Let $a : [0,1] \to D^2 = \{z \in \mathbb{C} : |z| \leq 1 \}$ be a continuous map with $|a(0)| = |a(1)|$ and further assume that $a$ is an embedded - so $a$ is an embedded arc. Let $f : D \to D$ given by $f(z) = z^2$. Under what conditions is $f \circ a$ still injective? It seems to me that a ne...

I am currently completing an investigation assignment on modelling the growth of a virus inside of the host. There are 3 ODEs that I am using in the system, all determined by change in t. The differential equations I am trying to solve are below: $𝑑𝐼/𝑑𝑡=𝛽𝑉[1−(𝐼/S)]−𝑘_𝑎*𝐼*(𝐴_3^2/(𝐴_3^2...

The notion of minimal surface (i.e. having vanishing mean curvature) is not "affinely invariant" in the following sense: if $M\subset\Bbb R^n$ is an ($m$-dimensional) minimal surface, then $T(M)$ is not a minimal surface for most $T\in\mathrm{GL}(\Bbb R^n)$. This is because having zero mean curva...

I am reading a book that has the following text which I don't understand. Let $F$ be a finite field and $\alpha \in K$ where $K$ is an extension of $F$. Then we write $F[\alpha]$ to indicate all sums of the form $\sum x_i \alpha^i$ where $x_i \in F$ and where all but a finite number of the coeff...

I'm looking for a combinatorial argument to complete a proof (below) of the following: Claim: If $(\Omega,2^\Omega,P)$ is a probability space with finite $\Omega,$ then $\sum_{A\in2^\Omega}P(A)=2^{|\Omega|-1}.$ In other words, if $\Omega$ is finite and every subset of $\Omega$ is considered an ev...

I want to know why if $\mu$ is a haar measure on a compact $G$ and $\mu(A)=\mu(G)$ then $A$ is dense in $G$. This fact is mentioned in the wikipedia page, but I couldn't find a proof for it.

I am wondering if such a result exists (or similar) and or if there is a "simple" proof. Let $\mathcal X$ be a bounded and closed subset of a topological vector space, let $\Sigma$ be the Borel $\sigma$-algebra associated and let $\Gamma$ be the set of bounded convex continuous function on $\math...

I have read an article in Wikipedia. https://en.wikipedia.org/wiki/Prenex_normal_form $(\forall x\phi)\lor\psi\Leftrightarrow\forall x(\phi\lor\psi)$ $(\exists x\phi)\land\psi\Leftrightarrow\exists x(\phi\land\psi)$ $(\exists x\phi)\rightarrow\psi\Leftrightarrow\forall x(\phi\rightarrow\psi)$ $\...

Let $X$ be a nonempty compact convex subset of $\mathbf{R}^n$. Suppose this subset has the following property: for every $x = (x_1, \dots, x_n) \in X$, for every $1 \le i< j \le n$, $$({x_1}, \ldots, {x_{i - 1}}, \frac{{x_i} + {x_j}}{2}, {x_{i + 1}}, \ldots, {x_{j - 1}}, \frac{{x_i} + {x_j}}{2}, ...

AB is a rod which is held such that $A=(1,-2,3)$ and $B=(2,3,-4)$ . A source of light is at the origin. Find the length of the shadow of the rod on a plane screen whose equation is $x+y+2z=1$ I figured out that origin and point B are on one side of given plane and point A is on other side. I foun...

is it possible to find a function $f$ such that $f \in L^{\infty}(\mathbb{T}^1)$ and $$\sum_{j=0}^{+\infty} \left(\sum_{n \in \mathbb{Z}: 2^j \le |n|<2^{j+1}}|\widehat{f}_{n}|^{2} \right)^{1/2}=+\infty \hspace{0.5cm}\text{?}$$

Based on this: https://math.stackexchange.com/a/4454826/595084 I want to find the integral $$I_1=\int^{\infty}_0\frac{\sin(x)}{e^x+1}\text{ d}x$$ In the method the answer uses, it converts this integral into $$I(n)=\int^{\infty}_0\frac{\sinh(nx)}{e^x+1}\text{ d}x$$ and solves for $-iI(i)$. In oth...