyedpeasfrogeyedpeas
7,56072053
asked Dec 15 '18 at 21:13
frogeyedpeasfrogeyedpeas
7,56072053
asked Dec 15 '18 at 21:13
frogeyedpeasfrogeyedpeas
7,56072053
7,56072053
add a comment |
add a comment |
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