Math ASCII Notation Demo

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.

Give the function which shows that the Cantor set is uncountable.
I = [0,1]. If J = [a,b], let J(0) = [a , a+(b-a)/3], let J(1) = [b-(b-a)/3 , b].
If AcN and n:-N, let I(A(n)) = I(1A(1), 1A(2), ..., 1A(n)).
If n:-N then let C[n] = U{I(A(n)) : AcN}. Let's define C = //\\n:N [ C[n] ].
For AcN, let F(A) be the one point belonging to //\\n:-N I(A(n)).
Give a description of the Cantor set using series.
1) +(n=1 to oo) [ 2 * 1A(n) * 3^(-n) ]
For every Ac|N, the sum of the series above
uniquely determines an element of the Cantor set.
2) +(n=1 to oo) [ a[n] * 3^(-n) ]
Where a[n] is any sequence such that /\n:-|N [ a[n] :- {0,2} ].
Consider the metric space (P(|N),d), d(A,B) = 1/min(A+B).
Let {A[n]} be a sequence contained in P(|N).
(1) lim d(A[n],A) = 0
(2) lim A[n] = A (the lower and upper limits are both equal to A)
Show that (1) <=> (2).
page 169 in the palace notebook
Show that every function f:X->|R is the limit of some uniformly convergent sequence of functions f[n]:X->|R.
f[n] = 1/n * [ n*f(x) ]
[r] denotes the integer part of r:-|R.
r-1 <= [r] <= r
Can a countable dense subset of |R be written as a countable intersection of open sets?
NO.
Argue by contradiction.
Use the Baire Category Theorem.
page 48 in OLDTIMER
Let M be a countable dense subset of |R. Does there a exist a sequence of continuous functions on |R which converges pointwise to the characteristic function of M?
NO.
Let there be such a sequence {f[m]}. Put U[m] = f[m]-1 ((1/2,oo)). U[m] is the inverse image of (1/2,oo) under f[m]. U[m] is open because f[m] is continuous. Notice that
M = //\\n:-N \\//(m=n to m=oo) U[m].
M is written as the intersection of countably many open subsets of R. This is impossible. Contradiction. (see the previous item)
Let (X,d1) and (X,d2) be metric spaces.
What does it mean that d1 and d2 are similar metrics?
/\x:-X /\{x[n]}cX [ lim d1(x[n],x)=0 <=> lim d2(x[n],x)=0 ]
Let z be a complex number.
Express differently:
|z|*|z| = ?
|z|*|z| = z*conjugate(z)
Let X = (0,oo). Define d(x,y) = |x-y| / (x*y).
Show that (X,d) is a metric space.
d(x,y) = |1/x - 1/y|
page 173 in palace
Let X = (0,oo). Define d(x,y) = |x-y| / (x*y). (X,d) is a metric space.
Is is complete?
NO.
{n} is a Cauchy sequence. It does not converge.