In German known as Fünf-Punkte-Satz. This theorem is astounding. It says: If two meromorphic functions share five values ignoring multiplicity, then both functions are equal. Two functions, and , are said to share the value if and have the same solutions (zeros).
More precisely, suppose and are meromorphic functions and are five distinct values. If
For a generalization see Some generalizations of Nevanlinna’s five-value theorem. Above statement has been reproduced from this paper.
The identity theorem makes assumption on values in the codomain and concludes that the functions are identical. The five-value theorem makes assumptions on values in the domain of the functions in question.
Taking and as examples, one sees that these two meromorphic functions share the four values but are not equal. So sharing four values is not enough.
There is also a four-value theorem of Nevanlinna. If two meromorphic functions, and , share four values counting multiplicities, then is a Möbius transformation of .
According Frank and Hua: We simply say “2 CM + 2 IM implies 4 CM”. So far it is still not known whether “1 CM + 3 IM implies 4 CM”; CM meaning counting multiplicities, IM meaning ignoring multiplicities.
For a full proof there are books which are unfortunately paywall protected, e.g.,
- Gerhard Jank, Lutz Volkmann: Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen
- Lee A. Rubel, James Colliander: Entire and Meromorphic Functions
- Chung-Chun Yang, Hong-Xun Yi: Uniqueness Theory of Meromorphic Functions, five-value theorem proved in §3
For an introduction to complex analysis, see for example Terry Tao:
- 246A, Notes 0: the complex numbers
- 246A, Notes 1: complex differentiation
- 246A, Notes 2: complex integration
- Math 246A, Notes 3: Cauchy’s theorem and its consequences
- Math 246A, Notes 4: singularities of holomorphic functions
- 246A, Notes 5: conformal mapping, covers Picard’s great theorem
- 254A, Supplement 2: A little bit of complex and Fourier analysis, proves Poisson-Jensen formula for the logarithm of a meromorphic function in relation to its zeros within a disk
Abstract. This article will give a very simple definition of -forms or differential forms. It just requires basic knowledge about matrices and determinants. Furthermore a very simple proof will be given for the proposition that the double outer differentiation of -forms vanishes.
MSC 2010: 58A10
1. Basic definitions.
We denote the submatrix of consisting of the rows and the columns with
The conference will start in September 6-9, 2015, Krakow, Poland.
Parallel Programming and Applied Mathematics, PPAM for short, is a biennial conference started in 1994, with the proceedings published by Springer in the Lecture Notes in Computer Sciences series, see PPAM. It is sponsored by IBM, Intel, Springer, AMD, RogueWave, and HP. The last conference had a fee of 420 EUR.
It is held in conjunction with 6th Workshop on Language based Parallel Programming.
Prominent speakers are:
Day 2 of the conference had below talks. Prof. Dr. Bernd Brügmann gave a short introduction. He pointed out that Jena is number 10 in Physics in Germany, has ca. 100.000 inhabitants, and 20.000 students.
- Dr. Karl Rupp, Vienna, Lessons Learned in Developing the Linear Algebra Library ViennaCL. Notes: C++ operator overloading normally uses temporary, special trickery necessary to circumvent this, ViennaCL not callable from Fortran due to C++/operator overloading, eigen.tuxfamily.org, Karl Rupp’s slides, with CUDA 5+6 OpenCL and CUDA are more or less on par,
- Prof. Dr. Rainer Heintzmann, Jena, CudaMat – a toolbox for Cuda computations. Continue reading
As announced in Workshop Programming of Heterogeneous Systems in Physics, July 2014, I attended this two-day conference in Jena, Germany. Below are speakers and pictures with my personal notes.
- Dipl.-Ing. Hans Pabst from Intel, Programming for the future: scaling forward with cores and vectors. Continue reading