Mathematical content on Apronus.com is presented in Math ASCII Notation
which can be properly displayed by all Web browsers because it uses only
the basic set of characters found on all keyboards and in all fonts.
The purpose of these pages is to demonstrate the power of the Math ASCII Notation.
In principle, it can be used to write mathematical content of any complexity.
In practice, its limits can be seen when trying to write complicated formulas
(containing, for example, variables with many indexes or multiple integrals).
Despite its limitations the Math ASCII Notation has much expressive power,
as can be seen from browsing through these pages.
Define the taxi norm on |R^n and prove that it is a norm.
x = (x,x,...,x[n])
||x|| = +(k=1 to k=n) |x[k]|
What does it mean that two norms are equivalent?
Let W be a vector space. Let f,g be two norms on W.
\/A>0 \/B>0 /\x:-W A*f(x) <= g(x) <= B*f(x)
It is an equivalence relation.
Let a,b be real numbers.
sqrt(a*a + b*b) <= ???
sqrt(a*a + b*b) <= |a| + |b|
Let D be a connected subset of |C.
Let f,g:D->|C be continuous.
Suppose that /\z:-D exp(f(z)) = exp(g(x)).
What can we say about functions f and g?