Recent Articles

Construction of the Null Set

Making nothing out of something

Let  be a set with at least two members, and  be the collection of its nonempty subsets. Let , the collection of all ordered pairs of nonempty subsets of . Define an equivalence relation on  satisfying  if and only if   and  have no common members and  and  have no common members, or . Define  to be the collection of equivalence classes induced by . Denote by […]

The J Transform

 where  is the Mellin transform.[1] , where  if , else . so if  converges uniformly to  on , then  converges uniformly to  on .

The Euler Infinite Product Representation of the Factorial

I have no idea how Euler thought of it, but I did figure out how to prove it.

To solve the problem of extending the factorial to non-integer arguments, Leonhard Euler gave a representation of the factorial as an infinite product. In this knol we give a derivation of this product.

The Erdős-Straus Conjecture

The conjecture states that for every n>1, there exist positive integers a, b and c satisfying 4/n=1/a+1/b+1/c.
In this Knol we give verifying examples for n<127.

An Interesting Function

Let  be any enumeration of the rational numbers. Define the function . Then  has the following properties:  is strictly increasing, and thus has a derivative almost everywhere.  for   for  If we define , then we have for    and

A Definition of a Geometrical Point

Consider a space of any dimension. For convenience I will define a point in three dimensions. We take for granted that the concept of ball of positive radius is understood. Now consider a nest of balls. This is simply any nonempty collection of balls  with the property that for any  and , we have either  or . We define a […]

Semigroup cleanup on aisle 5

The mathematical definition of a semigroup is, “an algebraic structure consisting of a nonempty set S together with an associative binary operation.” (Wikipedia). If you are not sufficiently put off by that definition, and still curious as to what a semigroup really is, here goes. An easy example of a semigroup can be found in […]

The Difference Between {} and {{}}

To the mathematically naïve, these two entities can cause some perplexity. In this knol we show the significant differences between these two mathematical objects.

The Cardinality of A^A Not Greater Than That of 2^(2^A)

INTRODUCTION Two sets  and  are said to be disjoint if , i.e. they have no members in common. If  and  are sets, the meanings of  and  are identical. However,  means the same thing as , i.e. the set consisting of the sets  and .  means the same thing as , i.e. the set of members of  which are not members of .  means the collection of mappings of  into .  is shorthand for .  means […]