# 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.

Let F be a s-algebra in X.
(1) f : X -> |R*.
(2) n:-|N
(3) a[1], ..., a[n] :- |R
(4) A[1], ..., A[n] :- F , some of them may be empty
(5) X = \\*//(k=1 to n) A[k]
(6) /\x:-X f(x) = +(k=1 to n) a[k]*1(A[k])(x)
f is simple and F-measurable
page 109 in 2nd measure
Let F be a s-algebra in X.
Let f : X -> [0,oo] be F-measurable.
In this setting, how can we interestingly approximate function f ?
(In this item, prove the existence of this interesting approximation.)
There exists a sequence of functions { s[n] }(n:-|N) such that
(1) /\n:-|N s[n] : X -> [0,oo) is simple and measurable
(2) /\x:-X /\n:-|N s[n](x) <= s[n+1](x)
(3) /\x:-X lim(n->oo) s[n](x) = f(x)
page 113 in 2nd measure
f : X -> [0,oo]
In this broad setting, how can we interestingly approximate function f ?
(In this item, do not prove the existence of this interesting approximation.)
There exists a sequence of functions { s[n] }(n:-|N) such that
(1) /\n:-|N s[n] : X -> [0,oo) is simple
(2) /\x:-X /\n:-|N s[n](x) <= s[n+1](x)
(3) /\x:-X lim(n->oo) s[n](x) = f(x)
page 110 in 2nd measure
Let f : X -> [0,M] , M < oo.
In this broad setting, how can we interestingly approximate function f ?
There exists a sequence of functions { s[n] }(n:-|N) such that
(1) /\n:-|N s[n] : X -> [0,oo) is simple and bounded
(2) /\x:-X /\n:-|N s[n](x) <= s[n+1](x)
(3) /\x:-X lim(n->oo) s[n](x) = f(x)
(4) sequence s[n] converges uniformly to function f
page 113 in 2nd measure
======================
lim(x,y)->(0,0) y*(x-sin(x)) / (x^4 + y^2) = ???
lim(x,y)->(0,0) y*(x-sin(x)) / (x^4 + y^2) = 0
hint:
x-sin(x) = x^3 * u(x), where lim(x->0) u(x) = 1/6
|xy/(x^2+y^2)|<=1/2
page 100 in OLDTIMER
Let a : |N -> |C.
Let z be a complex number.
Suppose that series a[n]*|z|^n converges.
Can we conclude that series a[n]*z^n converges?
NO
(1) Let a[n] = 1/n * (-1)^n. Let z = (-1).
(2) Consider series (-1)^n / n * z^(3n). Let z = -1.
(1) W is a ring of sets in X
(2) y : W -> [0,oo) is additive
(3) A[1] , ... , A[n] :- W (disjoint sets)
(5) B[1] , ... , B[m] :- W (disjoint sets)
(6) X = U(k=1 to n)_A[k] = U(j=1 to m)_B[j]
(7) a[1], ... , a[n] , b[1] , ... , b[m] are positive real numbers
+(k=1 to n) a[k]*y(A[k]) = +(j=1 to m) b[j]*y(B[j])
page 147 in 1st measure
Let W be a s-algebra in X.
Let y : W -> [0,oo] be additive, y(O) = 0.
Let f : X -> [0,oo) be a simple and W-measurable function.
Define the Lebesgue integral of function f with respect to y.
LebS(X,f,dy) = ???
Since f is simple and W-measurable, there exist:
(1) disjoint sets A[1], A[2], ..., A[n] :- W, with X = U(k=1 to n) A[k]
(2) a[1], a[2], ..., a[n] :- [0,oo)
such that (5) /\x:-X f(x) = +(k=1 to n) a[k]*1(A[k])
Define LebS(X,f,dy) := +(k=1 to n) a[k]*y(A[k]).
Let W be a s-algebra in X.
Let y : W -> |R* be additive.
Let A :- W.
LebS(X,1(A),dy) = ???
LebS(X,1(A),dy) = y(A)
page 149 in 1st measure
Let W be a s-algebra in X.
Let f,g : X -> |R be simple and W-measurable.
Prove that there exist:
(1) E[1], ... , E[n] :- W (disjoint, having union X)
(2) a[1], ... , a[n], b[1], ... , b[n] :- |R
such that
(3) f = +(k=1 to n) a[k]*1(E[k])
page 149 in 1st measure
Notice the technical difficulty here.
One day I want to go deeper into this theorem.
MRW 2001.02.24; 2001.06.02