When storing big text contents in a database, it is better to store after compressing the data. The following is an example storing HTML contents in a database cache. We first create a table mapping URLs to HTML contents. The…

When storing big text contents in a database, it is better to store after compressing the data. The following is an example storing HTML contents in a database cache. We first create a table mapping URLs to HTML contents. The…

PHP provides str_pad() to add a string to a certain length with another string. Another way is to use sprintf() with formatting patterns. Pad leading zeros to get exactly 3 digits:

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Let \(S\) be a family of subsets of \(\{1,\ldots,n\}\) such that no set in \(S\) contains another. This is often called an antichain. Theorem. [Sperner's Theorem] \[ |S| \leq {n \choose \lfloor n/2 \rfloor}. \] The proof of Sperner’s Theorem…

Kolmogorov complexity measures the complexity of describing an object, such as a binary string. The description is considered as a program which can be run on a computer. By Church’s thesis, all non-trivial computational models are equivalent; thus, one can…