Let $\Omega$ be a convex, closed, compact set in $\mathbb{R}^d$ with a smooth boundary.
Given a data $(x_i,d_i)$, $x_i \in \Omega$,$d_i \in \mathbb{R}$, $i = 1,2,3...N$, and $\sum\limits_{i=1}^N d_i = 0$
I want to find a function $f:\Omega \to \mathbb{R}$, such that,
$\int_{\Omega}f(x)dx = ...
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I'm trying to prove that the group below isn't cyclic:
$G = \{ (1, 2, 3, ... n)^a\cdot(n+1, n+2, n+3, ... 2n)^b \mid 0 \leq a,b \leq n-1 \}$
To do this, I'm trying to show that none of the elements of $G$ have order $n^2$ - however I've made little progress.
Any help would be much appreciated,...
How would one denote generated sets from a given cover within a set $X$ utilizing the definition in $(1.1)$
$$(1.1)$$
A cover of a set $X$ is a collection of sets whose union contains $X$ as a subset. Rigorously speaking we have if:
$$C = \Big\{U_{a} : a \in A \Big\}$$
is an indexed family of ...
There are two common ways to measure the length of a solution to Rubik's Cube. The first is to count the number of quarter turns. The second is to count the number of outer-layer twists, called "face turns". A move to turn an outer layer two quarter (90°) turns in the same direction would be counted as two moves in the quarter turn metric (QTM), but as one turn in the face metric (FTM, or HTM "Half Turn Metric", or OBTM "Outer Block Turn Metric").
The maximum number of face turns needed to solve any instance of the Rubik's Cube is 20, and the maximum number of quarter turns is 26. These numbers...
I am working on the following problem:
Let $R$ be an additive abelian group. Turn $R$ into a ring by defining $xy = 0$ for all $x$, $y$ $\in$ $R$. Under which conditions is an ideal $I$ of $R$ prime?
I have already shown that if $I$ is prime, then $R - I$ is at most a singleton by showing ...39 mins ago, by search
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