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How can you prove that if there are two countable functions, their composition is also countable? please show solution.

I know that G-δσ is not contained in G-δ, so there exists a sequence of open sets G_n,m such that ⋃⋂G_n,m is not a G-δ set. However, I am not sure whether ⋃⋂G_n,m can be expressed as the limit inferior of some suitable sequence of open sets. Since the set of rational numbers is not a G-δ set, I tried to obtain open sets B_n such that lim inf B_n = Q , but I failed. Any suggestions?

I have a problem with a hacky solution that's been bugging me for a long time. I'm hoping someone can help me with a proper solution and put my mind at peace. Imagine you have a computer screen, where (0,0) is at the top-left corner. Going right, x increases. Going down, y increases. I want to render tiles in an isometric map that's represented by an N x M array. However I am having trouble converting from a given grid coordinate to an on-screen coordinate. For example, this is my 2D array: tiles = [[42, 42, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [42, 42, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 42, 42, 42, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 42, 42, 42, 42, 42, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 42, 42, 42, 42, 42, 42, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 42, 42, 42, 42, 42, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 42, 42, 42, 42, 42, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 42, 42, 42, 42, 42, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 42, 42, 42, 42, 42, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 42, 42, 42, 42, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 42, 42, 42, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 42, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 42], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 42], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 42], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 42, 42, 42, 42]] Which represents this isometric map: On the image, I've labeled where the grid (0,0) is rendered on-screen (where the 4 square green island is). In this case, I have a 20x30 grid, and I tried the below formula to convert from grid -> screen: # Try converting at grid (0,0) grid_x=0 grid_y=0 # NxM grid dimensions rows=30 cols=20 # Each tile size in pixels tile_w=128 tile_h=64 # Total Map size in pixels map_pixel_w = (rows + cols) * floor(tile_w / 2) map_pixel_h = (rows + cols) * floor(tile_h / 2) # Calculate offsets origin_x = floor(screen_w / 2) # <-- Pretty sure this is WRONG!! origin_y = 0 # Apply isometric transform screen_x = (grid_y - grid_x) * floor(tile_w / 2) + origin_x screen_y = (grid_y + grid_x) * floor(tile_h / 2) + origin_y # Final Screen calculation screen_x, screen_y = floor(screen_x), floor(screen_y) The above calculation gets me close, but it's off by (-4,0). ie, objects calculated at grid (0,0) in the calculation will actually show up on-screen incorrectly at grid (4,0). If I then manually adjust my x offset by -4, (ie, if I manually change the above to): grid_x = -4 grid_y = 0 ... Then the above calculation will correctly render my object on-screen at grid (0,0). So somehow everything is shifted down by 4 for my x offset. So my hacky solution was to offset everything by -4 in my grid_x coordinates, which is not a proper nor elegant solution. I think it's because the map is an uneven N x M grid, so calculating the origin_x = floor(screen_w/2) is wrong, since the origin for x is not necessarily exactly half of the total map width. But how would I calculate this offset from just knowing the tilesize and map width? Would someone help me fix my equation to provide the correct conversion from a 2D grid to on-screen coordinates?

If I have an n$\times$n orthogonal matrix $\mathbf{Q}$ where the sum over each row and each column equals 1, is it a permutation matrix? A permutation matrix has a single 1 in each row and column, with the rest of the entries in that row and column equal to zero. I'm not sure where to start. I know I can express these conditions as: $\mathbf{Q}^T \mathbf{Q}=I$ and $\mathbf{Q} \mathbf{1} = \mathbf{1}$ and $\mathbf{Q}^T \mathbf{1} = \mathbf{1}$ where $\mathbf{1}$ is a column vector of ones and $I$ is the identity matrix.

Background. Consider a nonlinear system of equations $$F(u, p) = 0$$ for a vector $u$, where $p$ are some parameters and $F$ is smooth. The implicit function theorem tells us that there is a mapping $G$ such that $$F(G(p), p) = 0$$ and moreover $$\mathrm dG\cdot \delta p = -\frac{\partial F}{\partial u}^{-1}\cdot\frac{\partial F}{\partial p}\cdot\delta p.$$ The problem. Now suppose instead of a nonlinear system of equations I have a nonlinear complementarity problem $$F(u, p) + w = 0, \quad u, w \ge 0,\quad \langle u, w\rangle = 0. \qquad (*)$$ I'd expect that I can compute the derivatives of $u$ and $w$ by solving a linear complementarity problem. If I replace $u \mapsto u + \delta u$, $w \mapsto w + \delta w$, $p \mapsto p + \delta p$, and subtract off parts of equation (*), I get $$\frac{\partial F}{\partial u}\delta u + \frac{\partial F}{\partial p}\delta p + \delta w = 0, \quad u + \delta u,\;w + \delta w \ge 0, \quad \langle u, \delta w\rangle + \langle \delta u, w\rangle + \langle\delta u, \delta w\rangle = 0.$$ It kind of looks like a non-standard linear complementarity problem for $\delta u$, $\delta w$ but I'm stuck here. What is the right analogue of the implicit function theorem for nonlinear complementarity problems? I searched google scholar and came across this paper but I found it hard to understand. It feels like there should be a simple approach based on what I wrote above. I tried doing the same calculation starting from a linear rather than a nonlinear complementarity problems and came up short there too.