Let $G$ be a lie group and $M$ a manifold. Let $A : G \times M \to M$ be a proper Lie group action. It seems to be a well known result that in this case $M$ admits an invariant metric $g$. That is for $h \in G$, $A(h, -)_*g=g$. However, I only know a proof (thm 3.0.2) in case $G$ is compact. More...

Recently I posted this question here about different scenarios where two Random Walks can intersect: Bounded Random Walks vs Non-Bounded Random Walks I realized that this question might be too complicated and decided to ask a simpler question involving the Hitting Times of a single Random Walk. F...

Suppose $X_n$ is a sequence of positive random variables with $\mathbb{E} X_n=f(n) = C n^\delta$ with $C>0$ and $Var(X_n)=M=\text{const}$. What can we say about: $$ \lim_\limits{n\to\infty}\mathbb{P}(X_n>n^{\alpha}) $$ for different values of $\alpha$? It seems straightforward that for $\alpha > ...

I encountered a post on locales and geometric theory here https://grossack.site/2022/05/22/locale-basics.html In about the middle of this blog, the author gives a geometric theory defining a function from $\Bbb N$ to $\Bbb R$, and asked: Do you see why this is a nontrivial theory with no classica...

Let $a_n$ be a sequence of real numbers such that $1=a_1 \le a_2 \le a_3 \le \cdots \le a_n.$ Additionally, we have that $a_{i+1}-a_i \le \sqrt{a_i},$ for all $1 \le i <n.$ Then prove that $$\sum_{i=1}^{n-1}\frac{a_{i+1}-a_i}{a_i} \le 2 \log_2 n.$$ My attempt: Consider the LHS of the given inequa...

The so called Renyi parking constant gives the covering density of an infinite interval $[0,L]_{L\to\infty}$ that was randomly covered by unit intervals. Covering is only allowed if the place is not occupied by a previously covered interval. The process finishes if the largest uncovered segment ...

The paper Dividing a polygon into two similar polygons provides that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original). Right angled triangles $1:\sqrt{2}$ parallelograms The Golden Bee I wish to find examples of polyhedra t...

Say we have $N$ spheres indexed as $1,2,3,\dotsc, N$ such that all of them have identical weight apart from one, and we don't know if that one is heavier or lighter. We have to determine which sphere has the odd weight using just a balance scale. We could solve this problem by weighing repeatedl...

The below result is from Linear Algebra Done Right, 4th edition, Sheldon Axler: 5.11 linearly independent eigenvectors Suppose $T \in \mathcal{L}(V)$. Then every list of eigenvectors of $T$ corresponding to distinct eigenvalues of $T$ is linearly independent. Firstly, cool result. Secondly, I'm...

Problem: Let $A\in \mathbb C^{n\times n}$ be a matrix. Determine the smallest number $k\in \mathbb N$ (dependent on $A$), such that $\operatorname{rank} A^m=\operatorname{rank} A^k$ for all $m>k$. So far: My intuitive thought was to somehow decompose $A$ and use the fact that similar matrices hav...

A right-angled triangle has sides of integer length. Its area (in square metres) is twice its perimeter (in metres). What are the lengths of the sides? The equations I have made so far is: Using Pythagoras' Theroem and Area Formula: $$\frac{ab}{4}=a+b+\sqrt{a^2+b^2}$$ Using Heron's formula: $$s(s...

In an recent test I was asked to evaluate the integral $$ \int_0^1 \frac{\sqrt[3]{x^2(1-x)}}{(1+x)^3} \text{d}x$$ in 8 minutes, but I didn't have a clue what to do with it. After the test, I tried the following approach: substituting $x=\frac{1}{t}, u=\sqrt[3]{t-1}$ gives $$ \int_0^1 \frac{\sqrt[...

Let $ f $ be a smooth function defined on the sphere such that the set of points where $ f(x) - f(\tilde{x}_y) $ vanishes divides $\mathbb{S}^2$ into exactly four regions for all $y\in \mathbb{S}^2$, where $\tilde{x}_y $ is the reflection of $ x $ with respect to the plane $ P_y $ passing through...

I've learned that $x^ \frac nm$ should be turned into radical form like $\sqrt[m]{x^n}$. Therefore $10^ \frac32$ should be turned into radical form like $ \sqrt {10^3} $. What I don't understand is how this is then further simplified to $10\sqrt10$. Could anyone please explain the steps?

I have the following situation where the line $(AB)$ is orthogonal to the "vertical line" passing through $C$. Is there a way to determine the blue angle in terms of $\boldsymbol {AC}$, $\boldsymbol{BC}$ and the red angle, or is there a degree of freedom? I only get the equation (where $b$ is th...