turing machines – Recognizability and complements

I’m learning about Turing Machines, decidability, and recognizability, and read that if a language is recognizable, its complement is sometimes recognizable. I don’t really understand how this could be, can someone give me an example where a language has a recognizable complement and an unrecognizable complement? Thank you!

gt.geometric topology – Harmonic functions on knot complements

In Axler’s Harmonic Function Theory, he and his coauthors develop the theory of harmonic polynomials on $mathbb{R}^n$ by considering the restrictions of arbitrary polynomials on the sphere $S^{n-1} = {x in mathbb{R}^n : ||x||^2 = 1 }$ and taking the Poisson integral to get a harmonic polynomial in the interior ball. One can then take the Kelvin transform to get a harmonic polynomial on the exterior of the sphere. This process yields a canonical projection $mathscr{P}(mathbb{R}^n) to mathscr{H}(mathbb{R}^n)$, from the space of polynomials to the space of harmonic polynomials, factoring through the restriction map to $L^2(S^{n-1})$. By taking polynomial approximations via the Stone-Weierstrass theorem, and verifying some series convergence, I believe one can expand the domain and codomain of the projection to larger, more general function spaces. I have not verified this myself, so I could be wrong.

Does this theory generalize to knot complements? Say we have a knot $K subseteq mathbb{R}^3$, and we take a small tubular neighborhood $V$ around $K$, whose boundary is topologically a torus $T$. Given a function on the knot complement, one could restrict to $T$ and then solve the Dirichlet problem on the knot complement to get a projection like the one above. However, in the sphere case, there are many nice properties of the projection; namely it comes with an efficient algorithm for computation which involves repeatedly differentiating the function $f(x) = |x|^{2-n}$.

Is anyone aware of any theory along this vein? Are there any obstacles to generalizing what happens in the sphere case?

linear algebra – Show that there are infinitely many distinct subspaces which are complements

I don’t quite understand what is meant by the direct sum and complement. I am trying to grasp the definitions and was doing some problems. As such, I came across this one that I don’t quite understand:

Let $V=mathbb{R}^2$ and $W=$span$((1,0)^T)$. Show that there are infinitely many distinct subspaces $tilde W$ which are complements of $W$.

To be a complement, we must have that $V=Wboxplustilde W$. So one obvious complement is span$((0,1)^T)$. However, I don’t know what they mean by distinct complements. Is the answer simply span$((0,1)^T,v)$ where v is any other vector at all because it will just get washed away by the direct sum operator?

Geometry of complements to codimension 2 compacts

Let $ K subset R ^ n $ be a compact (non-empty) cover size $ n-2 $. In particular, K $ do not separate $ R ^ n $ (even locally). I will equip $ M = R ^ n-K $ with the distance function $ d $ associated with the limited flat Riemannian metric of $ R ^ n $. The metric space $ (M, d) $ is incomplete, leave $ ( bar {M}, bar {d}) $ denote its completion of Cauchy. Since the integration of identity $ (M, d) to R ^ n $ decreases the distance, it extends to a continuous map $ f: bar {M} to R ^ n $.

Question. is $ f $ injective?

Less formally: if points in $ M $ are close in the Euclidean metric, as a result they can be connected by a short path in $ M $?

Note. The answer is positive if I assume that the Hausdorff dimension of K $ is inferior to $ n-1 $.