Group presentation is a method to express groups as a quotient of a free group and a normal subgroup. In a presentation, there are generators and relators, and the presentation yields a ‘minimalist implementation’ of these generators and relations. Some cases of special interests are finitely generated/related/presented groups. From the definition, finite generation + finite relation is not linguistically equivalent to finite presentation. There has been some discussion on the relation of the three properties for several other algebraic structures. This entry proves that finite generation + finite relation imples finite presentation for groups.
Blog entries with tag “Maths”
What goes wrong when arrows are reversed in the diagrams for direct product/sum universal properties?
I’m taking an algebra course and regaining my algebra sense (non-sense?). Familiar to us is that direct products [resp. sums] of groups [resp. abelian groups] possess their versions of universal properties. When reviewing these things, I went curious on what would go wrong if the arrows were reversed. Let’s try and see.
Base conversion is the conversion between notations of number, and NOT the numbers themselves. I discuss a common mistake on methods of base conversion, and a common mistake in thinking about ‘base conversion’ in computer programs.
Lagrange’s four-square theorem is a beautiful result in number theory. However, to the best of my knowledge, I haven’t encountered it often in theoretical computer science. Today’s entry discusses two interesting stories related to this theorem.
Starting from the classic puzzles of finding patterns in a sequence, this entry explores the framework to define persuasive patterns. The framework is found to be quite self-contained in the sense that it is ‘asymptotically invariant’ to the choice of ‘language of expression’. The only short-coming of this framework is that it works only for computable sequences, yet the process of pattern discovery is incomputable.
In this entry, I discuss a pathological construction of general terms of a class of integer-valued sequences with the widely accepted concept of ‘elementary functions’. The idea here is to simply ‘concatenate’ the integers in a real number, then extract the appropriate digits for each term. It turns out that this definition characterises the elementariness very well (note that the definition is also self-referencing). The extended inspection leaves a problem open: Are all integer-valued sequences elementary? Update: The question is solved with an affirmative answer.
In this entry, I discuss a kind of widely posed exam problems for elementary calculus learners. It concerns a technique of application of (differential) mean value theorems. Speficially, a class of such problems can be solved dogmatically if the equation to prove resembles a ‘factorisable’ linear differential equation.
Let me explain the classic idea that ‘the algorithm of a crypto system should be considered and made public’ to you. Particularly, I will discuss what it means by ‘crypto system’, which tells us what information should be no secret at all.
A practical problem I encountered in 2016, which I insisted on implementing with pure CSS as long as it was possible. At first it was solved with mathematical logic tricks but later it turned out there was a much simpler solution.