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Jul
7
reviewed Leave Open In which book will I find these types of problems and theory too.
Jul
7
reviewed Close find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that : $ f(f(x))=x^2-2 $
Jul
7
reviewed Leave Open What is the left adjoint of the forgetful functor from fields to integral domains?
Jul
7
reviewed Leave Open Group of Unitaries: Strong Continuity
Jul
2
comment Combinatorial optimization - Bijections between duplicated numbers
Keep me posted. I'm very curious about this problem. It looks like random arrays (as described above) of up to about 35 rows (with elements between 0 and 30) are unlikely to have solutions, but that arrays with 40 or 50 rows do, most often, have solutions.
Jul
2
comment Combinatorial optimization - Bijections between duplicated numbers
Mathematica managed to solve an array of 50 rows. This was a random example that I created with elements between 0 and 30. The only constraint I imposed was that each element occur an even number of times. It seems that most such random systems have no solution, and Reduce is able to conclude this fairly quickly. For this particular example, however, there where 114 solutions. It took 6 or 7 hours of running to find them.
Jul
2
comment Combinatorial optimization - Bijections between duplicated numbers
There's a typo in my comment: the second-to-last constraint should be a[1,1]+a[2,2]==a[1,2]+a[2,3] instead of a[1,1]+a[2,2]==a[1,2]+a[2,1]. You should find that this system has four solutions.
Jul
2
comment Combinatorial optimization - Bijections between duplicated numbers
You need to create a system of equations/inequalities. Here's an example for the array $$ \begin{bmatrix} 1 & 1 & 2 & 3\\ 1 & 1 & 3 & 2 \end{bmatrix}. $$ system=(a[1,0]+a[1,1]+a[1,2]+a[1,3]==1 && a[2,0]+a[2,1]+a[2,2]+a[2,3]==1 && a[1,0]>=0 && a[1,1]>=0 && a[1,2]>=0 && a[1,3]>=0 && a[2,0]>=0 && a[2,1]>=0 && a[2,2]>=0 && a[2,3]>=0 && a[1,0]+a[1,3]+a[2,0]+a[2,3]==a[1,1]+a[1,0]+a[2,1]+a[2,0] && a[1,2]+a[1,1]+a[2,2]+a[2,1]==a[1,3]+a[1,2]+a[2,3]+a[2,2] && a[1,2]+a[2,1]==a[1,3]+a[2,2] && a[1,0]+a[2,3]==a[1,1]+a[2,0] && a[1,1]+a[2,2]==a[1,2]+a[2,1] && a[1,3]+a[2,0]==a[1,0]+a[2,1])
Jul
2
awarded  Curious
Jul
2
comment Block walking and Pascal's Triangle
You have to make $n$ decisions. Call them Decision 1, Decision 2, ..., Decision $n.$ Of this set of $n$ decisions, $k$ of them are to be right. How many ways are there to choose which $k$ of the $n$ decisions are right?
Jul
2
awarded  Benefactor
Jul
2
revised Combinatorial optimization - Bijections between duplicated numbers
added 1241 characters in body
Jul
2
comment Combinatorial optimization - Bijections between duplicated numbers
It would be $4\times500=2000$ variables, along with $2000$ nonnegativity constraints, $x_{ri}\ge0.$ Then there is one equation per row plus two equations per distinct array element. So $500+2\times31=562$ equations. Perhaps a further example will clarify what the equations for the distinct array elements look like. I'll add one. I agree that this is a big system, but I'm unsure where the limit of solvability is.
Jul
1
comment Combinatorial optimization - Bijections between duplicated numbers
I'm curious about the origin of this problem. Are you reasonably certain that your system has a solution? If not, ruling out a solution can be much easier than finding one. If you do believe it has a solution, can you generate similar problems of smaller size? What's the largest system you've been able to solve?
Jul
1
revised Combinatorial optimization - Bijections between duplicated numbers
edited tags
Jul
1
answered Combinatorial optimization - Bijections between duplicated numbers
Jun
30
comment Are there 2D analogues for integer division and modular arithmetic?
What I wanted to say was getting too long for comments, so I've posted an answer.
Jun
30
answered Are there 2D analogues for integer division and modular arithmetic?
Jun
30
comment Ulam spiral: Is there an “unusual amount of clumping” in prime-rich quadratic polynomials?
I think the constant is actually closer to 6.6395, but I don't have high confidence in that. This was obtained using Hardy and Littlewood's formula, but I don't have a fast-converging method, hence my doubts. If correct, however, this would make things worse, I believe?
Jun
29
comment Are there 2D analogues for integer division and modular arithmetic?
I'm interested in your question, but I'm not sure how to give an answer that doesn't tell you things that are either obvious or that you already know. As mentioned in @AnuaragA's comment, you are looking at lattices. The fundamental region can be chosen in infinitely many different ways, which means there are infinitely many choices for the matrix $Q$ and for the set $P$ that describe the same lattice. Finding a nice fundamental region is a well-studied problem.