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.

Contemplate the fact that from every strictly monotonic function f:R->R we can construct a metric on R which is similar to the Euclidean metric.
d(x,y) = |f(x)-f(y)|
It's a metric because f is injective.
It's similar to the Euclidean metric because f is continuous and so is its inverse.
0 <= a <= x
0 <= b <= x
Contemplate the fact that |b-a| <= x.
_____
Prove that for a,b >= 0, sqrt(a+b) <= sqrt(a) + sqrt(b).
Square it.
Investigate the uniform convergence of f(n,x) = x/(1+nx), on [0,1].
YES.
hint: 1 + nx >= nx
Investigate the uniform convergence of
f[n](x) = sqrt(n) * x * (1-x*x)^n on [0,1].
No uniform convergence on [0,1].
Use: x[n] = 1 / sqrt(n)
If 0<M<1, then uniform convergence on [M,1].
Investigate the uniform convergence of
the series x / ( (1+nx)*(1+(n+1)x) ).
(x>=0)
It does not converge uniformly on [0,oo).
It converges uniformly on [M,oo) for all M>0.
hint: f[n](x) = 1 / (1+nx); f[n](x)-f[n+1](x)
f(n,x) = x^n / (n + x^n), x>=0.
Investigate the uniform convergence of f.
It does not converge uniformly on [0,oo).
It does not converge uniformly on (1,oo).
It converges uniformly on [0,1].
It converges uniformly on [M,oo) for all M>1.
Given two complex numbers A and B, B != 0, write the equation of a line passing through points A and A+B.
Im( (z-A)/B ) = 0
see page 188 in the palace notebook
What are the three equivalent definitions of a line on the complex plane?
1) Im( (z-a)/b ) = 0, where a,b are complex numbers, b != 0
2) A*Re(z) + B*Im(z) + C = 0, where A,B,C are real numbers, A and B are not both zero
3) |z-A| = |z-B|, where A,B are distinct complex numbers
Prove that for every complex number A,
it is false that lim cos(n*A) = 0.
Use: lim cos(2*n*A) = 0.
page 189 in the palace notebook