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Let $X$ be a set and suppose $\mathbb{B}$ is a collection of subsets of $X$.
$\mathbb{B}$ is a basis for some topology on $X$ $\textbf{if and only if}$
(1)$\bigcup_{B\in \mathbb{B}}B = X$ (Covering property)
(2) If $B_1, B_2 \in \mathbb{B}$ and $x\in B_1 \cap B_2$ then $\exists$ $B_3 \in \ma...
Let's assume the probability distributions are Gaussian (or normal) distributions. In other words, in the Bayes' rule
\begin{align}
p(z|x)=\frac{p(x|z)p(z)}{p(x)}
\tag{1}\label{1}
\end{align}
The posterior $p(z|x)$, the likelihood $p(x|z)$, the prior $p(z)$ and the evidence (or marginal) $p(x)$...
https://en.wikipedia.org/wiki/Semimodule
So every $R$-semimodule $N$ over a ring $R$ is a module.
The axioms are:
$$
\begin{matrix}
(1) & r(m + n) = rm + rn \\
(2) & (r+s)m = r m + s m \\
(3) & (rs)m = r(sm) \\
(4) & 1m = m \\
(5) & 0m = r0 = 0 \\
\end{matrix}
$$
for all $r, s \in R, \ \ m,n ...0
Say you take a disk model of $\Bbb R^2$ and deform it into a square model of $\Bbb R^2$ and then glue six copies together such that they form the six faces of a cube. If each face of the cube is projected towards the opposite face, can a unique three manifold in the interior of the cube be built ...
