bound for the size of an epsilon packing

[[concept]]

Topics

const fieldName = "theme"; // Your field with links
const oldPrefix = "Thoughts/01 Themes/";
const newPrefix = "Digital Garden/Topics/";
 
const relatedLinks = dv.current()[fieldName];
 
if (Array.isArray(relatedLinks)) {
    // Map over the links, replace the path, and output only clickable links
    dv.el("span",
        relatedLinks
            .map(link => {
                if (link && link.path) {
                    let newPath = link.path.startsWith(oldPrefix)
                        ? link.path.replace(oldPrefix, newPrefix)
                        : link.path;
                    return dv.fileLink(newPath);
                }
            })
            .filter(Boolean).join(", ") 
	// Remove any undefined/null items
    );
} else {
    dv.el(dv.current().theme);
}
 
 

Proposition

Suppose $\mathcal{X}$ is an epsilon packing of $B^d$. Then $|\mathcal{X}|\le (1+\frac{1}{ϵ}{)}^d$

NOTE

The true value is is $\exp \left(\mathcal{O}\left(\frac{d}{ϵ}\right)\right)$

Proof

Since $\mathcal{X}$ is a packing of $B^d$, we know

\bigcup_{x \in {\cal X}} B^d(x, \varepsilon) &\subseteq B^d(0, 1+\varepsilon) \\ \implies \lvert {\cal X} \rvert \cdot \text{vol}(B^d(0, \varepsilon)) &\leq \text{vol}(B^d(0, 1+\varepsilon)) \\ \implies \lvert {\cal X} \rvert &\leq \frac{\text{vol}(B^d(0, 1+\varepsilon))}{\text{vol}(B^d(0, \varepsilon))} \\ &=\left( \frac{1+\varepsilon}{\varepsilon} \right)^d \end{align}$$ Note in the first line the LHS union is disjoint, which is how we get the second line. $$\tag*{$\blacksquare$}$$

References

References

See Also

Mentions

Mentions

const modules = await cJS()
 
const COLUMNS = [  
	{ id: "Name", value: page => page.$link },  
	{ id: "Last Modified", value: page => modules.dateTime.getLastMod(page) },
];  
  
return function View() {  
	const current = dc.useCurrentFile();
// Selecting `#game` pages, for example. 
	let queryString = `@page and linksto(${current.$link})`;
	let pages = dc.useQuery(queryString);
	
	// check types
	pages = pages.filter( (p) => !modules.typeCheck.checkAll(p, current) ).sort()
	
	
	return <dc.Table columns={COLUMNS} rows={pages} paging={20}/>;  
}  

const { dateTime } = await cJS()
 
return function View() {
	const file = dc.useCurrentFile();
	return <p class="dv-modified">Created {dateTime.getCreated(file)}     ֍     Last Modified {dateTime.getLastMod(file)}</p>
}