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.

(C,|x-y|) is the Cantor set with the Euclidean metric.

(P(N),d) : d(A,B) = 1/min(A+B).

Show that (P(N),d) and (C,|x-y|) are homeomorphic metric spaces.

(P(N),d) : d(A,B) = 1/min(A+B).

Show that (P(N),d) and (C,|x-y|) are homeomorphic metric spaces.

Show these two:

(1) d(A,B) < 1/m ==> |F(A) - F(B)| <= 3^(-m)

(2) |F(A) - F(B)| < 3^(-m) ==> d(A,B) < 1/m

F is the homeomorphism.

It's defined in the previous item.

page 73 in the palace notebook

(1) d(A,B) < 1/m ==> |F(A) - F(B)| <= 3^(-m)

(2) |F(A) - F(B)| < 3^(-m) ==> d(A,B) < 1/m

F is the homeomorphism.

It's defined in the previous item.

page 73 in the palace notebook

Let a[n] be a decreasing sequence converging to zero.

Show that if |z|=1 and z!=1, then the series a[n]*z^n converges.

Show that if |z|=1 and z!=1, then the series a[n]*z^n converges.

Show that the series z^n is bounded.

Then use the Dirichlet test.

Then use the Dirichlet test.

series (-1)^n / log(n) * z^(3*n-1)

Investigate convergence on |z| = 1.

Investigate convergence on |z| = 1.

For |z|=1,

z*z*z != -1 <=> the series converges

z^3 = -1 <=> the series diverges

z*z*z != -1 <=> the series converges

z^3 = -1 <=> the series diverges

Given a natural number k, show a power series that converges on
its circle except at k points.

series 1/n * z^(k*n)

Let G be an open subset of the complex plane.

Let f : G -> |C , z:-G.

What is the necessary condition for f being diffable at z?

State the condition without proof.

Let f : G -> |C , z:-G.

What is the necessary condition for f being diffable at z?

State the condition without proof.

There exist four limits Ux, Uy, Vx, Vy with

Ux = Vy

Uy = -Vx

See page 18 in OLDTIMER for details.

Ux = Vy

Uy = -Vx

See page 18 in OLDTIMER for details.

G is an open subset of the complex plane. f:G->C. Give a
sufficient condition for f being continuously diffable on G.

1) f has continuous partial derivatives on G

2) Ux(z) = Vy(z) and Uy(z) = -Vx(z), for all z:-G

page 21 in OLDTIMER

2) Ux(z) = Vy(z) and Uy(z) = -Vx(z), for all z:-G

page 21 in OLDTIMER

f(z) = |z|, for all complex z

Show that f is nowhere diffable.

Show that f is nowhere diffable.

hint: the Cauchy-Riemann equations are violated except at (0,0)

Investigate the limit

lim (x+y)/(x*x+y*y) as (x,y) approaches (oo,oo)

lim (x+y)/(x*x+y*y) as (x,y) approaches (oo,oo)

= 0

hint: (x,y) = t * exp(i*a)

hint: (x,y) = t * exp(i*a)

What is a monotone class of sets?

M is a monotone class of sets

iff

1) the union of every increasing sequence of sets from M belongs to M

2) the intersection of every decreasing sequence of sets from M belongs to M

iff

1) the union of every increasing sequence of sets from M belongs to M

2) the intersection of every decreasing sequence of sets from M belongs to M

{ E c |R^n : E is convex }

Is this class of sets monotone?

Is this class of sets monotone?

Yes.

The intersection of convex sets is a convex set.

The union of an increasing sequence of convex sets is convex.

The intersection of convex sets is a convex set.

The union of an increasing sequence of convex sets is convex.