Consider a stochastic approximation process with $$x_{t+1} = x_t + \frac{1}{t} (g(x_t)+u_t)$$ where $(u_s)_s$ is a sequence of i.i.d. shocks. Assume $g$ is Lipschitz, $u_t$ has finite variance, and that $(x_s)_s$ is bounded with probability one. Furthermore, assume that $$ C = \{ x \in \mathbb...

Let $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space; $f:E\to[0,\infty)^3$ be a bounded Bochner integrable function on $(E,\mathcal E,\lambda)$ and $p:=\alpha_1f_1+\alpha_2f_2+\alpha_3f_3$ for some $\alpha_1,\alpha_2,\alpha_3\ge0$ with $\alpha_1+\alpha_2+\alpha_3=1$ and $$\int p\:{\r...

Let $I\subseteq\mathbb N$ be finite and nonempty, $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space, $$\lambda f:=\int f\:{\rm d}\lambda$$ for $\lambda$-integrable $f:E\to\mathbb R$, $p:E\to(0,\infty)$ be $\mathcal E$-measurable with $c:=\lambda p\in(0,\infty)$, $r:(I\times E)\times E\t...

On this article starting from equation $(30)$ it's presented a derivation of the first moment for the solution the logistic SDE: $$dx=x\left[\mu\left(1-\frac{x}{\tilde{x}}\right)dt+\sigma dW\right]$$ The solution for this equation is given by: $$x(t)=x_0e^{(\mu-\sigma^2/2)t+\sigma W_t}\left(1+\...

Where is the complexity of the problem 'Given two bounded compact convex integral polyhedra in $\mathbb R^n$ presented by $O(poly(n))$ integer valued halfspaces given by linear inequalities with coefficients of size $O(poly(n))$ bits with promise that they are equal volume is there a scisso...

I was thinking about the following some time ago. My question is whether such things have been studied before. Let $E_n$ be the abelian group with a generator for each (bounded) euclidean polytope of dimension at most $n$ and relations (1) $P=Q$ if $P$ and $Q$ are congruent, (2) $P\cup Q=P+Q-P\c...

EDIT: as pointed-out below, this has been posted on math.stackexchange. I'll leave it up to the community whether or not to delete this question, but I do think there is room for a more technical answer than the one posted on math.stackexchange. A famous question related to Hilbert's third pro...

I have a couple of questions concerning existence and description of transfers for Bloch groups and scissors congruence groups/pre-Bloch groups. To fix notation and recall definitions: From the general algebraic K-theory machinery, we get transfers on $K_3$. In particular, for a finite field...

I will call two graphs $G$ and $H$, $r$-equidecomposable (in analogy with Hilbert's third problem) if they can be written as unions of disjoint subgraphs $$G\cong \bigsqcup_{i=1}^r G_i\quad ,\quad H\cong \bigsqcup_{i=1}^r H_i$$ (disjoint here means they have no common edges), and $G_i\cong H_i ...

It is easy to see that an equilateral triangle can be cut into 2 identical 30-60-90 degrees right triangles which can then be patched together to form a 30-30-120 degrees triangle. So, via 2 intermediate pieces, we can dissect an equilateral triangle into the 30-30-120 triangle of the same area. ...

Consider a right square pyramid whose base has side length $2r$ and whose height is $h$. Let the dihedral angle between the base and each triangular side be $\theta$, and the dihedral angle between adjacent triangular sides be $\phi$. We have $$\cos(\theta)=\frac{r}{\sqrt{r^2 + h^2}}$$ and $$\cos...

I am currently writing a geometry paper "Rectifications of Convex Polyhedra" and I am confused to have discovered what appears to be a remarkable discrete geometric fact: Conjecture: Let $P$ be a convex polyhedron. Then $P \cong P^{\circ} \sqcup R_{1}[P]$. The relation $\cong$ is to denote scis...

The following questions seem related to the still open question whether there is a point(s) whose distances from the 4 corners of a unit square are all rational. To cut a unit square into n (a finite number) triangles with all sides of rational length. For which values of n can it be done if at...