$x\sqrt{x\sqrt x}=2$, the answer is $\sqrt[7]{16}$ but I got $2\sqrt[3]{2}$ by: $$\begin{align}x\sqrt{x\sqrt x}=2&\to \sqrt{x\sqrt{x}}=\frac2x\\&\to x\sqrt{x}=\frac{4}{x^{2}}\\&\to \sqrt{x}=\frac{4}{x}\to x=\frac{16}{x^{2}}\\&\to x^{3}=16\\&\to x=\sqrt[3]{16}\\&\to x=2\sqrt[3]2\end{align}$$ Where...

Here is the theorem from Steven Abbot's Understanding Analysis. Theorem. Two real numbers $a$ and $b$ are equal if and only if for every real number $\epsilon > 0$ it follows that $|a - b| < \epsilon$. I have a two part question. How do I write the theorem using quantifiers? How is contradict...

A shooter shoots at eight identical targets in a shooting gallery. The probability of hitting each target with each shot is the same. It takes the shooter 11 shots to hit all 8 targets. What is the probability that the shooter hit fewer than 4 targets with the first five shots? I did the followin...

Lee's Smooth Manifolds primarily defines tangent vectors as derivations. That is, functions $v: C^\infty(M) \to \mathbb{R}$ that satisfy the Leibniz rule. By appealing to a chart $(U, \phi)$, we can write down a derivation $v$ concretely as a linear combination of basis vectors given by $d\phi^{-...

In a pile you have 100 stones. A partition of the pile in $k$ piles is good if: 1) the small piles have different numbers of stones; 2) for any partition of one of the small piles in 2 smaller piles, among the $k + 1$ piles you get 2 with the same number of stones (any pile has at least 1 stone...

Consider a standard mollifier $\rho_\delta$ on $\mathbb{T}^2$ (the 2 dimensional torus) and let $$ \hat{\rho_\delta}(m): = \int_{\mathbb{T}^2} \rho(z) e^{2 \pi i m \cdot z} dz $$ for any $m \in \mathbb{Z}^2$. Let $|m|:=m_1+m_2$ and $\alpha \in (0,1]$. I'm trying to obtain an estimate for $$ || (...

Let $(X_t)_{t\ge0}$ be a time-homogeneous shift-ergodic Markov process with transition semigroup $(\kappa_t)_{t\ge0}$ and invariant measure $\mu$. One implication of a Poincaré inequality is a $L^2$-contractivity $$\operatorname{Var}_\mu\left[\kappa_tf\right]\le e^{-2\lambda t}\operatorname{Var}_...

I got a expression that is related to the combinatorics, and it looks like this: $$\sum_{k=1}^n \frac{2^{2k-1}}{k}\binom{2n-2k}{n-k}\cdot \frac{1}{\binom{2n}{n}}$$ it is from a question I've been studying, and with another approach, I got the answer: $$\sum_{k=1}^n\frac{1}{2k-1}$$ so I think the ...

I am studying a population of $N$ bits, comprising $K$ ones and $N-K$ zeros. For sampling $n$ bits without replacement, the situation conforms to a hypergeometric distribution. The sum of these $n$ bits, $S_n$, yields a mean of $n\frac{K}{N}$ and a variance of $n \frac{K}{N} \frac{N-K}{N} \frac{N...

Question: how to evaluate $$\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x$$ MY try to evaluate the integral $$ \begin{aligned} & I=\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x \\ & e^{-\frac{\pi x}{2}} \stackrel{\rig...

I saw that a topological group $G$ is profinite if and only if it is compact, Hausdorff and totally disconnected. In such groups, an open subgroup is closed but not vice-versa. From this, I came to a question as a curiosity: In a profinite group $G$, if every subgroup is closed, can we say about...

Evaluate the integral $$\int_{-\infty}^{\infty}\binom{n}{x}dx$$ This question came in Cambridge Integration Bee and I have no clue what to do in this. I rewrote $\binom{n}{x}$ as $\frac{n!}{{x!}{(n-x)!}}$ but I don't know how to integrate those factorials. Also what to do if $n$ isn't an intege...

Apologies for the unclear title, I have no idea if the property I'm looking for has a better name. I'm wondering if there exists a pair of functions $f, g : \mathbb{R} \rightarrow \mathbb{R}$ such that : $g$ is a bijection and is nowhere continuous (for an example, see this answer). $f$ is conti...

I know that $L_p$ is defined when p > 1, yet I am wondering if we were to define $L_i$ which turns out to be: $$ \|f\|_i​=(∫_D​\|f(x)\|^idx)^{-i}$$ Then, you can basically calculate it for any vector and which gives you a complex value. I was wondering if this makes any sense at all or there woul...

Let ($X_n$)$_{n \in \mathbb N}$ be a sequence of random variables defined on a probability space ($\Omega,F,P$). Suppose that $X_n \geq 0$ and $E(X_n ^4) < \frac{1}{n^\frac{3}{2}}$ , for all $n \in \mathbb N$. Show that $X_n$ converges to $0$ almost surely as $n \rightarrow \infty $ I know I need...

$M \subset \mathbb{R}^3$ is a surface created by rotation of set $\{ x^2 + z^2 = 4x-3, \ 1<x<2, \ -1<z<1 \}$ by $\pi$ around axis OZ. Find $\int_M \max(x,y,z) d \lambda_2$. I know that $x^2 + z^2 = 4x-3 \iff x^2 - 4x + 4 + z^2 = 1 \iff (x-2)^2 + z^2 = 1$ So, we have a circle with radius equal t...

I consider any set $A$ in $\mathbb{R}^n$. Note that $A$ is not necessary to be a convex set. I define the extreme point as follows: a point $a\in A$ is an extreme point of the set $A$ if we cannot find any two different points $a_0\in A,a_1\in A$ and $a_0\neq a_1$ such that $a=\lambda a_0 + (1-\l...

For my profession I have to calculate the weighted average number of days it takes a business to review contracts. The count of each contract types reviewed is the weight, and the sum of days it takes to review is the number. The weighted average for this below data set is 35.6 days, however the...