Betweenness is an algorithmic problem in order theory about ordering a collection of items subject to constraints that some items must be placed between others. It has applications in bioinformatics and was shown to be NP-complete by Opatrný (1979). Problem statement The input to a betweenness pr...

Suppose that $\Omega \subseteq \mathbb R^n$ non-empty, open, and $n \ge 1$. We have an increasing collection $(K_i)_{i \in \mathbb N}$ of compact subsets of $\Omega$ with union $\Omega$. For $K \subset \Omega$ fixed and compact, is it necessarily the case that $K \subseteq K_i$ for $i$ sufficient...

Question Prove that for all $d\geq 1, \mathbb{E}^d\nrightarrow (l_4)_3$. That is, we need to prove that there is no way to colour the points in $\mathbb{E}^d$ with 3 colours such that there will always exist 4 collinear points, evenly spaced 1 unit apart, that are all the same colour. Note: Here,...

Could someone please clearly write out how we get from this expression $$\log\left[\sum_{\mathbf Z}\left(\prod_{n=1}^N\prod_{m=1}^M\pi_m^{\mathbf{1}(z_n=m)}\mathcal{N}\left(\mathbf x_n;\mathbf{\mu}_m,\mathbf{\Sigma}_m\right)^{\mathbf{1}(z_n=m)}\right)\right]$$ to this one $$\sum_{n=1}^N\log\left[...

Assume that $X\in \mathrm{PSD}(n)$ is a symmetric, positive semi-definite matrix with real entries and $C>0$ is a positive real number. If $X$ satisfies $$ \forall \alpha\in\mathbb{Z}^n\setminus\{0\},\qquad \alpha^T X \alpha \geq C, $$ then is it true that $X$ is positive definite? That is, if $X...

Progressive Tax Rate Explanation Progressive taxation works by taxing income within a certain bracket at different rates. For example: Bracket # % tax rate within bracket Min amount (exclusive) Max amount (inclusive) 1 10% 0 50,000 2 20% 50,000 60,000 3 25% 60,000 n/a (>60,000) In t...

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with good reduction at $p$, i.e., there exists an elliptic curve $\mathcal{E}$ over $\mathbb{Z}_p$ whose generic fiber is isomorphic to $E$. I have seen on page 76, just above Theorem 5.38, of a course note by Schraen that the action of $G_{\mathbb...

Let $S$ be a finite set and $n,m\in\mathbb N$. Consider a random variable $R$ uniformly distributed on $S^{n+m}$. I am trying to tackle the probability of the event \begin{equation} A_m(n):=\{\exists_{1\leq i<j\leq m}: (R_i,\dots,R_{i+n-1}) = (R_j,\dots,R_{j+n-1})\} \end{equation} as $m\to\infty$...

In this hexahedron, define the planes $\pi_0 = ABCD$, $\pi_1 = ABFE$, $\pi_2 = BCGF$ , $\pi_3 = CDHG$ , $\pi_4 = DAEH$, $\pi_5 = EFGH $. You are given that $ABCD$ is an isosceles trapezoid, with $AB = 10, CD = 4 $, and $AD = BC = 3\sqrt{5} $. In addition, $\angle(\pi_0, \pi_1) = 70^\circ, \ang...

In the following figure, the line that touches the ellipse at only one point, called A, is the tangent line to the ellipse at that point. C is the center of the ellipse. Point $L'$ is the point where the perpendicular passing through C to the tangent line intersects the tangent line. Point L, ins...

alright, this question is philosophical and somewhat fuzzy. i also admit to knowing little about logic. all in all, this question can possibly be easily resolved by either pointing to (perhaps even well-known) literature i haven’t found or by pointing out a fault in my reasoning. joel david hamki...

In a metric space, we have that the intersection of countably many open sets is open: Let $A_1, A_2, \ldots, A_n$ be open sets and $A = \bigcap_{i=1}^n A_i$. For $x \in A$, $x \in A_i$ for all $i$, and since each $A_i$ is open, $\exists r_i > 0$ with $B_{r_i}(x) \subseteq A_i$. Set $r = \min\{r_1...

I am trying to prove (in a strict way) that $X/A$ inherits a natural CW-complex structure from a CW-pair $(X,A)$. I start with the following inductive definition of a CW-complex: Let $\{\mathcal A_n\}_{n=0}^\infty$ be a sequence of disjoint sets such that $\mathcal A_0\ne\emptyset$. Starting wit...

As a continuation of this question relating the Minimum $k$ for which every positive integer of the interval $(kn, (k+1)n)$ is composite and this other one on the divisibility of numbers in intervals of the form $(kn, (k+1)n)$, I have been looking for strategies to prove the following Conjecture:...

Let $(X_i, Y_i)_{i=1}^{\infty}$ be iid continuous random vectors with continuous joint density, where $X_1$ have support $\mathcal{X}$. Let $B_n\subset \mathcal{X}$ be decreasing subsets such that $\cap B_n= x_0\in\mathcal{X}$. Let $S = \{i\leq n: X_i\in B_n\}$. I want to show that $$ \frac{1}{|S...