## 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 $$n-1$$.