A specialized mathematician: Julius Hurwitz, and an application …oswald/julius.pdf · 2016-05-24 · Return to continued fractions. An application from a modern point of view. In

$Page 1: A specialized mathematician: Julius Hurwitz, and an application …oswald/julius.pdf · 2016-05-24 · Return to continued fractions. An application from a modern point of view. In$
Würzburg

N. Oswald

A specialized mathematician: Julius

Hurwitz, and an application of his

complex continued fraction.

Nicola Oswald

July 23, 2014

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What happened on July 13, 1895?Who was J. Hurwitz?

”A special kind of con-tinued fraction expansionof complex numbers.”

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What happened on July 13, 1895?Who was J. Hurwitz?

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What happened on 13 July 1895?The detours before.

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Complex Continued FractionsIn general...

The continued fraction expansion of a complex number

z = a0 +1

a1 +1

...+1

an +1

...

∈ C

is based on partial quotients ai ∈ C, i = 0, ..., n.

The algebraic structure of the partial quotients influencesarithmetical characteristics of the continued fraction.

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Complex Continued FractionsJulius vs. Adolf Hurwitz

(Ueber die Entwicklung Complexer Größen in

Kettenbrüche, 1887)

A. Hurwitz consideredthe set of the Gaussianintegers

Z[i ] = Z+ iZ

as possible partial quoti-ents.

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Complex Continued FractionsJulius vs. Adolf Hurwitz

(Ueber eine besondere Art der Kettenbruch-Entwicklung

Complexer Größen, 1895)

J. Hurwitz considered thesubset

(1 + i)Z[i ]

= (1 + i)Z+ (1 − i)Z

of the Gaussian integersZ[i ] as possible partialquotients.

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Julius’ Complex Continued FractionsA first example of the arithmetical differences.

Let ∆ be the volume of the fundamental domainand let I be set of possible partial quotient.Then we know for the approximation quality:

NO 2013

For any z ∈ C \ Q(I ) there exist infinitely many p, q ∈ I suchthat

∣

∣

∣

∣

z − p

q

∣

∣

∣

∣

<

√6∆

π|q|2 .

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Childhood in HildesheimFamily.

Salomon Hurwitz and Sons(1813 - 1885)

(D.E. Rowe; Felix Klein, Adolf Hurwitz, and the ”Jewish

question” in German academia, Math. Intelligencer, 2007)

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Childhood in HildesheimFamily.

Salomon Hurwitz and Sons(1813 - 1885)

(D.E. Rowe; Felix Klein, Adolf Hurwitz, and the ”Jewish

question” in German academia, Math. Intelligencer, 2007)

”Moreover he set a

high value that the

young boys started

smoking since he

could not imagine a

proper man without

cigar [...] ”

(Biographical dossier of Ida

Hurwitz, Archive ETH Zürich.)

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Childhood in HildesheimEducation.

School certificate,April 07, 1870

(Municipal archive of Hildesheim)

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School certificate,April 07, 1870

(Municipal archive of Hildesheim)

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Once [Julius] H. visited(unothaurized) a tavern. No otherimpertinent behaviour is noted.

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”All of the three brothers showed a particular

talent for mathematics, which was documented at a

very early stage [...] concerning Adolf.”

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Childhood in HildesheimEducation: Schubert.

Hermann Caesar HannibalSchubert

(1848 - 1911)13 / 45

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Childhood in HildesheimEducation: Schubert.

Hermann Caesar HannibalSchubert

(1848 - 1911)

Dissertation in Halle, 1870(Archive of the University of Halle)

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Childhood in HildesheimEducation: The end.

”Schubert even visited the father to convince him

to let both sons choose the studies of mathematics

[...]. [Salomon] talked to a very wealthy and

childless friend E. Edwards, who offered to bear

the costs of the studies for one of the sons.

Dr. Schubert elected Adolf.”

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Julius’ detours begin.Some documents about Julius in Nordhausen ...

Julius became banker inNordhausen, Thuringia.

(Salomon to Julius, ETH archive)

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Julius’ detours begin.... in Hamburg and Hanover.

”[...] the second brother, Julius, lives in Hamburg working as anexporter.”(Adolf Hurwitz to Luigi Bianchi, October 30, 1885.)

”Since his uncle Adolph’s death Max and Julius

were owning the bank ’Adolph M. Wertheimer’s

Nachf.’ in Hanover, however, they felt

uncomfortable in this business. [...]”

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A professorship changes Julius’ life.1884.

Salomon to Julius Hurwitz in Hanover on April 1, 1884

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A professorship changes Julius’ life.1884.

”It is an extraordinary event, and we cannot thank enough thedestiny, that our Adolf is so gifted, so acclaimed and is alreadyrecognized by the most important mathematicians as excellentperson. [...]So your youngest brother is professor by the age of 25 and afteronly two years of being private lecturer [in Göttingen]!”(Salomon to Julius, April 1, 1884, ETH library.)

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Entrance to an academic career.Quakenbrück.

”[...] Hence, first Julius quitted in order to

return to school at the age of 33 years and

finished with the school examination [...]”

