Physlib.SpaceAndTime.Space.Derivatives.MultiIndex

For any dimension $d$, it is decidable whether two multi-indices $I, J$ on $d$ coordinates are equal. This provides an algorithm to determine the truth value of the proposition $I = J$.

This definition provides a coercion that allows a multi-index $I$ of dimension $d$ to be treated as a function from the set of indices $\{0, \dots, d-1\}$ to the natural numbers $\mathbb{N}$. Under this coercion, for any $i \in \{0, \dots, d-1\}$, the expression $I(i)$ denotes the $i$-th component of the multi-index.

The zero multi-index $0$ in the space of multi-indices on $d$ coordinates is defined as the multi-index where each component is $0$. That is, for a multi-index viewed as a function $I: \{0, \dots, d-1\} \to \mathbb{N}$, the zero element satisfies $I(i) = 0$ for all $i$.

The addition of two multi-indices $I$ and $J$ in the space of multi-indices on $d$ coordinates is defined by the component-wise sum of their underlying functions from $\{0, \dots, d-1\}$ to $\mathbb{N}$. For each $i \in \{0, \dots, d-1\}$, the $i$-th component of the sum is given by $(I + J)(i) = I(i) + J(i)$.

For a multi-index $I$ on $d$ coordinates, represented as a function $I: \{0, \dots, d-1\} \to \mathbb{N}$, its order $|I|$ is defined as the sum of its components: $$|I| = \sum_{i=0}^{d-1} I_i$$

Given a multi-index $I$ on $d$ coordinates, represented as a function $I: \{0, \dots, d-1\} \to \mathbb{N}$, and a specific coordinate index $i \in \{0, \dots, d-1\}$, the increment function returns a new multi-index by increasing the value of the $i$-th coordinate by one. This is equivalent to adding the unit multi-index $e_i$ (which has 1 at the $i$-th position and 0 elsewhere) to $I$.

Two multi-indices $I$ and $J$ on $d$ coordinates are equal if all their components are equal. That is, if $I_i = J_i$ for all $i \in \{0, \dots, d-1\}$, then $I = J$.

For any index $i \in \{0, \dots, d-1\}$, the $i$-th component of the zero multi-index $0$ is $0$.

For any multi-indices $I$ and $J$ on $d$ coordinates and any index $i \in \{0, \dots, d-1\}$, the $i$-th component of the sum $I + J$ is equal to the sum of the $i$-th components of $I$ and $J$. That is, $(I + J)_i = I_i + J_i$.

Let $I$ be a multi-index on $d$ coordinates, represented as a function $I: \{0, \dots, d-1\} \to \mathbb{N}$. For any coordinate index $i \in \{0, \dots, d-1\}$, the $i$-th coordinate of the multi-index obtained by incrementing $I$ at position $i$ is equal to $I_i + 1$. That is, $(\text{increment}(I, i))_i = I_i + 1$.

Let $I$ be a multi-index on $d$ coordinates, which is defined as a function $I: \{0, \dots, d-1\} \to \mathbb{N}$. For any two coordinate indices $i, j \in \{0, \dots, d-1\}$ such that $j \neq i$, the $j$-th coordinate of the multi-index obtained by incrementing $I$ at position $i$ is equal to the original $j$-th coordinate of $I$. That is, $(\text{increment}(I, i))_j = I_j$.

For the zero multi-index $0$ on $d$ coordinates, its order $|0|$ is equal to $0$. Here, the order of a multi-index is defined as the sum of its components.

For any two multi-indices $I$ and $J$ on $d$ coordinates, the order of their sum is equal to the sum of their individual orders, denoted as $|I + J| = |I| + |J|$. Here, the order $|I|$ of a multi-index $I$ is defined as the sum of its components.

For any coordinate index $i \in \{0, \dots, d-1\}$, the multi-index on $d$ coordinates that has the value $1$ at the $i$-th position and $0$ at all other positions has an order equal to $1$. In other words, if $e_i$ is the unit multi-index in the $i$-th direction, its order is $|e_i| = 1$.

For a multi-index $I$ on $d$ coordinates and a coordinate index $i \in \{0, \dots, d-1\}$, the order of the multi-index obtained by incrementing $I$ at the $i$-th coordinate is equal to $|I| + 1$. Here, the order $|I|$ is the sum of the components of the multi-index, and the increment operation increases the $i$-th component of $I$ by one.

