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Part I - memories Part II - talk Appendix - after the talk
Dual Padé iterations andsentimental memories
Krystyna Ziętak
Institute of Mathematics and Computer ScienceFaculty of Fundamental Problems of Technology
Wrocław University of Technology
Manchester, April 11, 2013
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Part I - memories Part II - talk Appendix - after the talk
Outline
1 Part I - memoriesImportant people in my lifeConferencesVisitors in WrocławCastle
2 Part II - talkIdea born in Leuvenp-sector function and pth rootDual Padé iterationsResiduals and convergenceSeries expansion of iteratesReferences
3 Appendix - after the talk
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Part I - memories Part II - talk Appendix - after the talk
Important people in my life
Important people in my life
Wrocław – town hall
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Part I - memories Part II - talk Appendix - after the talk
Important people in my life
My first boyfriend (on the left), KZ (on the right)Warsaw 1943/1944
He is 2 months older.Chemist, Polish Academy of Sciences, WarsawVice-President of Polish Academy of Sciences, 2003-2006Honorary Professor of Siberian Branch of RussianAcademy of Sciences
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Part I - memories Part II - talk Appendix - after the talk
Important people in my life
Witold Roter (first row, first from the right)KZ (second row, third from the left) Nowa Ruda 1957
My first teacher of mathematics in primary schoolLater, professor at University of Wrocław
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Part I - memories Part II - talk Appendix - after the talk
Important people in my life
Stefan Paszkowski, Wrocław 1992
Supervisor of my PhD dissertation1962 – he established Cathedra of Numerical Methods,where an English computer Elliott 803 was installedsoftware: for Polish computers produced by Elwrobook: Zastosowania numeryczne wielomianów i szeregówCzebyszewa
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Part I - memories Part II - talk Appendix - after the talk
Important people in my life
Stefan Paszkowski (on the left)Andrzej Kiełbasiński (on the right)
Kiełbasiński (professor at University of Warsaw) hastranslated into Polish Wilkinson’s bookbook: A. Kiełbasiński, H. Schwetlick, NumerycznaAlgebra Liniowa (German and Polish editions)
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Part I - memories Part II - talk Appendix - after the talk
Important people in my life
Warsaw 1975 German edition 1988Polish editions 1992, 1994
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Part I - memories Part II - talk Appendix - after the talk
Important people in my life
Helmut Späth, Oldenburg (1986) wife of Späth and sea (1988)
My first visit at a “western university" (4 days)The second visit (1988) was 2 days longer.
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Part I - memories Part II - talk Appendix - after the talk
Important people in my life
Jacek Cichoń
Professor at Wrocław University of TechnologyHe applies advanced mathematics in computer science.The most popular teacher of informatics in ourdepartment.Financial support of my participation at conferencesdepends on him.
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Part I - memories Part II - talk Appendix - after the talk
Important people in my life
At last my parents, sister and brother
KZ on the right KZ on the right
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Conferences
Wrocław – opera
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Karl-Marx-Stadt (1982) – my first foreign conferenceOberwolfach (1988)my first conference in “western country”invitation by Manfred von Golitschek
G. Meinardus, W. Hayman, F. Deutsch, L. CollatzOberwolfach 1988
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Oberwolfach 1988
William Light and platform
Ward Cheney – Manfred von Golitschek – William Light
approximation in tensor product spaces
AX + YB = C , Chebyshev norm
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Letter and picture: Lothar Collatz, Oberwolfach 1988
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Part I - memories Part II - talk Appendix - after the talk
Conferences
1989 – William Light (Lancaster)Alistair Watson (Dundee Conference)1991 – Manchester (Nick Higham and Kung Kiu Lau)
Householder Symposium, Lake Arrowhead (1993)invitation by Gene Golub
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Householder Symposium 1993
W. Proskurowski D. and Ch. LawsonLos Angeles, Beverly Hills father day in Pasadena
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Northeast Africa and Arabian Peninsula from 500 000 kmby Galileo spacecraft (NASA)
