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 […]
where is the Mellin transform. , where if , else . so if converges uniformly to on , then converges uniformly to on .
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 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.
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
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 […]
Lorenz’s equations determine x, y and z as functions of time and are used as a classic example of chaotic behavior.
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 […]
To the mathematically naïve, these two entities can cause some perplexity. In this knol we show the significant differences between these two mathematical objects.
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 […]