Given a multi-index $I$ of dimension $d+1$, which can be viewed as an element of $\mathbb{N}^{d+1}$ or a sequence $(I_0, I_1, \dots, I_d)$, the tail function maps $I$ to a multi-index of dimension $d$ by dropping the $0$-th coordinate. Specifically, the resulting multi-index is $(I_1, I_2, \dots, I_d)$, where the $i$-th component of the tail corresponds to the $(i+1)$-th component of the original multi-index $I$.

For a multi-index $I$ of dimension $d$, which can be viewed as an element $(I_0, I_1, \dots, I_{d-1}) \in \mathbb{N}^d$, the function `toList` returns the canonical ordered list of coordinate directions. This list contains each coordinate index $i \in \{0, \dots, d-1\}$ exactly $I_i$ times, sorted in non-decreasing order. Recursively, if $d=0$ the list is empty, and if $d > 0$, the list consists of $I_0$ copies of the index $0$ followed by the indices of the tail multi-index $(I_1, \dots, I_{d-1})$ shifted by $1$.

For the zero multi-index $0$ on $d+1$ coordinates, its tail—the multi-index of dimension $d$ obtained by dropping the first coordinate—is equal to the zero multi-index $0$ on $d$ coordinates.

For any multi-index $I$ of dimension $d+1$, the tail of the multi-index obtained by incrementing its $0$-th coordinate is equal to the tail of the original multi-index $I$. That is, if $\text{increment}(I, 0)$ is the multi-index where the $0$-th component is increased by $1$ and $\text{tail}(I)$ is the multi-index of dimension $d$ formed by dropping the $0$-th coordinate, then $\text{tail}(\text{increment}(I, 0)) = \text{tail}(I)$.

For any multi-index $I$ of dimension $d+1$ (represented as an element $(I_0, I_1, \dots, I_d) \in \mathbb{N}^{d+1}$) and any coordinate index $i \in \{0, \dots, d-1\}$, the tail of $I$ after incrementing its $(i+1)$-th coordinate is equal to the tail of $I$ incremented at its $i$-th coordinate. Specifically, if $\text{increment}(I, k)$ denotes the operation of increasing the $k$-th component of a multi-index by one, and $\text{tail}(I)$ denotes the operation of dropping the $0$-th component $(I_1, \dots, I_d)$, then: $$\text{tail}(\text{increment}(I, i+1)) = \text{increment}(\text{tail}(I), i)$$

For the zero multi-index $0$ of dimension $d$ (where every component is zero), its canonical list of coordinate indices $\text{toList}(0)$ is the empty list $[\,]$. This is consistent with the definition that the list contains each coordinate index $i$ exactly $I_i$ times; since $I_i = 0$ for all $i$ in the zero multi-index, the resulting list is empty.

For a multi-index $I$ of dimension $d$ (an element of $\mathbb{N}^d$), the length of its canonical list of coordinate indices, `toList I`, is equal to its order $|I|$. The order $|I|$ is defined as the sum of the components of the multi-index: $$|I| = \sum_{i=0}^{d-1} I_i$$ The list `toList I` is the sequence containing each index $i \in \{0, \dots, d-1\}$ exactly $I_i$ times.

For any multi-index $I$ of dimension $d+1$, the canonical sorted list of coordinate indices for the multi-index obtained by incrementing the $0$-th coordinate of $I$ is equal to the list obtained by prepending the index $0$ to the canonical list of $I$. Mathematically, $$\text{toList}(\text{increment}(I, 0)) = 0 :: \text{toList}(I)$$ where $\text{increment}(I, 0)$ increases the first component $I_0$ by $1$, and $\text{toList}(I)$ is the sorted list containing each index $i \in \{0, \dots, d\}$ exactly $I_i$ times.

For any coordinate index $i \in \{0, \dots, d-1\}$, let the multi-index $I$ be the unit multi-index obtained by incrementing the $i$-th component of the zero multi-index $0$. The canonical list of coordinate directions for $I$, denoted as $\text{toList}(I)$, is the singleton list containing only the index $i$. Mathematically, $$\text{toList}(\text{increment}(0, i)) = [i]$$ where $\text{increment}(0, i)$ is the multi-index with $1$ at position $i$ and $0$ elsewhere.