phimuemue

Since Emacs needs quite an amount of time to start, I decided to give vim a try and am quite happy with it. However, I missed a feature that I appreciated since I came to know it in AucTeX: Showing and browsing the table of contents. So, I implemented a basic table of contents myself.
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I came across the following identity recently: . This can be proved in various ways. Here, I will describe a “combinatorial proof”.
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I described how to encode tuples of natural numbers into natural numbers. Now, I wrote (hacked) a small piece in Haskell to try it in practice.
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Here’s a 290-characters interpreter for the brainf*** programming language I wrote.
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A while ago, a friend of mine told me that he once knew a number with the following property: “You can prepend any number before that number and will always obtain a prime number.”

Of course, he was just kidding: Such a number can’t exist, because a number is divisible by 3 if and only if its digit sum is divisible by 3. So, it is no problem to find a number to prepend such the digit sum of the resulting number is divisible by 3. That implies that the resulting number is not a prime.

However, it’s possible to prove a similar theorem…
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We consider a propositional logic formula in conjunctive normal form, containing clauses, where each clause has exactly k (distinct) variables.

We prove that this formula is satisfiable.
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I have always been interested in how to encode certain things into simple natural numbers. This post shows a way to encode sets and tuples of natural numbers, that I personally find quite nice.
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In predicate logic, skolemization is a central method to manipulate logic formulas. Since we had to do some homework on those formulas, and I always mixed up the parentheses, I decided to have it done by the computer.
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The following is a very simple unification algorithm in Haskell.
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Consider the following problem: You are celebrating a festival and want to invite guests. The only problem is the following: You want to distribute all guests in a single row, but you can only place to guests next to each other if they get along with each other. The question is if there’s an efficient (meaning poly-time) algorithm computing such an order of guests.
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