Ch. Lawson – Jet Propulsion Laboratory in Pasadena 1993
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Lake Arrowhead (1993)
Tony Chan, Nick Higham, Valerie Fraysse, Haesun Park
polar decomposition, approximation of matricesspectral portraits – MATLAB
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Charles Van Loan, Sabine Van Huffel Valeria SimonciniLake Arrowhead 1993 Whistler 1999
shared roomsduring Householder Symposiums in Peebles and Whistler
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Whistler (1999) Pontresina (1996)V. Simoncini and B. Parlett Nick Higham and Francoise Tisseur
car in 1993: Gragg, Parlett in Alps
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Whistler (1999)
Walter Gander with wife Heidi Chandler Davis with wifeSwiss chocolates conjectures
son Aaron, concert of Holly Cole in Wrocław
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Part I - memories Part II - talk Appendix - after the talk
Conferences
M. Overton and his mother N. Trefethen, Michele (wife of Overton)“Overton” hotel in Vancouver M. Overton, Z. Strakos
Vancouver 1999 Berlin-Zeuthen 2008
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Nick Trefethen Peebles 2002Zeuthen - Berlin, 2008 photo done by Nick HighamChebyshev polynomials
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Francoise Tisseur and Nick HighamHouseholder Symposium, Berlin-Zeuthen 2008
gift: new book, SIAM 2008the workshop New Directions in Functions of Matrices
Manchester 2008
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Part I - memories Part II - talk Appendix - after the talk
Conferences
J. Liesen, Berlin-Zeuthen 2008 P. Tichy, Valencia 2012
Application of approximation of matrices to GMRES
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Van Dooren, Bhatia Poloni, Meini, IannazzoILAS, Chemnitz 1996 Valencia 2012
matrix analysis some inspirationspositive definite matrices
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Valencia 2012
Jennifer Scott Dopico, Ipsen, Holodnak, MiędlarMackey, Mackey
walks and talks structure preserving
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Leuven 2012
Dario Bini, Daniel Kressner Froilán Dopico, Francoise Tisseur
structured matrices
GAMM in Gdańsk 2009 (minisymposium, Kressner and KZ)
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Part I - memories Part II - talk Appendix - after the talk
Conferences
Boston 2001strict spectral approximation of matrices – conjecture
Gilbert Strang (see his home page)
I wish all of you to have such cupcakes!!!
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Part I - memories Part II - talk Appendix - after the talk
Visitors in Wrocław
Visitors in Wrocław
University of Wrocław – assembly hall
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Part I - memories Part II - talk Appendix - after the talk
Visitors in Wrocław
H. Späth K.-K. LauWrocław - Książ (1987) Wrocław (1992)
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Part I - memories Part II - talk Appendix - after the talk
Visitors in Wrocław
Helena Krupicka, Alistair Watson Alistair and his familyWrocław (1988)
book: Approximation Theory and Numerical Methods
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Part I - memories Part II - talk Appendix - after the talk
Visitors in Wrocław
William Allan Light (Wrocław 1989)
(1950–2002)
gift: book of Cheney and Kincaid
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Part I - memories Part II - talk Appendix - after the talk
Visitors in Wrocław
Dottie and Chuck Lawson, and Poland in 1994
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Part I - memories Part II - talk Appendix - after the talk
Visitors in Wrocław
Ludwig Elsner, Jaroslav Zémanek (2000) Jesse Barlow (2000, 2002)
Wrocław museum of honey near Krynicaroots in ...
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Part I - memories Part II - talk Appendix - after the talk
Visitors in Wrocław
Charles Johnson Ala Smoktunowicz, Miro RozloznikWrocław Novi Sad (2008)
Matrix analysis Gram-Schmidt orthogonalization
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Part I - memories Part II - talk Appendix - after the talk
Visitors in Wrocław
German–Polish Workshop for Young Researches
Będlewo 2006GAMM – Heike Fassbender, Volker Mehrmann, KZ
Volker Mehrmann I.Wróbel, B.Laszkiewicz, A.MiędlarWhistler 1999 Będlewo 2006
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Part I - memories Part II - talk Appendix - after the talk
Castle
Welcome to Książ Castle on rock cliffPearl of Lower Silesia
picture done by P. Kobylański
I would like to thank all of youI have met on my “scientific” way!!!
