Lars Ahlfors

Lars Ahlfors

From Wikipedia, the free encyclopedia

Lars Ahlfors

Lars Ahlfors - MFO.jpg

Lars Ahlfors

Born 18 April 1907

Helsinki, Finland

Died 11 October 1996 (aged 89)

Pittsfield, Massachusetts, U.S.A.

Nationality Finnish

Fields Mathematics

Institutions University of Helsinki

ETH Zurich

Harvard University

Alma mater University of Helsinki

Doctoral advisor Ernst Lindelöf

Rolf Nevanlinna

Doctoral students Paul Garabedian

James A. Jenkins

Robert Osserman

Henry Pollak

Halsey Royden

George Springer

Eoin Whitney

Known for Riemann surfaces

Notable awards Fields Medal (1936)

Wihuri Prize (1968)

Wolf Prize in Mathematics (1981)

Lars Valerian Ahlfors (18 April 1907 – 11 October 1996) was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his text on complex analysis.

Contents [hide]

1 Background

2 Career

3 Personal life

4 See also

5 Bibliography

6 References

7 External links

Background[edit]

Ahlfors was born in Helsinki, Finland. His mother, Sievä Helander, died at his birth. His father, Axel Ahlfors, was a professor of engineering at the Helsinki University of Technology. The Ahlfors family was Swedish-speaking, so he first attended a private school where all classes were taught in Swedish. Ahlfors studied at University of Helsinki from 1924, graduating in 1928 having studied under Ernst Lindelöf and Rolf Nevanlinna. He assisted Nevanlinna in 1929 with his work on Denjoy's conjecture on the number of asymptotic values of an entire function. In 1929 Ahlfors published the first proof of this conjecture, now known as the Denjoy–Carleman–Ahlfors theorem. It states that the number of asymptotic values approached by an entire function of order ρ along curves in the complex plane going toward infinity is less than or equal to 2ρ.

He completed his doctorate from the University of Helsinki in 1930.

Career[edit]

Ahlfors worked as an associate professor at the University of Helsinki from 1933 to 1936. In 1936 he was one of the first two people to be awarded the Fields Medal. In 1935 Ahlfors visited Harvard University. He returned to Finland in 1938 to take up a professorship at the University of Helsinki. The outbreak of war led to problems although Ahlfors was unfit for military service. He was offered a post at the Swiss Federal Institute of Technology at Zurich in 1944 and finally managed to travel there in March 1945. He did not enjoy his time in Switzerland, so in 1946 he jumped at a chance to leave, returning to work at Harvard where he remained until he retired in 1977; he was William Caspar Graustein Professor of Mathematics from 1964. Ahlfors was a visiting scholar at the Institute for Advanced Study in 1962 and again in 1966.[1] He was awarded the Wihuri Prize in 1968 and the Wolf Prize in Mathematics in 1981.

His book Complex Analysis (1953) is the classic text on the subject and is almost certainly referenced in any more recent text which makes heavy use of complex analysis. Ahlfors wrote several other significant books, including Riemann surfaces (1960)[2] and Conformal invariants (1973). He made decisive contributions to meromorphic curves, value distribution theory, Riemann surfaces, conformal geometry, quasiconformal mappings and other areas during his career.

Personal life[edit]

In 1933, he married Erna Lehnert, an Austrian who with her parents had first settled in Sweden and then in Finland. The couple had three daughters.

See also[edit]

Ahlfors finiteness theorem

Ahlfors measure conjecture

Bibliography[edit]

Complex Analysis: an Introduction to the Theory of Analytic Functions of One Complex Variable (1953,[3] 1966, 1979) (ISBN 0-07-000657-1)

Contributions to the Theory of Riemann Surfaces: Annals of Mathematics Studies (1953) (ISBN 0-691-07939-0)

Ahlfors, Lars (February 1966). "FUNDAMENTAL POLYHEDRONS AND LIMIT POINT SETS OF KLEINIAN GROUPS". Proceedings of the National Academy of Sciences 55 (2): 251–254. doi:10.1073/pnas.55.2.251. PMC 224131. PMID 16591331.

References[edit]

Jump up ^ Institute for Advanced Study: A Community of Scholars

Jump up ^ Springer, George. "Review of Riemann surfaces. By Lars V. Ahlfors and Leo Sario". Bull. Amer. Math. Soc. 67 (2): 170–171.

Jump up ^ Schaeffer, A. C. (1953). "Review: Complex analysis. By Lars V. Ahlfors". Bull. Amer. Math. Soc. 59 (5): 464–467.