Excerpts of the Abitur certificate of Julius Hurwitz(Archive of the Univerisity of Halle.)

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Entrance to an academic career.Königsberg.

”[...] Afterwards, in 1890, at the age of 33, he

finally followed his brother to Königsberg.”

”Warmest greetings to you and Hurwitz major and minor natu.”(H. Minkowski to D. Hilbert, February 9, 1892.)

(Archive of the ETH.)

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Working on the dissertation.Subject from his brother.

In 1892 Adolf Hurwitz was offered the Georg Frobenius’ chair atthe Eidgenössische Polytechnische Hochschule Zürich, apolytechnic, and Hermann Amandus Schwarz’s chair at theUniversity of Göttingen. Adolf Hurwitz chose Zurich.

”[Also] his brother Julius followed him soon to

Zurich, where he wrote his doctoral thesis for

which he had received the subject from his

brother.”

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Working on the dissertation.Subject from his brother: my guess!

(Mathematical diaries of A. Hurwitz, no.5., 1886, Archive ETH.)

Theorem A. Hurwitz (1886)

The root of a quadratic (irreducibel) equation with complexinteger coefficients has a periodic continued fraction expansion.

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An assumption about Julius’ task: Transferring the theorem tothe case that the set of possible partial quotients is restricted.

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Julius as a doctoral student.University of Halle.

New University of Halle, 1836(Lithography of W. Breye.)

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Working on the dissertation.Help from the brother.

(Mathematical diaries of A. Hurwitz, no.9., November 04 1894, Archive ETH.)

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(Mathematical diaries of A. Hurwitz, no.9., November 04 1894, Archive ETH.)

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(Excerpt of the dissertation of J. Hurwitz.)

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Finally: the dissertation in June 1895.Who was the real advisor?

”Eight days ago the news frommy brother came that he passedthe doctoral exam very good.The dissertation concerns con-tinued fraction expansions ofcomplex numbers. I determinedthe subject”

A. Hurwitz to D. Hilbert, June 19, 1895(Collected letters from Adolf Hurwitz, Archive of the Mathematical Institute in Göttingen.)

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Julius’ time in Halle.A plausible answer: Wangerin.

Albert Wangerin(1844 - 1933)

Wangerin advised the remar-kable number of 53 studentsto their dissertation, JuliusHurwitz being number 28 ofthem.The supervised topics ran-ge from calculus, in particular,differential equations, via ana-lytic and differential geometryto mathematical physics; thereare only two theses from num-ber theory.

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Return to continued fractions.An application from a modern point of view.

In 1985 the Japanese mathematician Shigeru Tanaka published(nearly) the same complex continued fraction considering acompletely different point of view.

Our task.

On behalf of methods from ergodic theory we want to deriveinformation of the approximation characteristics of JuliusHurwitz’s respectively Shigeru Tanaka’s complex continuedfraction algorithm.

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Basics from ergodic theory.Some definitions on the time-process.

Our ingredients are a probablility space (X ,Σ, µ,T ) withnon-empty set X , a σ-algebra Σ on the set X , a probabilitymeasure µ on (X ,Σ) and a measure preserving transformationT : X → X , i.e.

µ(T−1E ) = µ(E )

for E ∈ Σ.

Notice! Here, we model a time-process and assume T as ’shiftinto the future’ whereas T−1 can be considered as ’shift intothe past’.

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Complex Continued FractionsOn behalf of a transformation.

We consider the fundamental domain

X := {z = (1 + i)x + (1 − i)y :−1

2≤ x , y ≤ 1

2}

and the transformation

T : X → X , Tz :=1

z−

[

1

z

]

T

,

with [ . ]T : C → (1 + i)Z[i ],

[z]T :=

[

x +1

2

]

(1 + i) +

[

y +1

2

]

(1 − i).

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Complex Continued FractionsOn behalf of a transformation.

Setting

an :=

[

1

T n−1z

]

T

∈ (1 + i)Z[i ]

the transformation T serves as operator in Tanaka’s complexcontinued fraction. We receive

z = [0; a1a2, . . . , an + T nz].

Here, T can be considered as a shift n the sequence of partialquotients.

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Basics from ergodic theory.What is/ how helps ergodicity?

This transformation T is ergodic which means that for eachµ-measurable, T -invariant set E either

µ(E ) = 0 or µ(E ) = 1.

Birkhoff, 1931

Let T be a measure preserving, ergodic transformation on aprobablitiy space (X ,Σ, µ). If f is integrable, then

limN→∞

1

N

∑

0≤n≤N

f (T nx) =

∫

X

fdµ

for almost all x ∈ X .

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Tools from ergodic theory.Dual transformation.

We define a dual transformation S : Y → Y by

Sw =1

w−

[

1

w

]

S

for w 6= 0.

WithY = {w ∈ C : |w | ≤ 1}

and [w ]S = a ∈ (1 + i)Z[i ] its convergents rnsn

satisfy

qn−1

qn= [an, an−1, . . . , a1] =

rnsn

=: Vn.

