computing infimum by an example

Sat, 21 Sep 2024 15:25:30 +0000

Let us consider a problem where you are asked to find infimum of the set

\[\{\int_0^{1}\sqrt{1+f'(x)^2}dx\}_{f\in S}\]

where \(S\) is the set of all \(f\in C^1(\mathbb{R})\) with the property that \(f(0)=10\) and \(f(1)=0\).

When we see integral and differential together, that should remind us the famous fundamental theorem of calculus, which says that

limit/limsup/liminf of a sequence (by an example)

Fri, 13 Sep 2024 05:29:07 +0000

Let us check for limit/limsup/liminf of the sequence \(\frac{n}{10^{\lceil \log_{10}n \rceil}}\), where the notation \(\lceil x \rceil\) means the smallest integer greater than or equal to \(x\).

For example, \(\lceil 0.1 \rceil=1, \lceil 0.9 \rceil=1, \lceil -1.2 \rceil=-1, \lceil -2.5 \rceil=-2\)

To compute limit (to have a hope of computing), we need to know it converge (which we can check by checking it is Cauchy sequence).

Analysis on Geometry and some category theory

computing infimum by an example

limit/limsup/liminf of a sequence (by an example)