I'm reading this paper and considering the following notations/definitions: Let $G$ be a finite group, $V,W$ finite dimensional vector spaces and consider two maps $\phi:V \rightarrow \mathbb{R}$, $\Phi:V \rightarrow W$. Now we define: $$\begin{align} \psi(X) &:= \frac{1}{|G|}\sum_{g \in G}\phi(g...

Where $\angle{A}=84^{\circ}, \angle{ACD}=42^{\circ}, BD=AC$, find $\angle{BCD}$. Wonder if there is solution without using trigonometric functions. I tried with getting circumcenter of triangle ABC, but seems hard to prove it forms an equilateral triangle with side AC. Also if trying from equila...

Here is a really tough problem. If $$\boldsymbol{a\cos\alpha\cos\beta+b\sin\alpha\sin\beta+c=0}$$ $$\boldsymbol{a\cos\gamma\cos\delta+b\sin\gamma\sin\delta+c=0}$$ $$\boldsymbol{a\cos\beta\cos\gamma+b\sin\beta\sin\gamma+c=0}$$ $$\boldsymbol{a\cos\delta\cos\epsilon+b\sin\delta\sin\epsilon+c=0}$$ $$...

Among functions $f$ satisfying $\forall x\in\mathbb R\, f(x+p) = f(x)$ with $p>0$ are sinusoids $f(x) = A\sin(\omega x +\varphi)$ with $p=2\pi/\omega.$ These also satisfy the function equations $$ \forall n\in\{2,3,\ldots\}\, \forall x\in\mathbb R \, \sum_{k\,=\,0}^{n-1} f\left(x + \frac {kp} n ...

In my algebraic geometry course I found the following problem: Let $E$ an elliptic curve over the field $\mathbb{F}_5$ given by the equation: $$y^2z=x^3+xz^2-z^3$$ Find the group asociated to $E$. Doing some computations and with quadratic residues I found that $E$ has the following points: $$[0:...

Definition: let $n, a \in \mathbb{N}$. The root $\sqrt[n]{a}$ is reducible if it can be written as $\sqrt[m]{b}$ for some natural numbers $b$ and $m<n$, or if it can be written as $M\sqrt[n]{c}$ for naturals $M$ and $c<a$. Otherwise we say the root is reduced. The condition that the roots be red...

In Henson's Model Theory lecture notes I found an exercise quite early on (1.30, p. 12) that prove too difficult for me. It goes like this: Let $L$ be the first order language whose only nonlogical symbol is the binary predicate symbol $<$. Let $\mathcal{A}=(\mathbb{N},<)$ and let $\mathcal{B}=\...

I have a question that reads: "An insurance company has initial capital of £$20,000$ and receives claims from clients with a homogeneous Poisson process $X(t),t\geq 0$. The intensity of claims is $6$ claims a month. Assume that the amount $Z_k,k=1,2,...$ of successive claims are independent rando...

Given a field $k$, a valuation map $v$ is a surjective group homomorphism $k^\times\twoheadrightarrow H$, where $H$ is an ordered group (called the valuation group), such that $v(a+b)\ge \operatorname{min}(v(a),v(b))$. I must prove that if I have another valuation $v':k\twoheadrightarrow H'$, wit...

Professor gave me the problem that calculates below real integral using complex analysis. $\int_0^{2\pi} \tan \frac{\theta}8 d\theta $ Actually this integral can easily be calculated just substituting $t=\cos\frac{\theta}8$, but the professor requested me to calculate this integral using complex ...

Let $S$ be a smooth oriented surface in $\mathbb R^3$ with boundary $C$, and let $f: \mathbb R^3 \to \mathbb R^3$ be a continuously differentiable vector field on $\mathbb R^3$. Stokes's theorem states that $$ \int_C f \cdot dr = \int_S (\nabla \times f) \cdot dA. $$ In other words, the line int...

Looking at the pefsu problem of the Moscow Mathematical Papyrus here I don't understand why the algorithm takes 1/2 of the calculated grain measure to produce beer. Why aren't the 5 heqats multiplied by 4 to get 20 quantities of beer which would be a better deal in exchange? The problem transcrib...

A subway car has $10$ individual seats, with $5$ in front and $5$ in back. From $10$ ​passengers, $4$ prefers front seat, $3$ prefers back seat and the others have no preference. In how many ways can passengers be seated, respecting preferences? Attempt: I want to solve using only the notion of F...

I have the roots for a very large order polynomial (>100), and from those alone wish to recreate the polynomial and run into numerical challenges when using the poly function for doing this available in Matlab and Python (numpy). I attempted to convolve the coefficients for the roots and still se...