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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.

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a : |N x |N -> [0,oo]
Consider this line:
+(n=1 to n=oo) +(k=1 to k=oo) a(n,k) = +(k=1 to k=oo) +(n=1 to n=oo) a(n,k)
Suppose that infinity is assumed by one of the values pre
page 59 in 2nd measure
Determine if it is true:
/\x,y:-|R\{0} |x/y + y/x| >= 2
YES.
Investigate
lim (x,y)->(0,0) x*y / (x + x^2 + y^2)
doesn't exist
smart hint: -y^2 = x
1) x[n] = 1/n, y[n] = 1/n, lim = 0
2) x[n] = -1/n^2, y[n] = -1/n, lim = oo
page 68 in OLDTIMER
Let s be a complex number other than -1.
s / (1+s) = ???
s / (1+s) = 1 - 1/(1+s)
Let s be a complex number other than -1.
1 - 1/(1+s) = ???
1 - 1/(1+s) = s / (1+s)
Let y be a nonnegative function on P(X), such that
(1) if A c B c X, then y(A)<=y(B);
(2) if A is a disjoint countable union of sets A[n] c X,
then y(A) <= y(A[1]) + y(A[2]) + ... + y(A[n]) + ...
What further can we conclude about function y?
It is countably subadditive.
page 43 in 2nd measure
Let M,B,A be arbitrary sets.
(A n M) \ (B n M) = ???
(A n M) \ (B n M) = (A \ B) n M
Let W c P(X) be a ring in X.
Let y : W -> [0,oo] be additive.
y(A u B) <= ???
y(A u B) <= y(A) + y(B)
Recall that y(A u B) + y(A n B) = y(A) + y(B).
Notice that y(A n B) >= 0.
Hence y(A u B) <= y(A) + y(B).
page 104 in 1st measure
Let A be a bounded connected subset of |R. Let E>0.
Prove that there exists a closed interval F c A such that d(A) <= d(F) + E.
d(A) and d(F) denote the length of those intervals.
page 105 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.
Let A:-W. Let E>0.
There exists a closed F:-W with FcA and m(A)<=m(F)+E.
How can that be proven?
The proof is tedious.
page 106 in 1st measure

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