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Part I - memories Part II - talk Appendix - after the talk
Dual Padé iterations
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Part I - memories Part II - talk Appendix - after the talk
Idea born in Leuven
My questionsWhy some properties of Newton’s and Halley’s methods,shown by Guo (2010) for computing matrix pth root,are common?Why Guo applies similar residuals to Newton’s andHalley’s methods for investigation of convergence?
My answer“Newton” is not Padé iteration, but “Halley” is.However, Newton’s and Halley’s methods are(new) dual Padé iterations.
Question of IannazzoHow to generalize the reciprocal Padé iterationsof Greco-Poloni-Iannazzo (for sign) to p-sector function?
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Part I - memories Part II - talk Appendix - after the talk
Idea born in Leuven
Padé iterations
sign functionKenney, Laub (1991)square rootHigham (1997)square rootHigham, Mackey, Mackey, Tisseur (2004)p-sector function and pth rootLaszkiewicz, Zietak (2009)sign functionreciprocal Padé iterationsGreco-Poloni-Iannazzo (2012)p-sector function and pth rootdual Padé iterationsZietak (2012)
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Part I - memories Part II - talk Appendix - after the talk
p-sector function and pth root
scalar p-sector function
sectp(z) =z
p√zp
the nearest pth root of unity to z
p = 2 sign function
Re(z)
Im(z
)
The sector regions, p=2
−1.6 −1.2 −0.8 −0.4 0.0 0.4 0.8 1.2 1.6 2.0
1.6
1.2
0.8
0.4
0.0
−0.4
−0.8
−1.2
−1.6
−2.0
Φ0 Φ
1
π
Re(z)
Im(z
)
The sector regions, p=3
−1.6 −1.2 −0.8 −0.4 0.0 0.4 0.8 1.2 1.6 2.0
1.6
1.2
0.8
0.4
0.0
−0.4
−0.8
−1.2
−1.6
−2.0
Φ0
Φ1
Φ2
2π/3
Re(z)
Im(z
)
The sector regions, p=4
−1.6 −1.2 −0.8 −0.4 0.0 0.4 0.8 1.2 1.6 2.0
1.6
1.2
0.8
0.4
0.0
−0.4
−0.8
−.12
−1.6
−2.0
Φ0
Φ1
Φ2
Φ3
π/2
sectors
Φk =
{z ∈ C :
2kπp− π
p< arg(z) <
2kπp
+π
p
}
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Part I - memories Part II - talk Appendix - after the talk
p-sector function and pth root
Principal pth root and p-sector function
matrix principal pth root A1/p of Ano eigenvalue of A lies on closed negative real axis
X p = A, λj(X ) ∈ Φ0
matrix p-sector function of nonsingular Aarg(λj(A)) 6= (2q + 1)π/p, q = 0, 1, . . . , p − 1
sectp(A) = A(Ap)−1/p
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Part I - memories Part II - talk Appendix - after the talk
Dual Padé iterations
Padé approximants
Pkm(z)
Qkm(z)=
2F1
(−k, 1
p −m;−k −m; z)
2F1
(−m,−1
p − k;−k −m; z)
[k/m] Padé approximant to f (z) = 1(1−z)1/p
Qkm(z)Pkm(z)
[m/k] Padé approximant
to1
f (z)= (1− z)1/p
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Part I - memories Part II - talk Appendix - after the talk
Dual Padé iterations
Matrix principal pth rootPadé
Xj+1 = XjPkm(I − A−1X p
j )
Qkm(I − A−1X pj ), X0 = I
Halley k = m = 1
dual Padé
Yj+1 = YjQkm(I − AY −p
j )
Pkm(I − AY −pj )
, Y0 = I
Newton k = 0,m = 1, Halley k = m = 1Schröder k = 0,m arbitrary
coupled iterations
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Part I - memories Part II - talk Appendix - after the talk
