Five-Value Theorem of Nevanlinna

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, f(z) and g(z), are said to share the value a if f(z) - a = 0 and g(z) - a = 0 have the same solutions (zeros).

More precisely, suppose f(z) and g(z) are meromorphic functions and a_1, a_2, \ldots, a_5 are five distinct values. If

\displaystyle{      E(a_i,f) = E(a_i,g), \qquad 1\le i\le 5,  }

where

\displaystyle{      E(a,h) = \left\{ z | h(z) = a \right\},  }

then f(z) \equiv g(z).

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 e^z and e^{-z} as examples, one sees that these two meromorphic functions share the four values a_1=0, a_2=1, a_3=-1, a_4=\infty but are not equal. So sharing four values is not enough.

There is also a four-value theorem of Nevanlinna. If two meromorphic functions, f(z) and g(z), share four values counting multiplicities, then f(z) is a Möbius transformation of g(z).

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.,

  1. Gerhard Jank, Lutz Volkmann: Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen
  2. Lee A. Rubel, James Colliander: Entire and Meromorphic Functions
  3. 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:

  1. 246A, Notes 0: the complex numbers
  2. 246A, Notes 1: complex differentiation
  3. 246A, Notes 2: complex integration
  4. Math 246A, Notes 3: Cauchy’s theorem and its consequences
  5. Math 246A, Notes 4: singularities of holomorphic functions
  6. 246A, Notes 5: conformal mapping, covers Picard’s great theorem
  7. 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
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Why does deep and cheap learning work so well?

Very interesting.

the morning paper

Why does deep and cheap learning work so well Lin & Tegmark 2016

Deep learning works remarkably well, and has helped dramatically improve the state-of-the-art in areas ranging from speech recognition, translation, and visual object recognition to drug discovery, genomics, and automatic game playing. However, it is still not fully understood why deep learning works so well.

So begins a fascinating paper looking at connections between machine learning and the laws of physics – showing us how properties of the real world help to make many machine learning tasks much more tractable than they otherwise would be, and giving us insights into why depth is important in networks. It’s a paper I enjoyed reading, but my abilities stop at appreciating the form and outline of the authors’ arguments – for the proofs and finer details I refer you to the full paper.

A paradox

How do neural networks with comparatively…

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Methods of Proof — Diagonalization

Very clear presentation on the uncountability of the real numbers, and the halting problem.

Further keywords: Cantor, natural numbers, real numbers, diagonalization, bijection, Turing halting problem, proof by contradiction.

Math ∩ Programming

A while back we featured a post about why learning mathematics can be hard for programmers, and I claimed a major issue was not understanding the basic methods of proof (the lingua franca between intuition and rigorous mathematics). I boiled these down to the “basic four,” direct implication, contrapositive, contradiction, and induction. But in mathematics there is an ever growing supply of proof methods. There are books written about the “probabilistic method,” and I recently went to a lecture where the “linear algebra method” was displayed. There has been recent talk of a “quantum method” for proving theorems unrelated to quantum mechanics, and many more.

So in continuing our series of methods of proof, we’ll move up to some of the more advanced methods of proof. And in keeping with the spirit of the series, we’ll spend most of our time discussing the structural form of the proofs…

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On Differential Forms

Abstract. This article will give a very simple definition of k-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 k-forms vanishes.

MSC 2010: 58A10

1. Basic definitions.

We denote the submatrix of A=(a_{ij})\in R^{m\times n} consisting of the rows i_1,\ldots,i_k and the columns j_1,\ldots,j_k with

\displaystyle{          [A]{\textstyle\!{\scriptstyle j_1\atop\scriptstyle i_1}          \!{\scriptstyle\ldots\atop\scriptstyle\ldots}          \!{\scriptstyle j_k\atop\scriptstyle i_k}} := \begin{pmatrix}                  a_{i_1j_1} & \ldots & a_{i_1j_k}\\                  \vdots     & \ddots & \vdots\\                  a_{i_kj_1} & \ldots & a_{i_kj_k}\\          \end{pmatrix}  }

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Announcement: 11th International Conference on Parallel Programming and Applied Mathematics

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:

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Day 2, Workshop Programming of Heterogeneous Systems in Physics

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.

  1. 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,
  2. Prof. Dr. Rainer Heintzmann, Jena, CudaMat – a toolbox for Cuda computations. Continue reading

Day 1, Workshop Programming of Heterogeneous Systems in Physics

As announced in Workshop Programming of Heterogeneous Systems in Physics, July 2014, I attended this two-day conference. Below are speakers and pictures with my personal notes.

  1. Dipl.-Ing. Hans Pabst from Intel, Programming for the future: scaling forward with cores and vectors. Hans Pabst Continue reading