External links[edit]

Lars Ahlfors at the Mathematics Genealogy Project

Ahlfors entry on Harvard University Mathematics department web site.

Papers of Lars Valerian Ahlfors : an inventory (Harvard University Archives)

Lars Valerian Ahlfors The MacTutor History of Mathematics page about Ahlfors

The Mathematics of Lars Valerian Ahlfors, Notices of the American Mathematical Society; vol. 45, no. 2 (February 1998).

Lars Valerian Ahlfors (1907–1996), Notices of the American Mathematical Society; vol. 45, no. 2 (February 1998).

Frederick Gehring (2005). "Lars Valerian Ahlfors: a biographical memoir". Biographical Memoirs 87.

National Academy of Sciences Biographical Memoir

http://en.wikipedia.org/wiki/Lars_Ahlfors

documentclass[12pt]{article}

\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}

\numberwithin{equation}{section}

\begin{document}

\title{\bf Announcement 185　: The importance of the division by zero $z/0=0$}

\author{{\it Institute of Reproducing Kernels}\\

Kawauchi-cho, 5-1648-16,\\

Kiryu 376-0041, Japan\\

E-mail: kbdmm360@yahoo.co.jp\\}

\date{\today}

\maketitle

{\bf Abstract: } In this announcement, we shall state the importance of the division by zero $z/0=0$. The result is a definite one and it is fundamental in mathematics.

\bigskip

{\bf Introduction}

\bigskip

%\label{sect1}

By {\bf a natural extension of the fractions}

\begin{equation}

\frac{b}{a}

\end{equation}

for any complex numbers $a$ and $b$, we, recently, found the surprising result, for any complex number $b$

\begin{equation}

\frac{b}{0}=0,

\end{equation}

incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices, and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the case of real numbers. The result is a very special case for general fractional functions in \cite{cs}.　

The division by zero has a long and mysterious story over the world (see, for example, google site with division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,

Sin-Ei, Takahasi (\cite{taka}) (see also \cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing some full extensions of fractions and by showing the complete characterization for the property (1.2). His result will show that our mathematics says that the result (1.2) should be accepted as a natural one:

\bigskip

{\bf Proposition. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that

$$

F (b, a)F (c, d)= F (bc, ad)

$$

for all

$$

a, b, c, d \in {\bf C }

$$

and

$$

F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.

$$

Then, we obtain, for any $b \in {\bf C } $

$$

F (b, 0) = 0.

$$}

\medskip

Furthermore, note that Hiroshi Michiwaki gave the important interpretation of the division by zero $z/0=0$ by the intuitive meaning of the division, independently of the concept of the product (see \cite{ann}).

We shall state the importance of the division by zero $z/0=0$.

\bigskip

\section{}

On AD 628, the zero was appeared in India, and the zero division $z/0=0$ was discovered on Feburary 2, 2014, definitely with the clear definition and motievation. The uniquess and the natural interpretation were given in \cite{taka, ttk,kmsy} and \cite{ann}, respectively. Several physical interpretations of the division by zero were given in \cite{kmsy}.

\bigskip

\section{}

By the introduction of the division by zero $z/0=0$, four arithmetic operations; that is,

addition, subtraction, multiplication, and division are always possible; note that for division, we were not able to divide by zero. There was one exceptional case for the division by zero.

\section{}

For the Euclidean (B.C. 3 Centuary ) geometry, two non-Euclidean geometries were appered about 2 hundred years ago, and in particular, in the elliptic type non-Euclidean geometry, the point at infinity was introduced by the stereoprojection of the Euclidean plane to the sphere and the concept is a standard one in complex analysis around over one hundered years. And then we have considered as $1/0= \infty$ (\cite{ahlfors}). However, surprisingly enough, the division by zero means that $1/0=0$.

\section{}

We shall recall the fundamental law by Newton:

\begin{equation}

F = G\frac{m_1 m_2}{r^2}

\end{equation}

for two masses $m_1, m_2$ with a distance $r$ and a constant $G$. Of course,

\begin{equation}

\lim_{r \to +0} F =\infty,

\end{equation}

however, we obtain the important interpretation:

\begin{equation}

F = 0 = G \frac{m_1 m_2}{0}.

\end{equation}

Of course, here, we can consider the above interpretation for the mathematical formula (4.1) as the new interpretation (4.3). We can find many physical formulas with the division by zero.