Tanaka, Duality

The sequence of partial quotients a1, . . . , an is T -admissible ifand only if the reverse sequence of partial quotients an, . . . , a1

is S-admissible.

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Tools from ergodic theory.Natural extension.

We construct a natural extension R containing information ofT as well as information of S . Thus, R includes ’future’ and’past’ of the sequence of partial quotients. We defineR : X × Y → X × Y by

R(z ,w) =

(

Tz ,1

a1(z) + w

)

,

where a1 =[

1

z

]

T.

How does Rn(z ,w) look like?

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Tools from ergodic theory.Natural extension.

Tanaka proved

Natural Extension

R is a natural extension; in particular T and S are ergodic andthe function h : X × Y → R defined by

h(z ,w) =1

|1 + zw |4

is the density function of an absolutely continuous R-invariantmeasure.

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Approximation Quality.Approximation Coefficients.

We examineθ′n := |qn|2|z − pn

qn|,

an analogue to the so called approximation coefficients fromreal theory. Tanaka stated the equation

θ′n|qn|2

=

∣

∣

∣

∣

z − pn

qn

∣

∣

∣

∣

=|2i(−1)nT nz |

|qn(qn + qn−1T nz)| .

Hence,

ϕn =θ′n2

=|T nz |

|1 + VnT nz | ≤1

(1 − |VnT nz |) . (1)

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Application of ergodic theory.Döblin-Lenstra-Conjecture.

Our aim is to transfer the Döblin-Lenstra-Conjecture to thecomplex case:

Complex Conjecture

The limiting distribution function

l(g) := limn→∞

1

n|{j ; j ≤ n, ϕj (z) ≤ g}|

exists.

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Döblin-Lenstra-Conjecture.Applying ergodic methods.

For approximation coefficients, we know

ϕj(z) ≤ g ⇔∣

∣

∣

∣

T jz

1 + VjT jz

∣

∣

∣

∣

≤ g , (2)

and

R(z , y) =

(

Tz ,1

a1 + y

)

,

where a1 := [ 1z]T is the first partial quotient of z . We have

R i(z , y) =(

T iz , [0; ai , ai−1, . . . , a2, a1 + y ])

, and in particular

R i (z , 0) =(

T iz ,Vi

)

.

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Döblin-Lenstra-Conjecture.Applying ergodic methods.

Because of (2), this leads to a new equivalence:We have ϕi (z) ≤ g if and only if

R i (z , 0) ∈ Ag := {(u, v) ∈ X × Y :

∣

∣

∣

∣

u

1 + uv

∣

∣

∣

∣

≤ g},

respectively

R i(z , 0) ∈ Ag = {(u, v) ∈ X × Y :

∣

∣

∣

∣

1

u+ v

∣

∣

∣

∣

≥ 1

g}.

As shown by Tanaka, there exists a probability measure µ withdensity function h(z , y) = 1

|1+zy |4, such that the quadrupel

(X × Y ,Σ, µ,R) forms an ergodic system with Ag ∈ Σ.

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Döblin-Lenstra-Conjecture.First Statement.

Applying the Ergodic Theorem of Birkhof leads to

Theorem 1

For almost all z the distribution function exists and is given by

l(g) = limn→∞

1

n

∑

i≤n

1Ag (Ri (z , 0)) = µ(Ag ) =

1

G

∫ ∫

Ag

dλ(u, v)

|1 + uv |4 ,

where G :=∫ ∫

X×Y

dλ(u,v)|1+uv |4

.

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Döblin-Lenstra-Conjecture.Explicit Calculation.

Just some remarks about the explicit calculation ofG :=

∫ ∫

X×Y

dλ(u,v)|1+uv |4

:

We simplify |1 + uv |2 = (1 + uv)(1 + uv ).

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∫ ∫

X×Y

dλ(u,v)|1+uv |4

:


The formula 1

(1−x)2=

∑

m≥1mxm−1, being valid for

|x | < 1, is used forward and backward.

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∫ ∫

X×Y

dλ(u,v)|1+uv |4

:


The formula 1

(1−x)2=

∑



To prevent difficulties from singularities, we firstly considersubdomains Xr and Yr inheriting the symmetric structureof X and Y .

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∫ ∫

X×Y

dλ(u,v)|1+uv |4

:


The formula 1

(1−x)2=

∑



To prevent difficulties from singularities, we firstly considersubdomains Xr and Yr inheriting the symmetric structureof X and Y .

We split X into four domains of equal sizes along the axesand ficiliate the calculation of the integral over X .

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Finally, we receive

Theorem 2

The normalizing constant G is given by

G = 4π

∞∑

n=0

(−1)n

(2n + 1)2,

where∑ (−1)n

(2n+1)2is known as Catalan constant.

Remark: The explicit calculation of∫ ∫

Agis still in its infancy.

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Thank you...... for yout attention.

Thank you for your attention!

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Announcement.... excursion and barbecue!

This afternoon, there will be an excursion and in the evning wewill have BBQ at the Schönstattzentrum!

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Documents

A specialized mathematician: Julius Hurwitz, and an application …oswald/julius.pdf · 2016-05-24 · Return to continued fractions. An application from a modern point of view. In