Suppose $\Omega \subset \mathbb{R}^d$ is a $C^{1,1}$ domain. Consider the biharmonic boundary value problem (BVP): $$ \begin{cases} \Delta^2 u = f \\ \nabla u \cdot \nu = g \\ u = u_D \end{cases} $$ wherein $\Delta^2 u = \Delta\Delta u$ is the application of the Laplace operator twice. (a) Determ...

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 pro...

I came across this example in an old book. I have this question here. What is this point P, how is it defined? I did some calculations (implicit differentiation) and it seems to me it's the point where $x = \sqrt[3]{4}, y = \sqrt[6]{4}$. Why? Because it seems to me that's when the derivative $...

I was trying to evaluate the following integral, $$I=\int_\limits {-\infty}^{\infty} \dfrac{\sin(x)}{\sinh(x)}\,dx$$ but had no success. I first expanded the the hyperbolic sine: $$I=2\int_\limits {-\infty}^{\infty} \dfrac{\sin(x)}{e^{x}-e^{-x}}\,dx=2\Im \int_\limits {-\infty}^{\infty} \dfrac{e...

I started with a $40^\circ$ angle: I then copied that to the other side with a compass, I then added a perpendicular line on each side to make it a right triangle, making sure both are the same length: I then drew a triangle with the width of the other two triangles, and with a height doub...

I'm looking for general methods for solving equations of elementary functions of one variable in closed form. Definition: The elementary functions are generated by applying finite numbers of $\text{exp}$, $\ln$ and/or unary or multiary $\mathbb{C}$-algebraic functions (according to Ritt 1925 and ...

I'm studying general topology and a question has come to my mind. By definition, in a topological space, a neighbourhood basis of a point x is a subset of its neighbourhoods such that every neighbourhood of x contains one element of that subset, where a subset is neighbourhood of x if it contains...

In the midterm of probability in my university, they demand for us to calculate the derivative of $ h(x)=8\times\log_e{(\frac{x}{10})}$ and i find $ h(x)^`=\frac{8}{10x}$ but in the solution they write $ h(x)^`=\frac{8}{x}$ , i repeat the operation a lot of time but i can't find the same result, ...

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(\log\mu)}{\mu^n},...

$$A(0)= x \in\mathbb{Z}^+,\ A(n) = A(A(n-1) \bmod n)^2$$ At first glance, one would think that such sequence would grow very fast. But my testing suggest that this sequence actually ends with $x^4$ repeating indefinitely. To make testing faster, i set $B(n) = log_x(A(n))$. $$B(0) = 1$$ $$B(n) = 2...

I'm working on a calculus problem and need help solving the following definite integral: I'm struggling to simplify the integrand or find a substitution that makes the integral easier to evaluate. Here's what I've tried so far: Attempt at simplifying the inner square root. Checking if the funct...

Fix $n \geq 3$ and take any orthonormal vectors $x,y,z \in \mathbb{R}^n$. Let also $A \in M_n(\mathbb{R})$ be a symmetric matrix with positive entries ($A_{ij} = A_{ji} > 0$). Is the following inequality true? $$\sum_{i<j}A_{ij}\begin{vmatrix}x_i & x_j \newline y_i & y_j \end{vmatrix}^2 \leq \sum_{i

Let $\mathcal{C}$ be a small category. Functoriality for presheaves says that for any functor $u\colon\mathcal{C}\to\mathcal{D}$ the precomposition functor $u^*\colon PSh(\mathcal{D})\to PSh(\mathcal {C})$ has two adjoints $u_!, u_*$ on the left and on the right correspondingly, see SGA 4, Exposé...

This is the theorem to prove. Below is my proof that I consider rather long and complex. The given data is on this drawing: Construct $\angle DCE = \angle DCB$. The point $E$ on ray $CE$ is chosen in such way that $CE = CB$, and that is always possible by segment construction axiom. Connect...

Given a probability distribution $F$: Let $X=F^{−1}(U)$ and $Y=F^{−1}(1−U)$, where $U$ is uniformly distributed on $[0,1]$. How to prove $\operatorname{Cov}(X, Y)<0$? I think $E[X]=E[Y]$ and the focus should be on proving $E[XY]<E[X]E[Y]$, but how to deal with the $E[XY]$ term $?$.

I am working on calculating the expected value of the reciprocal of the square of a variable $X$ that follows an Inverse Gaussian distribution with parameters $\mu$ (mean) and $\lambda$ (shape). The probability density function (PDF) of the Inverse Gaussian distribution is given by: $f(x; \mu, \l...