1

From the low-discrepancy wiki I have:

$$\displaystyle \prod_{i=1}^s[a_i, b_i) = \{ \in \mathbb{R}^s : a_i \le x_i \le b_i\}$$ where $$0 \le a_i < b_i \le 1$$

How can I read it? Should I first make do a cartesian product of $[a_i,b_i)$? For example for $s=3$ I would have: $$ [a_1, b_1) \times [a_2, b_2) \times [a_3, b_3) = \{ \{a_1, a_2, a_3\}, \{a_1, a_2, b_3\}, \{a_1, b_2, a_3\}, \{a_1, b_2, b_3\}, \{b_1, a_2, a_3\}, \{b_1, a_2, b_3\}, \{b_1, b_2, a_3\}, \{b_1, b_2, b_3\}\} $$

then I would put it into set-builder notation.. but for first $\{a_1, a_2, a_3\}$ I don't have $b_i$ value.

Community
  • 1
Darek Nędza
  • 113

1 Answers1

1

You should read, $\displaystyle \prod_{i=1}^s[a_i,b_i)$ as, "the product of all intervals closed $a_i$, open $b_i$ from $i=1$ to $s$". And $\displaystyle \{\tilde x\in \Bbb R^s:\tilde x=(x_1,x_2,\dots x_s) \text{ where }x_i\in[a_i,b_i),\forall i=1,2\dots s\}$ as, "the collection of all $\tilde x$ ($x$ tilde) in $\Bbb R^s$ such that $x$ tilde is of the form of $s$-tuple $(x_1,x_2,\dots x_s)$ where $x_i$ belongs to $[a_i,b_i)$ for all $i=1,2,... s$".

Hope it works.

Further, $\displaystyle \prod_{i=1}^s[a_i,b_i]=[a_1,b_1]\times [a_2,b_2]\times ... \times [a_s,b_s]$ where $\times$ is the cartesian product defined here.

Sujit Bhattacharyya
  • 3,030
  • After I compute $\displaystyle \prod_{i=1}^s[a_i, b_i) = { \in \mathbb{R}^s : a_i \le x_i \le b_i}$ I will get collection of s-tuples: ${ s-tuple, s-tuple...}$. So, for $s = 1$ I will get ${a_1, a_1 + v, a_1 + 2v, ...a_1 + kv}$ where $v$ is some "small value" and $a_1 + k*v$ is smaller than $b_1$. So, for $v=0.4$, $a_1=0$ and $b_1=1$ I will get ${0,0.4,0.8}$. For $s=2$ I need to compute the Cartesian product: ${0,0.4,0.8}\times{0,0.4,0.8| = {{0,0},{0,0.4},{0,0.8},{0.4,0},{0.4,0.4},{0.4,0.8},{0.8,0},{0.8,0.4},{0.8,0.8}}$. Is this correct? How do I find $v$? – Darek Nędza Apr 29 '19 at 11:40
  • I think you have confused with https://math.stackexchange.com/questions/2115752/what-is-an-interval-in-mathbbrn/2115757 this one. Further for $s=1$ you simply get an interval which cannot be written in your form ${a1,a1+v...}$ etc. – Sujit Bhattacharyya Apr 30 '19 at 02:33
  • As for link: instead of ${{0,0},...}$ I should write ${(0,0)}$ and ${0, 0.4, 0.8}$ as ${(0), (0.4), (0.8)}$. That is what you mean? As for $s=1$ do you mean that I have forgotten parenthesis ($()$)? – Darek Nędza May 09 '19 at 16:06
  • 1
    Definitely yes. Thats what I mean. – Sujit Bhattacharyya May 10 '19 at 01:59