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.

f:[0,1]->|R, f(x)=x*sqrt(1+x)
Calculate Integral([0,1],f)
4*(sqrt(2)+1) / 15
hint1: g:[1,sqrt(2)]->|R, g(t)=t*t-1
hint2: t=sqrt(1+x)
g : |R -> |C
h : |C -> |C
h(z) = g(Re(z))
Suppose that g is diffable. Does h have to be diffable?
NO.
Choose g(x)=x.
Then h(z)=Re(z), and h is nowhere diffable!
Let A be a connected subset of |R.
Let g : A -> |R.
Suppose that g is diffable and /\x:-A g'(x)=0.
What can we say about function g?
It is constant.
Use Lagrange's mean value theorem.
Let X be a metric space.
Let x[n] be a sequence in X.
Let B = \\//(k=1 to oo) //\\(n=k to oo) B(x[n], 1/n).
Suppose that B is not empty.
What can we conclude?
sequence x[n] converges
Let X be a set, and let W c P(X).
Define W-convergence in X.
Let (T,>) be a directed set. Let x:T->X be a net in X. Let k:-X.
The net x W-converges to k
iff
/\B:-W [ k:-B ==> \/t0:-T /\t>t0 x(t):-B ]
page 44 in the General Topology notebook
Let S be a set. Give a collection of sets, W c P(|R^S), such that pointwise convergence of functions S->|R is equivalent to W-convergence.
If s:-S, and Uc|R is open, define U(s) = {f:-|R^S : f(s):-U}.
Let W = {U(s): s:-S and Uc|R open}.
page 44 in the General Topology notebook
Let X be a set, and W1,W2 c P(X).
Prove that if W1 c W2, then W2-convergence implies W1- convergence.
page 45 in gen top
Let X be a set, and W c P(X). Let d be a metric on X.
Define what it means that W is compatible with d.
W-convergence is the same as d-convergence
page 46 in the Gen Top notebook
Let X be a set, and W c P(X), A c X, p:-X.
Define "p is W-adherent to A".
\/(T,>)directed set \/a:T->A [ net a W-converges to p ]
Notice the formal difficulty here.
page 46 in the Gen Top notebook
"p is W-adherent to A" =
= "there exists a net contained in A that converges to p"
Let X be a set, and W c P(X), A c X.
Define "the boundary of A" in the language of generalized convergence.
the set of all points which are W-adherent to A and W-adherent to X\A
page 46 in the Gen Top notebook
Let Fr(A) denote the boundary of A.