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Jul
29
revised Block of integers: Divisibility
added 18 characters in body
Jul
27
comment Why does $n \choose r$ where $r = 1,n$ track $2^n$?
In your array application, consider marking each array element with a 0 for "not taken" or a 1 for "taken". There are two choices for each array element and therefore $2^n$ ways to mark the entire array.
Jul
26
awarded  Necromancer
Jul
23
comment Ignoring the workspace in quantum computation
Does Mermin assume anything about the initial state of the output bits? It seems to me that for the computation shown in the diagram to be at all useful, you would at least have to know what the initial state of the output bits was. If one further assumes a particular initial state for the output bits, one may be able to say even more.
Jul
13
revised Maximal determinant of a $\{1,−1\}$ matrix of size $n$ is $2^{n−1}$ times the maximal determinant of a $ \{0,1\}$ matrix of size $n−1$.
added 3 characters in body
Jul
13
comment Maximal determinant of a $\{1,−1\}$ matrix of size $n$ is $2^{n−1}$ times the maximal determinant of a $ \{0,1\}$ matrix of size $n−1$.
See Connection of the maximal determinant problems for {1, −1} and {0, 1} matrices in the Wikipedia article on Hadamard's maximal determinant problem.
Jul
13
revised Proving $n^3 + 3n^2 +2n$ is divisible by $6$
edited tags
Jul
1
comment Hadamard matrices and sub-matrices (Converse of Sylvester Construction)
By the way, for size 16, there are five equivalence classes, and all five have Hadamard submatrices of size 8. I have no idea what the situation is for size 32. I'm not sure whether I can be helpful to you. Can you say why you refer to $B$ as a Hadamard submatrix?
Jul
1
comment Hadamard matrices and sub-matrices (Converse of Sylvester Construction)
The belief stated in my earlier comment - that it would be relatively easy to say something about Hadamard submatrices of Hadamard matrices whose size is an odd multiple of 8 - appears to have been unfounded. There are 60 equivalence classes of Hadamard matrices of size 24, and I expected that only a small subset of these with simple structure would have Hadamard submatrices of size 12. This is completely wrong. Now that I've done the calculation, I find that 58 of the 60 equivalence classes have such Hadamard submatrices, and that the structure can be more complicated than I thought.
Jun
30
comment How to prove infinite solution vs no solution for singular matrix problem.
As for the general question: write your system as an augmented matrix and row-reduce. If the coefficient matrix is singular, you will end up with a all-zero row on the left. The system is consistent only if you also have 0 on the right.
Jun
30
comment How to prove infinite solution vs no solution for singular matrix problem.
Should the $a$s in your definition of $B$ be $\alpha$s? Is your coefficient matrix written correctly? It doesn't look singular to me (except when $\alpha=0$).
Jun
25
comment Hadamard matrices and sub-matrices (Converse of Sylvester Construction)
I see that you changed to question in response to Jyrki Lahtonen's comments by restricting $d$ to powers of $2$ instead of letting it be any multiple of $4$. You could have eliminated his counterexample simply by requiring $d$ to be a multiple of $8$. My belief is that it's probably easier to say something about odd multiples of $8$ than about powers of $2$. In fact, I suspect that the higher the power of $2$ dividing $d$, there more difficult things become. If you are interested, I can try to elaborate.
Jun
24
comment Relation between number of unique values in Gramian Matrix (G) and the matrices that created it
I've been iteratively improving the answer, and now think I'm satisfied with it, unless there are questions. I plan to submit the sequence to OEIS.
Jun
24
revised Relation between number of unique values in Gramian Matrix (G) and the matrices that created it
nonrecursive formula
Jun
24
revised Relation between number of unique values in Gramian Matrix (G) and the matrices that created it
completed proof (sketch); removed extra equalities from deflation rules, leading to faster evaluation and greater compatibility with proof of the rules; modified examples accordingly
Jun
24
revised Relation between number of unique values in Gramian Matrix (G) and the matrices that created it
replaced proof of deflation rule 2 with a correct one; sketched proof of deflation rule 3
Jun
18
revised Relation between number of unique values in Gramian Matrix (G) and the matrices that created it
added 2463 characters in body
Jun
18
revised Relation between number of unique values in Gramian Matrix (G) and the matrices that created it
reformatted; corrected significant error in deflation rule 3; eliminated possibility of empty string as it is never used; started the proof
Jun
17
answered Relation between number of unique values in Gramian Matrix (G) and the matrices that created it
Jun
15
comment Visualising finite fields
The link seems no longer to work. Perhaps this is one of the papers you meant? If you know of other links that could be added, that would be very helpful.