Is there a theorem which characterises which bijective maps $f:\mathbb{GP}^1 \to \mathbb{GP}^1$, where $\mathbb{G:F}$ is a field extension of degree 2, have the property that whenever the cross ratio of four points of $\mathbb{GP}^1$, $(a,b;c,d)$, is in $\mathbb{FP}^1$, then the same is true for ...

Is there a theorem which characterises which bijective maps $f:\mathbb{GP}^1 \to \mathbb{GP}^1$, where $\mathbb{G:F}$ is a field extension of degree 2, have the property that whenever the cross ratio of four points of $\mathbb{GP}^1$, $(a,b;c,d)$, is in $\mathbb{FP}^1$, then the same is true for ...

Say I have a commutative monoid $M$ that is generated by three elements $A,B,C$, where I have that $A+C=2B$. I want to write this a free (does that even mean anything?) monoid $\mathbb N^3$ with generators $X$, $Y$, $Z$ quotiented out by some equivalence relation $\sim$ that yields $[X]+[Z]=2[Y]$...

During the re-tagging of questions, tags sometimes become orphaned from existing questions. Are these zombie tags ever removed from the tags list? What if a tag is misspelled and needs to be removed? How do we get rid of it? Return to FAQ Index

You should edit out the tag from all the questions that use it. Note that you should probably seek consensus on the per-site meta before doing so, if the tag was somehow added to a significant number of questions without anyone noticing the misspelling. Tags not associated with any question are ...

For context, I have been trying to find the subfactorials of numbers lately and am currently stuck on evaluating $!5$. The subfactorial of a number is defined as$$\dfrac{\Gamma(n+1,-1)}e$$however can be represented in integral form as$$!n=\int_{-1}^\infty x^ne^{-(x+1)}dx=\dfrac1e\int_{-1}^\infty ...

Consider the time-homogeneous optimal stopping problem $$V(x) = \sup_{\tau} E(f(X_\tau^x))$$ where $X^x$ is a diffusion starting at the point $x \in \mathbb{R}$. Typically, optimal stopping problems have an associated optimal stopping time $$\tau(x) \equiv \inf \{t \ge 0 : X^x_t \in S\}$$ where $...

As the title say see the reference for the Hilbert's curve : Say a creature start at some point in the Hilbert curve and have as code a rational number less than one to go away . Moreover each point are indexed like : And We consider for a sufficient large number of iteration a random walk se...

Definition: A family of functionals $\{F_n\}$ on a metric space $X$ is said to be equi-coercive if, for every $\alpha \in \mathbb{R}$, there is a compact set $K_\alpha$ of $X$ such that for all $n$ we have $$\{F_n \leq \alpha\} \subseteq K_\alpha.$$ Question: If $\{F_n\}$ is a family of equi-coer...

We know, If $X$ is path connected,locally pathconnected,semi locally simply connected then $X$ has a universal cover $\overline X$.Now, By Galois Correspondence for any subgroup $H$ of $π_{1}(X)$ we can find subcover $X_H$ of $\overline X$ such that $p_*(X_H)=H$. My Question is : If I know univer...

Suppose $K = \mathbb{Q}(\alpha)$ for some $\alpha$ algebraic, and let $f$ be the minimal polynomial of $\alpha$ over $\mathbb{Q}$. Say $d = [K:\mathbb{Q}]$. I want to show that $$\mathrm{disc}(1, \alpha, \alpha^2, \dots, \alpha^{d-1}) = (-1)^{d(d-1)/2} N_{K/\mathbb{Q}} (f'(\alpha)).$$ I'm trying...

Is there a theorem which characterises which bijective maps $f:\mathbb{GP}^1 \to \mathbb{GP}^1$, where $\mathbb{G:F}$ is a field extension of degree 2, have the property that whenever the cross ratio of four points of $\mathbb{GP}^1$, $(a,b;c,d)$, is in $\mathbb{FP}^1$, then the same is true for ...

In a certain problem I am asked to find how many $4$-cycles are there in the graph $C_4 + K_9$, where $C_4,K_9$ are the $4$-cyclic graph and the complete graph of $9$ vertices, respectively. Here, the addition means that both graphs are connected by exactly and edge and nothing else. Here is my a...

Here the user has just edited a "Tag Info" item. They then read about a seemingly additional "Tag Wiki" that they might want to edit. As Tag Info and Tag Wiki are probably in fact referring to the same thing, the language on the page needs to be tightened up to avoid confusing users.

Consider a stationary discrete Gaussian process $(X^{\lambda}_n)_{n\in\mathbb N}$ with mean function $m(x) = 0$ and covariance function $$k(n,m) = e^{-\lambda|m-n|}.$$ You can think of it as a discretization of the stationary Ornstein--Uhlenbeck process with drift coefficient $\lambda$. Fix $N \i...

Throughout this section $\Omega$ will be a fixed open bounded subset of $\mathbb{R}^n$ with Lipschitz countinuous boundary, and $W:\mathbb{R}\to [0 , +\infty )$ will be a fixed non-negative continuous functon with exactly two zeros $\alpha, \beta$ ($0< \alpha < \beta$). Definition Let $(F_j)$ be ...

Definition: A family of functionals $\{F_n\}$ on a metric space $X$ is said to be equi-coercive if, for every $\alpha \in \mathbb{R}$, there is a compact set $K_\alpha$ of $X$ such that for all $n$ we have $$\{F_n \leq \alpha\} \subseteq K_\alpha.$$ Question: If $\{F_n\}$ is a family of equi-coer...