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.

A[n], B[n] are sequences of subsets of X. A,B c X.
Given that lim A[n] = A and lim B[n] = B
show that
lim ( A[n] \ B[n] ) = A \ B.
Use:
(1) lim B[n] = B ==> lim X\B[n] = X\B
(2) lim A[n] = A and lim B[n] = B ==> lim ( A[n] n B[n] ) = A n B
A[n], B[n] are sequences of subsets of X. A,B c X.
Given that lim A[n] = A and lim B[n] = B
show that
lim ( A[n] + B[n] ) = A + B.
Use:
(1) lim B[n] = B ==> lim X\B[n] = X\B
(2) lim A[n] = A and lim B[n] = B ==> lim ( A[n] n B[n] ) = A n B
What does it mean that a complex function satisfies the Cauchy- Riemann equations at a point in its domain?
z:-A ; A c |C ; A is open ; f:A->|C ; U = Re(f) ; V = Im(f)
(1) all four partial derivatives exist at z: Ux, Uy, Vx, Vy
(2) Ux = Vy and Uy = -Vx
page 18 in OLDTIMER
Suppose that a complex function is defined on an open set and satisfies the Cauchy-Riemann equations at a point. Does it have to be diffable at that point?
f(x,y) = sqrt( |x*y| ) at (0,0)
C-R satisfied, not diffable
lim (x,y)->(0,0) [ x*y / sqrt( x*x + y*y) ] = ?
= 0
What is separate continuity?
Let (X,d) , (Y,g) be two metric spaces.
Consider f:X*Y->|R.
(1) /\y:-Y f is continuous as a function of x:-X
(2) /\x:-X f is continuous as a function of y:-Y
(1)and(2) <=> separate continuity
(X,d) is a metric space; a:-X; f:X->|R; f(x)=d(x,a).
Show that f is Lipschitz.
Use the second triangle inequality.
What is a directed set?
(X,>)
X is a set and > is a binary relation on X such that
(1) /\a,b:-X \/ c:-X [ c>a and c>b ]
(2) /\a,b,c:-X [ a>b and b>c ==> a>c ]
What is a net?
h is a net contained in X
means that
there is a directed set (T,>) (see the previous item)
h:T->X
Let T be the set of all finite subsets of |N.
For a,b:-T, a>b <=> b is contained in a
Prove that (T,>) is a directed set.
Very easy.