Mathematical Preprints
Generalized Vector Product
Additionally the terms "covariant" and "contravariant" will be explained.
A simple Date-Class for C++
With this class it is possible to compute the difference of two Date-Intances (in days).
The source is here:
"datetime.tar.gz".
A Suggestion
I want to modernize the vote counting for a democratically election
here in Germany.
There is a program in C:
"vote.tar.gz".
Several New Antiderivatives
The development of the Corona Calculator leads to an idea.
Corona Calculator
With this tool for MS EXCEL you can forecast the THEORETICAL evolution of a virus.
I cannot exclude mistakes.
(At this moment, no one can laugh about such a mistake.)
But I have the right of free speech.
Your feedback is welcome.
The theory can be found in
"CoronaCalculator.pdf".
A Striking Proof
I have got an incomplete proof. I believe, it is impossible for logical reasons.
I am really desperate.
I can proof the thesis with another method without any problem (it is very simple).
A Modification Of Euler's Polygonal Method
There are functions, which are hard to compute by the known Euler's Polygonal Method.
A corresponding FORTRAN90 source code for LINUX can be found in
"Euler.tar.gz".
Why is the torsion tensor T = 0?
It is difficult, to find a concrete affine and to the metric compatible covariant connection of a Riemannian manifold, that holds T <> 0.
I found out, why.
A New Bijection N → NxN
The proof is simple. The question is wether this bijection is known in literature or not.
A better Understanding of curl(V) = 0
In this document you can find an alternative approach to curl(V) = 0
(or rot(V) = 0 or ∇xV = 0).
The description of the result gets along without curl, rot,
grad or differential forms.
Problems with "canonical"
I have problems to grep the term "canonical" in a mathematical exact
manner. This problems are discussed in this paper.
One proof is omitted, because it is not necessary for understanding.
You can find it
here,
if you are interested.
An example:
Of course I know, what the "canonical" base of RxR is, but I cannot define the term ""canonical" base of RxR" in a mathematical exact manner.
A Way Out of the Halting-Problem
(result already known)
To say it explicitly:
The halting-problem is NOT DECIDABLE on Turing-Maschines.
This is only a solution of a problem RELATED to the halting-problem.
You can find the program-template in
"custos.zip".
The End of ZFC
There is a contradiction in the axioms of ZFC. For a further illustration
of the result you can find a mind theory
here.
The Geometry of Cartan's Derivation
Here you can find a new invariant description of Cartan's derivation
- or exterior derivation -
of a differential form.
News about Tensors and Projections
This document contains an extension of the results of
"The Geometry of Cartan's Derivation".
An Antiderivative of x^(α - 1)e^(-x) on R+
(α > 0)
The function x^(α - 1)e^(-x) is used at the computation of the Γ-function.
The idea for the construction of the antiderivative is a generalization of the
exponential series via the Γ-function. The result is a expansion in a series
of an antiderivative of x^(α - 1)e^(-x). This is not known to
Bronstein-Semendjajew, "Taschenbuch der Mathematik",
and similar to antiderivative Nr. 450 in this book.
With this antiderivative it is easy to get
an antiderivative of x^(α - 1)e^(-βx) (β > 0).
I want to present a new kind of power series.
The new kind of power series arises from a classical power series
through a transformation by a positive real number α and
through the use of the Γ-function.
Simple consequences:
Radius of Convergence
for the new kind of power series
Antiderivatives of x^(α - 1)sin(βx) and x^(α - 1)cos(βx) on R+
(α > 0, β ≠ 0)
an example for the use of the new kind of power series
Antiderivatives of x^(α - 1)sinh(βx) and x^(α - 1)cosh(βx) on R+
(α > 0, β ≠ 0)
a second example for the use of the new kind of power series
Antiderviatives of x^(α - 1)e^(βx) on R+
(α > 0, β > 0)
a third example for the use of the new kind of power series
Problems with the definition of the so called "Boolean Algebra"
I can define A "Boolean Algebra",
but I cannot define THE "Boolean Algebra".
An Antiderivative of x^(α - 1)/(x - 1) on ]0;1[
(α > 0)
Here is a another example for the use of the new kind of power series.
With the same method you can get an antiderivative of x^(α - 1)/(x + 1) on ]0;1[.
Standard Power Series / New Power Series
Here is the connection between the standard power series and the new power series.
News about the Binomial Formula
I thought a long time about this.
The proof is trivial, but I did not see it anywhere.
So I must try.
If you do know the result, please let me know.
Conclusion On Antiderivatives
Here my present results on antiderivatives are summarized.
I finally obtained 3 interesting equations.
The related proofs:
Sourcecode (PASCAL) for a stable Quicksort
Quicksort is a well-known efficient algorithm for sorting data
(s. Niklaus Wirth, "Algorithmen und Datenstrukturen").
Though it is not stable.
To make it stable, you have the choice between a substantially more
memory requirement or a substantially more time requirement.
I hope, my algorithms contain no errors anymore.
An Error In Delphi 5?
Just compile "error.pas" with "dcc32.exe" and execute the result.
You can't always win Solitaire
This ZIP-File contains a counterexample with proof.
You can't always win FreeCell
This ZIP-File contains a counterexample with proof.
Σ(i)?
I thought a long time about this.
The proof is trivial, but I did not see it anywhere.
So I must try.
If you do know the result, please let me know.