Prove that $$\int_0^{\frac{\pi}{2}}\frac{\log\left(\sin x\right)}{\cos^2\left(x\right)+y^2\sin^2\left(x\right)}{\rm d}x=-\frac{\pi}{2}\frac{\log\left(1+y\right)}{y}$$ where $y\ge0$. I came across this problem on a math forum, but after attempting to solve it, I didn't get any results. I attempted to solve the problem using the residue theorem but found it challenging to prove the expression. I believe the issue lies in how to handle $\log(\sin x)$, which might require some clever substitution. But I couldn't think of any. I hope someone can provide the correct solution. Thanks ..read more

The vertices of a pentagram are five uniformly random points on a circle. The areas of three consecutive triangular "petals" are $a,b,c$. (The petals are randomly chosen, but they must be consecutive, either clockwise or anticlockwise.) A simulation of $10^7$ such random pentagrams yielded a proportion of $0.5000179$ satisfying $a^2<bc$. Is the following conjecture true: $P(a^2<bc)=\frac{1}{2}$ Context Recently I have been exploring probabilities associated with such random pentagrams, and I stumbled upon this probability. Remarks Note that the three petals must be consecutive. Callin ..read more

Someone recently made a contribution on the french Wikipedia page for the Exponential integral. He claimed the following expansion for its compositional inverse function: $$ \forall |x| < \frac{\mu}{\log \mu},\quad \mathrm{Ei}^{-1}(x) = \sum_{n=0}^\infty \frac{x^n}{n!} \frac{P_{n+1}(\log\mu)}{\mu^n}, $$ with $\mu$ being the Soldner constant ($\log\mu$ being the positive zero of $\mathrm{Ei}$) and $(P_n)$ is defined by $$ P_0(x) = 1, \quad P_{n+1}(x) = x(P_n'(x) - (n-1)P_n(x)). $$ So $P_0(x) = 1, P_1(x) = P_2(x) = x, P_3(x) = x - x^2, P_4(x) = 2x^3 - 4x^2 + x, \ldots$ This is quite an incred ..read more

Two days ago, I tried to create an infinite series that might be able to generate a transcendental number, and when I checked the proper definition, it was mentioned that, it is a number that cannot be expressed as a root of any non zero polynomial with integral coefficients and that it is irrational. (Please tell me if I am right) And I came up with this series: $$\gamma = \sum_{r=0}^\infty\frac{1}{(5^r)(10^{f(r)})},f(r)=4^r$$ The basic idea that I had was to generate a number, such that its decimals will be randomised by the digits of the numbers of geometric progression of 2. $$\frac{1}{5^1 ..read more

The vertices of a pentagram are five uniformly random points on a circle. The areas of three consecutive triangular "arms" are $a,b,c$. (The arms are randomly chosen, but they must be consecutive, either clockwise or anticlockwise.) A simulation of $10^7$ such random pentagrams yielded a proportion of $0.5000179$ satisfying $a^2<bc$. Is the following conjecture true: $P(a^2<bc)=\frac{1}{2}$ Context Recently I have been exploring probabilities associated with such random pentagrams, and I stumbled upon this probability. Remarks Note that the three arms must be consecutive. Calling the ..read more

the question Consider the triangle $ABC$ and a point $M$ inside the triangle such that $\angle MAB = 10 ,\angle MAC = 40 ,\angle MCA = 30 $ and $\angle MBA = 20 $ . Show that triangle $ABC$ is isosceles. my idea the drawing As you can see I calculated the other angles we have in the triangle...I'm preatty sure we need an auxiliar construction, but I can't figure out which one to do. I dont know how to start. Hope one of you can help me! Thank you ..read more

Given as exercise 2.5.1: Show that the complete type realized by $1$ in $(\mathbb Z,+)$ is non-isolated. HINT Use the preceding exercise. The previous exercise: Let $\vec a$ and $\vec b $ be finite sequences from $\mathcal M.$ Prove that $\operatorname{tp}_{\mathcal M}(\vec a\vec b)$ is isolated iff $\operatorname{tp}_{\mathcal M}(\vec a/\vec b)$ and $\operatorname{tp}_{\mathcal M}(\vec b)$ are both isolated. Using this fact, show that when $\mathcal M$ is an atomic model and $\vec b\in M,$ then $\mathcal M$ is atomic over $\vec b.$ Conversely, if $\mathcal M$ is atomic over $\vec b$ and ..read more

This is a strengthening of a question another user asked, here: Are irrational numbers order-isomorphic to real transcendental numbers?. In the answer to that question, it was stated that the irrationals are order-isomorphic to the transcendental reals. My question is this. Suppose $S$ is a continuum-sized subset of the set of irrational numbers, which has the property that it is everywhere dense, meaning, between any two distinct reals, there exists a real number belonging to $S$. Must $S$ be order-isomorphic to the irrationals ..read more

Let $G$ be an abelian group and $M$ a $G$-module. The basic definitions: Let $H < G$ be a subgroup of finite index. We have a map $tr: H^0(H, M) \rightarrow H^0(G, M)$ on group cohomology defined by $m \mapsto \sum_{g \in G/H} gm$. This can be extended to a map $H^*(H, M) \rightarrow H^*(G, M)$ of cohomological functors. It is explicitly written down as following and called $$\text{Cor} : H^1(H,M)\to H^1(G,M)$$. Let $f$ be a cocyle for $H$. Take a set of representatives $X$ of $G/H$ in $G$. Then $\operatorname{Cor}(f)(g) = \sum_{x \in X} y\cdot f(y^{-1}gx)$ where $y\in X$ is the unique repr ..read more

In R.W. Sharpe's Differential Geometry, a flat fibre bundle is defined as a bundle whose transition functions are constant. I don't understand the difference between this and a trivial bundle. Because, surely if the transition functions are constant you can just perform a rescaling so that they all become the identity, making the bundle trivial ..read more