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 0 < b < a.

(a-b)/a < ??? < (a-b)/b

(a-b)/a < ??? < (a-b)/b

(a-b)/a < log(a/b) < (a-b)/b

Use Lagrange's Mean Value theorem.

page 153 in golden gate

Use Lagrange's Mean Value theorem.

page 153 in golden gate

Let X be a set, and J c P(X).

Let J be a semi-ring. Prove that every finite union of sets from J can be written as a disjoint finite union of sets from J.

Let J be a semi-ring. Prove that every finite union of sets from J can be written as a disjoint finite union of sets from J.

page 63 in 1st measure

page 66 in 2nd measure

page 66 in 2nd measure

Let X be a set, and J c P(X). Let J be a semi-ring.

Let H denote the set of all disjoint finite unions of sets from J.

What are the two steps that lead to the conclusion that R(J) = H?

Let H denote the set of all disjoint finite unions of sets from J.

What are the two steps that lead to the conclusion that R(J) = H?

1) R(J) = Js

2) Js = H

page 65 in 1st measure

2) Js = H

page 65 in 1st measure

Express differently:

( U(t:-T) A[t] ) x B = ???

( U(t:-T) A[t] ) x B = ???

( U(t:-T) A[t] ) x B = U(t:-T) ( A[t] x B )

Express differently:

U(t:-T) ( A[t] x B ) = ???

U(t:-T) ( A[t] x B ) = ???

U(t:-T) ( A[t] x B ) = ( U(t:-T) A[t] ) x B

Express differently:

A x U(t:-T) B[t] = ???

A x U(t:-T) B[t] = ???

A x U(t:-T) B[t] = U(t:-T) A x B[t]

Express differently:

U(t:-T) A x B[t] = ???

U(t:-T) A x B[t] = ???

U(t:-T) A x B[t] = A x U(t:-T) B[t]

Express differently:

A x //\\(t:-T) B[t] = ???

A x //\\(t:-T) B[t] = ???

A x //\\(t:-T) B[t] = //\\(t:-T) A x B[t]

Express differently:

//\\(t:-T) A x B[t] = ???

//\\(t:-T) A x B[t] = ???

//\\(t:-T) A x B[t] = A x //\\(t:-T) B[t]

Express differently:

( //\\(t:-T) A[t] ) x B = ???

( //\\(t:-T) A[t] ) x B = ???

( //\\(t:-T) A[t] ) x B = //\\(t:-T) ( A[t] x B )