# Tag Info

1 vote

### Compact-open Topology for Partial Maps?

There is a way to promote function space topologies so that convergence of nets behaves the way that one would expect it to behave. There are probably other ways of getting a topology for spaces of ...
• 27.4k

### Uniqueness of solutions of Young differential equations

We will follow the Lemma 8.10. (Rough Gronwall) and Proposition 8.12 from "a course in rough paths" but modify them for this particular setting of studying $$Y_{t}=Y_{0}+\int^{t}_{0} Y dX,$$ ...
• 3,833

### Functions that are Khinchin integrable but not Henstock-Kurzweil integrable

Let $F$ be the function from example 6.20 c) in [1]. That is, fix a perfect nowhere dense subset $E$ of $[0, 1]$ such that $0, 1\in E$ such that $0 < |E| < 1$, for example the fat Cantor set. ...
• 635
Accepted

### Can a nowhere locally Hölder function be differentiable almost everywhere?

Define $$\psi(x)=\begin{cases} 1/|\log x| &\text{if } x\in (0,1/2] \\ 0 &\text{if }x\leq 0\\ 1/\log 2& \text{ if } x>1/2.\end{cases}$$ Note that $\psi$ is increasing and bounded (and ...
• 1,220
Accepted

• 40.1k

### Is the hypergeometric function ${}_1F_2(1;a,a+\frac12;-x^2)$ an elementary function? How about its positivity, monotonicity, and convexity in $x$?

Might be useful - an expression in terms of the incomplete Gamma function. The series  {}_1F_2(1;a,a+\frac12;-x^2)=1-\frac{4x^2}{2a(2a+1)}+\frac{16x^4}{2a(2a+1)(2a+2)(2a+3)}-\frac{64x^6}{2a(2a+1)(2a+...
• 17.2k
### $f\in C(B_1)\cap W^{1,2}(B_1\setminus \{f=0\})$ implies $f\in W^{1,2}(B_1)$?
This should follow from the ACL (absolute continuity on lines) characterisation of Sobolev spaces, see for instance, Theorem 4.1.10 here. Indeed, since $f \in W^{1, 2} (B_1 \setminus \{f = 0\})$, it ...