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:|R->|R be continuous.
Does there exist a diffable function F:|R->|R
such that /\x:-|R F'(x) = f(x) ?
YES.
F(x) = Integral( 0, x, f(t), dt )
page 92 in OLDTIMER
Let f : |R -> |R be continuous. Let a:-|R.
Let g : |R -> |R be defined g(x) = Integral( a, x, f(t), dt ).
What can we conclude from this?
/\x:-|R g'(x) = f(x)
page 92 in OLDTIMER
Let a,b:-|R and a<b.
Let f : ]a,b[ -> |R be continuous and bounded.
Does there exist a diffable function g : ]a,b[ -> |R
such that g'(x) = f(x) for a<x<b ?
YES
page 92 in OLDTIMER
(1) a,b:-|R, a!=b
(2) A,B :- |R, A<B
(3) g : *[a,b]* -> |R continuously diffable
(4) f : [A,B] -> continuous
(5) g(*[a,b]*) c [A,B]
What can we conclude from this?
Integral( g(a), g(b), f(x), dx ) = Integral( a, b, f(g(x))*g'(x), dx )
page 93 in OLDTIMER
(1) a,b :- |R, a < b
(2) g : [a,b] -> |R continuously diffable
(3) f : |R -> |R continuous
(4) g(a) = g(b)
Integral( a, b, f(g(x))*g'(x), dx ) = ???
Integral( a, b, f(g(x))*g'(x), dx ) = 0
page 94 in OLDTIMER
Let a[1], a[2], a[3], ..., a[n], a[n+1], ..., a[2n] be numbers.
+(k=1 to k=2n) a[k] = +(k=1 to k=n) ???
+(k=1 to k=2n) a[k] = +(k=1 to k=n) a[2k-1] + a[2k]
Consider the p-dimensional Euclidean space |R^p.
Let W be the collection of all intervals in |R^p.
In the context of measure theory, let m be the "volume" function on W.
Complete this important theorem:
/\A:-W /\E>0 \/F:-W F is closed and ??????????????
/\A:-W /\E>0 \/F:-W F is closed and FcA and m(A) <= m(F) + E.
page 106 in 1st measure
Consider the p-dimensional Euclidean space |R^p.
Let W be the collection of all intervals in |R^p.
In the context of measure theory, let m be the "volume" function on W.
Complete this important theorem:
/\A:-W /\E>0 \/H:-W ???????????? and m(A) <= m(H) + E.
/\A:-W /\E>0 \/H:-W H is closed and HcA and m(A) <= m(H) + E.
page 106 in 1st measure
Let K be a s-algebra in X. Let f,g : X -> |R* be K-measurable.
Can we conclude that function max(f,g) is K-measurable?
YES: max(f,g) is K-measurable
Take any a:-|R. Since f,g are K-measurable, {x : f(x)>a}:-K and {x : g(x)>a}:-K. Since K is an algebra, {x : f(x)>a or g(x)>a}:- K. Hence {x : max(f(x),g(x))>a}:-K. We showed that max(f,g) is K-measurable.
Let K be a s-algebra in X. Let f,g : X -> |R* be K-measurable.
Can we conclude that function min(f,g) is K-measurable?
YES: min(f,g) is K-measurable
Take any a:-|R. Since f,g are K-measurable, {x : f(x)<a}:-K and {x : g(x)<a}:-K. Since K is an algebra, {x : f(x)<a or g(x)<a}:- K. Hence {x : min(f(x),g(x))<a}:-K. We showed that min(f,g) is K-measurable.