I understand the notion of Linearity typically applied to define Linear Programs (here I will capitalize "Linear" when and only when I use it in this sense). In contrast, this question is focused on the notion of linearity applied in math more broadly, with an eye toward scaling and adding toget...

Let $\Sigma$ be a surface and $\Omega$ be an open subset of $\Sigma$. Suppose that $\Omega$ is homeomorphic to the open unit disk $\mathbb{D}$ and is relatively compact in $\Sigma$. When $\Sigma$ is the complex plane or the Riemann sphere, then the Caratheodory-Torhorst Theorem can be used to sho...

Prove that: $\|v\|^2 \ge \sum _{i=1}^n \langle v,e_i\rangle^2$ for any $v \in V$, where $V$ is an inner product space and $S = \{e_1, e_2, \ldots , e_n\}$ is an orthonormal subset of $V$. I know from the decomposition theorem that \begin{align*} u &= \langle u, e_1\rangle^2 e_1+ \cdots + \lang...

Let $\Sigma$ be a smooth, $n$-dimensional homology sphere. In a paper that I am reading, the author states that there exists a smooth homotopy $n$-sphere $S$, such that the connected sum $\Sigma \# S$ can be embedded into $\mathbb{R}^{n+1}$. I honestly have no idea how to approach this problem. T...

I plotted the function $1000\sin{\frac{x}{1000}}$ on desmos and set it to the logarithmic scale but I am having trouble understanding the result which looks like this: Why does the graph go linear for the first part and the become so condensed right after that? Also if I set X axis as linear and...

This is a follow-up to this previous question, but a bit more specific. Suppose I have a simple Euclidean space $X = \mathbb{R}$, or $X = \mathbb{R}^2$, or $X = \mathbb{R}^3$, and a continuous real-valued function $f:X\to \mathbb R$. I can define sets like: \begin{align} A &= \{x \in X: f(x) = 0 ...

I am trying to find the differential equation which implies a smooth path $q:[a,b] \rightarrow \mathbb{R}^n$ subject to $|\dot{q}(t)| = 1$ (i.e. $q$ has unit speed) is a stationary point of $$ \int_a^b L(t,q(t), \dot{q}(t)) \ dt $$ for a certain $L(t,q(t), \dot{q}(t))$. This question and answer h...

Given a linear binary code $C$, the $r$-th generalized Hamming weight $d_{r}(C)$ is the minimal support size of an $r$-dimensional subcode of $C$ (so $d_{1}(C)$ is simply the code's distance). For a code length $n = 2^{m}-1$, and a designated distance $d=2t+1$, let's denote $\text{BCH}(n,d)$ the ...

Suppose we have a polynomial $$f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$$ where all the coefficients are whole numbers which includes Zero. Suppose we now randomly choose the values of these coefficients then: (A) What is the probability that the polynomial becomes a constant polynomial? (B) How wou...

The problems are from the proof of Theorem 1.5.12 in Leon Simon's book: Geometric Measure Theory Suppose $X$ is a locally compact Hausdorff space, $\mathcal{K}^{+}$ is the set of all non-negative continuous functions on $X$ with compact support, $\lambda:\mathcal{K}^{+}\to [0,\infty)$ is linear....

I am working with a "well-behaved" optimization problem of the form: \begin{equation*} \max_{x} f( g_{1}( x) ,g_{2}( x) ,g_{3}( x) ,\mathbf{y}) \end{equation*} where $\displaystyle f:\mathbb{R}^{n+1}\rightarrow \mathbb{R}$ is continously differentiable, $\displaystyle x\in \mathbb{R}$, $\displays...

Consider a second-order parabolic pde $$ u_t = \Delta u + F(t,x,u,\nabla u), \quad (t,x) \in [0,+\infty) \times M $$ where $M$ is a compact, smooth manifold with boundary. We assume that $F(x,t,u,p)$ is smooth everywhere except at one interior point $x_0 \in M$ and we set a smooth (parabolic) bou...

Let $\mathbb{K}\in\{\mathbb{Q},\mathbb{C}\}$ and let's consider polynomials over $\mathbb{K}$. If necessary, we can assume that the polynomials are $\mathbb{K}$-irreducible. My question is: Can a multivariate polynomial over $\mathbb{K}$ be rearranged into a univariate polynomial over $\mathbb{K}...

What is the „intuitive” reason behind the following statements... Let $\left(X_{n}\right)_{n}$ be a sequence of random variables. Let us assume $\mathbf{Q}\ll\mathbf{P}$, i.e. the $\mathbf{Q}$ probability measure is absolutely continuous with respect to the $\mathbf{P}$ probability measure. If $...

The usual definition of the Hodge star says that it maps $\Lambda^k(V)$ to $\Lambda^{n-k}(V)$ in such a way that for each pair $\omega, \eta \in \Lambda^k(V)$ holds $\omega \wedge *\eta = \langle \omega, \eta \rangle \operatorname{vol}$. I was curious whether this definition is equivalent to sayi...