\section{}

In complex analysis, linear fractional functions

$$

W = \frac{az + b}{cz + d}, \quad ad -bc \ne 0,

$$

map the extended complex plane onto the extended complex plane containing the point at infinity, one to one, conformally, beautifully. This beautiful property is changed as the beautiful formula that linear fractional functions map the whole complex plane onto the whole complex plane, one to one, however, at one point of the singular point, the linear fractional functions have strong discontinuity.

The division by zero excludes the infinity from the numbers.

\section{}

We did, essentially, not consider the division by zero, and so the property of the division by zero; that is, at the isolated singular points of analytic functions, to consider the analytic functions are new mathematics and new research topics, essentially.

\section{}

The impact to complex analysis is unclear, we, however, obtain a typical new theorem:

\medskip

{\bf Theorem :} {\it Any analytic function takes a definite value at an isolated singular point }{\bf with a natural meaning.} The definite value is given by the first coefficient of the regular part in the Laurent expansion around the isolated singular point.

\medskip

This will be the fundamental theorem on the division by zero in Complex Analysis and we have many applications for the Sato hyperfunction theory, generating functions theory and singular integral theory (\cite{mst}).

\section{}

In particular, the divison by zero gives new interpretations on the finite part of Hadamard

for singular integrals and the Cauchy's principal values. The division by zero will represent discontinuity properties on the universe.

\section{}

Even for middle high shool students, the division by zero may be accepted as the beautiful result with great pleasures:

For the elementary function

$$

y = f(x) = \frac{1}{x},　

$$

we have $f(0) = 0$; that is, $1/0=0$.　

\section{}

We can introduce the division by zero $100/0=0,0/0=0$ with the simple and natural definition for the division by the Hiroshi Michiwachi method (\cite{ann}) in the elementary school. The division by zero will request the change of all the related books and scientific books.

\section{Conclusion}

The division by zero $b/0=0$ is possible and the result is naturally determined, uniquely.

The result does not contradict with the present mathematics - however, in complex analysis, we need only to change a little presentation for the pole; not essentially, because we did not consider the division by zero, essentially.

The common understanding that the division by zero is impossible should be changed with many text books and mathematical science books. The definition of the fractions may be introduced by {\it the method of Michiwaki} in the elementary school, even.

Should we teach the beautiful fact, widely?:

For the elementary graph of the fundamental function

$$

y = f(x) = \frac{1}{x},

$$

$$

f(0) = 0.

$$

The result is applicable widely and will give a new understanding for the universe ({\bf Announcement 166}).

\medskip

If the division by zero $b/0=0$ is not introduced, then it seems that mathematics is incomplete in a sense, and by the intoduction of the division by zero, mathematics will become complete in a sense and perfectly beautiful.

\bigskip

\section{Remarks}

For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:

\medskip

{\bf Announcement 148} (2014.2.12):　 $100/0=0, 0/0=0$ 　--　 by a natural extension of fractions -- A wish of the God

\medskip

{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea

\medskip

{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?

\medskip

{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth

\medskip

{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.

\medskip

{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero

\medskip

{\bf Announcement 171} (2014.7.30): The meanings of product and division　--　The division by zero is trivial from the own sense of the division independently of the concept of product

\medskip

{\bf Announcement 176} (2014.8.9):　 Should be changed the education of the division by zero

\medskip

{\bf Announcement 179} (2014.10.22):　 Division by zero is clear as z/0=0 and it is fundamental in mathematics

\bigskip

\bibliographystyle{plain}

\begin{thebibliography}{10}

\bibitem{ahlfors}

L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.

\bibitem{cs}

L. P. Castro and S.Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.

\bibitem{kmsy}

S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,

An interpretation of the division by zero z/0=0 without the concept of product

(note).

\bibitem{kmsy}

M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,

New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,

Int. J. Appl. Math. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.

\bibitem{mst}

H. Michiwaki, S. Saitoh, and M. Takagi,

A new concept for the point at infinity and the division by zero z/0=0

(note).

\bibitem{s}

S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95.http://www.scirp.org/journal/ALAMT/

\bibitem{taka}

S.-E. Takahasi,

{On the identities $100/0=0$ and $ 0/0=0$}

(note).

\bibitem{ttk}

S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operators on the real and complex fields, Tokyo Journal of Mathematics (in press).

\bibitem{ann}

Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics,

Institute of Reproducing Kernels, 2014.10.22.

\end{thebibliography}

\end{document}