Here's mine. I'm answering late, I know that, but I am still answering it. We'll use the expansion of $\tanh^{-1}$: $$\frac{1}{2}\log\frac{1+y}{1-y}=\sum_{n\geq0}\frac{y^{2n+1}}{2n+1},\quad|y|<1$$ We start with this inequality: $$\int_{-1}^{1}\int_{-1}^{1}\frac{1}{1+2xy+y^2}dy\,dx=\int_{-1}^{1}\f...

Note:This is the same question, but it doesn't have any answers and it is very old and inactive, so it doesn't answer my question. This may also look similar, but doesn't answer my question, you can see the answers yourselves. While reading this section of the Wikipedia article on the Riemann hyp...

This forum brings together a broad enough base of mathematicians to collect a "big list" of equivalent forms of the Riemann Hypothesis...just for fun. Also, perhaps, this collection could include statements that imply RH or its negation. Here is what I am suggesting we do: Construct a more o...

See characterisation of the twin primes for a modular arithmetic viewpoint. Shift $m$ by $1$ so consider twin prime pairs $(m-1, m+1)$ and the formula becomes: $$ 4((n-2)! + 1) = -(n-1) \pmod {n^2 - 1} $$ Since $\gcd(n \pm 1, n) = 1$ for all $n \geq 2$ we have that we can faithfully multiply bo...

I have a question about showing that a Toeplitz operator is Fredholm: Show that the operator $T_{e^{inx}}:L_2^+\to L_2^+$ acting on $L_2^+=\sum_{k\ge 0}a_ke^{ikx}$ with $\sum_{k}|a_k|^2<\infty$ is Fredholm for every $n\in\mathbb{Z}$ and find its index. Denote $S^1$ to be the unit circle. The sp...

Congratulations robjohn for getting into the 300K club.

We truly appreciate your constant presence and insightful hints in the Mathematics chat room. Your dedication to learning is inspiring. Thank you also, for all your hard work as a moderator.

Let $\gamma_1,\gamma_2:I\to\mathbb{R}^3$ be a pair of regular space curves with the same domain, $I$. Define $\sigma:I\times(0,1)\to\mathbb{R}^3$ as $\sigma(t,u) = u\cdot\gamma_1(t)+(1-u)\cdot\gamma_2(t)$. Assume that $\gamma_1,\gamma_2$ are chosen such that $\sigma$ is regular. Prove that $K(t,...

Note $$(\sqrt[3]2+\sqrt[3]4 -1)^2= 5- \sqrt[3]{4} $$ which yields $$\sqrt[3]2+\sqrt[3]4 =\sqrt{5-\sqrt[3]{4}}+1$$

Explainable artificial intelligence (XAI) is concerned with the development of techniques that can enhance the interpretability, accountability and transparency of artificial intelligence and, in particular, machine learning algorithms and models, especially black box ones, such as artificial neu...