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By the Kirby–Paris theorem, Goodstein's theorem is independent of Peano arithmetic (PA). Therefore there are non-standard models in which every Goodstein sequence terminates. However, Tennenbaum's the …

I'm studying Sobolev spaces using Evans' PDE book. I can't figure out this simple fact. Let $L$ be an operator in this form: $$Lu= \sum{D_i(a_{ij}D_j(u))+\sum{b_iD_i(u)+cu}}.$$ I can't understand why …

Let $G$ be a compact Lie group acting freely on $X\times Y$ , product of two Hausdorff spaces. Is is true that $G$ must act freely on one of the factor spaces ($X$ or $Y$). For example the group $\mat …

I need to generate random gold ore channels for a game, I was thinking they would look kinda like lightning strikes. Anyone know any good fractals (recursive functions) that looks like it? Or otherwis …

Given an $A \in \mathfrak{su}(n)$, is it always possible to solve for $U,V \in SU(n)$ and $\lambda \in \mathbb{C}$ such that $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$? Cross posted from …

What is the rank (minimal number of group generators) of the group $GL(n,F)$, when $F$ is a finite field of odd order? I found that $SL(n,F)$ is $2$, but I can't find this information.

Is there some set of problems, for which determining if given problem is decidable or not is itself undecidable?

This is a somewhat vague and philosophical question. Consider the following three problems: Problem 1: Minimize over all real-valued $x,$ the function $f(x) = bx-ax^2$ where $a,b>0.$ Ans: $ …

Let $A,B$ be Banach algebras and $A\hat{\otimes}B$ be projective tensor product of them. Let $S$ be an ideal of $A\hat{\otimes}B$. Are there ideals $I$ of $A$ and $J$ of $B$ such that $S=I\hat{\oti …

Is the Laplacian operator $\Delta: C^2([a,b],\mathbb{R}) \to L^2(a,b)$ unbounded? Here $a,b \in\mathbb{R}$, $C^2([a,b],\mathbb{R})$ associated with $L^2$ norm. If yes, how can we "modify" these space …

I'm looking for a function that maps n variables to points on the surface of an n+1-dimensional cross polytope. For example, given one variable, the function would return a point on the perimeter of …

Let $f:[0,1]\to \ell_\infty[0,1]$ be defined by $f(t)=\chi_{[0,t]}$. Is it true that $f$ is weakly continuous almost everywhere w.r.t. Lebesgue measure ? Here $\ell_\infty[0,1]$ represents the functio …

We have the equations $$a_1x+b_1y+c_1z=d_1,$$ $$a_2x^2+b_2y^2+c_2z^2=d_2,$$ $$a_3x^3+b_3y^3+c_3z^3=d_3,$$ where $a_i,b_i,c_i \in\Bbb N$ at $i \in \{1,2,3\}$ are known. Is there an efficient ($O(\lo …

I asked this question in "math.stackexchange" but I did not get any response, so I put it here, maybe someone can help. Is it possible to write identity similar to the identity $$ (x^2+y^2)(u^2+v^2)= …

The necklace can be obtained from a circle by attaching $n$ 2-spheres $S^2$ along arcs, so the necklace $N(n,S^1,a_i)$ is homotopy equivalent to the space obtained by attaching $n$ 2-spheres $S^2$ to …