Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $q,N\in\mathbb{N}$. Points of $\mathbb{R}^{m}$ are tuples $x=(x_{1},\dots,x_{m})$ as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces, and $b(k,i)=(k-1)q+i$ is the block index of Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions, which lies in $[qN]$ for $k\in[N]$ and $i\in[q]$ by Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §range.

1. (Block maps)¶ For $k\in[N]$ the $k$-th block map is

and $\mathfrak{p}_{k}(x)$ is the $k$-th particle of $x$.

2. (Configurations)¶ For $y_{1},\dots,y_{N}\in\mathbb{R}^{q}$ the configuration $[y_{1},\dots,y_{N}]$ is the point of $\mathbb{R}^{qN}$ whose coordinate with index $b(k,i)$ is the $i$-th coordinate of $y_{k}$, for every $k\in[N]$ and $i\in[q]$. It is well defined, each index in $[qN]$ being $b(k,i)$ for exactly one such pair by Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection, and $\mathfrak{p}_{k}([y_{1},\dots,y_{N}])=y_{k}$ for every $k\in[N]$.

3. (Product maps)¶ For $p\in\mathbb{N}$ and a map $h:\mathbb{R}^{q}\to\mathbb{R}^{p}$, the product map is

the configuration of clause 2 being formed in $\mathbb{R}^{pN}$.

4. (Diagonal points)¶ For $a\in\mathbb{R}^{q}$ the diagonal point is $a^{\oplus}=[a,\dots,a]\in\mathbb{R}^{qN}$, the configuration all of whose particles equal $a$.