Blog entries with tag “Maths

Rethinking base ‘conversion’

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.

Is the pattern you found persuasive?

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.

Is the pattern you found persuasive?

Of elementary general terms

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.

Mean value theorem and heuristics from differential equation

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.

Mean value theorem and heuristics from differential equation



鸡贼地 bound 一个数列

我发现当我的生活充实的时候就有有趣的 blog 可以写!因为一忙起来,就要忙里偷闲,然后就会胡思乱想,然后就可能有有意思的东西。(已更新)

Of Kerckhoffs’ principle

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.