## co.combinatorics – How many ways are there to cover an N × N chess board with white and black boxes with some restrictions?

Suppose we have an N × N chess board and the boxes ■, □.

We should cover the chess board with these boxes, but there cannot be a 2 × 2 square $$scriptstyle { begin {array} {cc} square & square \ square & square end {array}}$$
on the chess board.

Can we calculate how to cover the chessboard?

(1) If we connect the upper (left) border and the lower (right) border together, it means that the upper (left) and lower (right) border form a 2 × N (N × 2) rectangle and it shouldn't have 2 × 2 square
$$scriptstyle { begin {array} {cc} square & square \ square & square end {array}}$$
to.
Can we calculate the routes?

(2) If we have k black boxes with the (1) chess board, can we calculate the routes?

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## Cover number for the unit ball in a reproducing kernel Hilbert space

I am looking for a reference for an upper limit of the cover number for the unit sphere $${f in mathcal {H}: || f || _ { mathcal {H}} || leq 1 }$$, Where $$mathcal {H}$$ is a reproducing kernel Hilbert space, the Sobolev space $$mathcal {W} ^ {p, 2} ((0,1))$$ in my application. I found some references like

• Coverage of numbers from Gaussian reproducing kernel-Hilbert spaces, T. Kuhn, Journal of Complexity,

but since this is only for Gaussian kernels, it is not exactly what I need. Therefore I would be happy about all the hints. Thanks in advance.

## Reference request – What do Sylow 2 subgroups for Schur look like that cover groups of finite simple groups?

But avoid

• Make statements based on opinions; Support them with references or personal experiences.

Use MathJax to format equations. MathJax reference.

## What happens to a demon's cover when the mortal who provides part of that cover dies?

Demon Dora buys his childhood experience in Townsville from mortal Fred in return for wealth and fame. Fred uses the money to buy a car, crashes and dies. Is something happening to the Dora cover made from these experiences?

## Polynomials – roots of combined equations – which theory does this cover?

We can make a polynomial with certain roots, r_n, as the product of all (x – r_n) = 0

Now let's assume:

y = x and
y = x + b

If we rewrite this as

0 = -y + x and
0 = -y + x + b

and then use them as roots (multiply them together):

(-y + x) (- y + x + b) = 0

We get an equation whose graph is both the original equations.

I think that's really neat and I want to explore it further. Finding an equation for any shape is a hobby of mine. If I want to study that more closely, which area of ​​mathematics should I study? Is there evidence that this works in general? There are some similar results that I noticed and I want to learn the general theory instead of just getting confused.

## Optimization – Which practical / efficient algorithms are there for the minimum weighted set cover problem (MIN-WSCP)?

For TSP, there are known heuristic and approximation solutions that run in a time with a low polynomial, such as Christofides / 2-OPT and so on.

I need a practical, fast algorithm, ideally sub-square complexity, to solve the minimally weighted set-cover problem (which is NP-heavy). It solves sub-problems in a genetic algorithm and therefore has to be fast rather than optimal. Are there standard algorithms to solve this?

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