3,258 reputation
1834
bio website jakitoimgeisterhaus.blogspot.…
location Munich, Germany
age 38
visits member for 3 years, 4 months
seen 12 hours ago

My past research interests included differential algebraic equations, nonlinear analysis and relations between symmetries and structural properties.

I recently investigated hierarchical structures, starting from group cohomology, continuing with semi-group theory and ending with lattices and universal algebra.


May
17
comment Can Goodstein's theorem be expressed as an axiom or axiom scheme in PA?
How an axiom scheme might really help here is by generating instances of $(\forall n)(\exists s)\Phi_\beta(n,s)$ for all (first-order PA) definable $\beta$-functions $\beta(x_1,x_2,x_3)$ (which can be proved in PA to be $\beta$-functions). This axiom scheme might be a stronger statement than an arbitrary individual instance of this scheme (also I don't know whether it is really stronger).
May
17
comment A consistent first-order theory whose impredicative second-order variant is inconsistent
@AsafKaragila Thanks, this really helped. I see now that the main obstacle to the second part of my question is the word "simple". If I drop it, I can just fix a Gödel numbering, and add $\lnot\mathsf{Con(PA)}$ as a single axiom to $\mathsf{PA}$. My best guess for an answer is then probably that either a "simple" independent statement for PA like Goodstein's theorem can be expressed as a single formula (so that I can add the negation of that formula as an axiom), or to see whether Robinson arithmetic $\mathsf{Q}$ allows for a simple formula (provably) independent of $\mathsf{Q}$.
May
17
comment Can Goodstein's theorem be expressed as an axiom or axiom scheme in PA?
Great, I know $\beta$-functions from lemma 10.6.11 in Ebbinghaus et. al. I will read it again now. An axiom scheme might replace $(\forall n)(\exists s)\Phi(n,s)$ with each specific instance of $(\exists s)\Phi(\mathbf{n},s)$ for $\mathbf{n}=0,S(0),S(S(0)), \ldots$. This helps if $\Phi(n,s)$ cannot be defined in PA, but $\Phi(\mathbf{n},s)$ can.
May
17
asked Can Goodstein's theorem be expressed as an axiom or axiom scheme in PA?
May
17
awarded  Talkative
May
17
comment Non-i.i.d Empirical Risk Minimization
Maybe you can learn something from the "final" resolution of my question How to sample numerically from an arbitrary smooth distribution? However, I have the impression that you are stuck much earlier, and wouldn't actually care too much about the accuracy issued that bothered me there. Still, it may be interesting as an example where the composition method was the most appropriate solution.
May
17
revised Non-i.i.d Empirical Risk Minimization
Mentioning only rejection sampling, but not also the composition method is not good...
May
15
awarded  Nice Question
May
13
reviewed Approve suggested edit on Continuous probability distribution with no first moment but the characteristic function is differentiable
May
13
reviewed Approve suggested edit on Finding expected value of E(Y^2)
May
12
revised A consistent first-order theory whose impredicative second-order variant is inconsistent
Fixed the question, to really ask the intended question. Also incorporated clarification into the question.
May
12
revised A consistent first-order theory whose impredicative second-order variant is inconsistent
edited tags
May
11
comment A consistent first-order theory whose impredicative second-order variant is inconsistent
@RickyDemer By "provably inconsistent", I just mean that I don't insist on a possibly extremely long syntactic derivation of $\phi\land\lnot\phi$, but that a "short" proof (possibly assuming ZFC) that such a derivation exist is also sufficient.
May
11
asked A consistent first-order theory whose impredicative second-order variant is inconsistent
May
7
comment Is nonlinear conjugate gradient a quasi-newton optimization technique?
How can I talk of approximating the Hessian, if I restart the approximation at every step? I rather would interpret the cited passage as a suggestion that BFGS might perform better than nonlinear conjugate gradient methods (ignoring memory consumption), because its a true quasi-Newton method.
May
7
comment Is nonlinear conjugate gradient a quasi-newton optimization technique?
@Dominique Interesting. Can you also tell me which part is erroneous? My guess is that you disagree with: "A quasi-Newton optimization technique use some approximation for the Hessian, or at least can be interpreted as implicitly using some approximation for the Hessian." Or do you disagree with my last "... not even implicitly"? In that case, could you please help me to also recognize the implicit Hessian you have in mind?
May
4
accepted Computing 2d radially symmetric Fourier transforms (with Wolfram Alpha)
May
4
asked Computing 2d radially symmetric Fourier transforms (with Wolfram Alpha)
May
3
reviewed Approve suggested edit on Proving a circuit of a graph will have an edge in common with a cycle of the same graph.
May
3
accepted Can the consistency of ZF be proved in MK?