Dual Padé iterations
Matrix p-sector function
Padé
Xj+1 = XjPkm(I − X p
j )
Qkm(I − X pj ), X0 = A
dual Padé
Yj+1 = YjQkm(I − Y −p
j )
Pkm(I − Y −pj )
, Y0 = A
matrix pure iterations
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Part I - memories Part II - talk Appendix - after the talk
Dual Padé iterations
reciprocal Padé for sign (Poloni-Greco-Iannazzo)
Yj+1 =Qkm(I − Y 2
j )
YjPkm(I − Y 2j ), Y0 = A
dual Padé for sign (p = 2)
Yj+1 =YjQkm(I − Y −2
j )
Pkm(I − Y −2j )
, Y0 = A
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Part I - memories Part II - talk Appendix - after the talk
Residuals and convergence
Residuals for pth root
Padé Xj+1 = XjPkm(I−A−1X p
j )
Qkm(I−A−1X pj ), X0 = I
S` = I − A−1X p`
S`+1 = fkm(S`)
dual Padé Yj+1 = YjQkm(I−AY −p
j )
Pkm(I−AY −pj )
, Y0 = I
R` = I − AY −p`
R`+1 = fkm(R`)
Guo (2010) – R` for Newton and Halley
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Part I - memories Part II - talk Appendix - after the talk
Residuals and convergence
Principal Padé iterations
for [k/k]principal Padé and principal dual Padé iterations
S` = I − A−1X p`
S`+1 = fkk(S`)
R` = I − AX−p`
R`+1 = fkk(R`)
answer to the question
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Part I - memories Part II - talk Appendix - after the talk
Residuals and convergence
Certain regions of convergenceλj(A) in regions:
Lp = {z ∈ C : |1− zp| < 1} Pade for p − sector ,X0 = A
L−p = {z ∈ C : |1−z−p| < 1} dual Pade for p − sector ,X0 = A
K =
{z ∈ C :
∣∣∣∣1− 1z
∣∣∣∣ < 1}
Pade for pth root,X0 = I
M = {z ∈ C : |1− z | < 1 } dual Pade for pth root,X0 = I
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Part I - memories Part II - talk Appendix - after the talk
Residuals and convergence
p = 5, Halley iterations for p-sector function,
L(Pade)−p (solid contour), L(Pade)
p (white flower)B(Hall)
p (dash contour)
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Part I - memories Part II - talk Appendix - after the talk
Residuals and convergence
Crucial function – arbitrary integers k,m
fkm(z) = 1− (1− z)
(Pkm(z)
Qkm(z)
)p
all Taylor coefficients of fkm(z) are positive !!!fkm(1) = 1
Oleksandr Gomilko, Minghua Lin, Dmitry Karp, KZ (2012)
Gomilko, Greco, KZ (2012)
Pkm(z)
Qkm(z)[k/m] Pade approximant
to1
(1− z)1/p
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Part I - memories Part II - talk Appendix - after the talk
Residuals and convergence
Positivity of coefficients of fkm(z)Particular cases:
Guo (2010)
Newton’s method f01(z) k = 0,m = 1
Minghua Lin (2010)
Halley’s method f11(z) k = m = 1
Cardoso and Loureiro (2011)
Schröder’s iterationsf0m(z) k = 0, m arbitrary
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Part I - memories Part II - talk Appendix - after the talk
Series expansion of iterates
Binomial expansion
(1− z)1/p =∞∑j=0
βjz j
dual [k/m] Padé iterate Y` for computing matrix pth root(I − B)1/p
Y` =∞∑j=0
ϕ(`)km,jB
j
ϕ(`)km,j = βj j = 0, . . . , (k + m + 1)` − 1
Guo (2010) – Newton and Halley
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Part I - memories Part II - talk Appendix - after the talk
Series expansion of iterates
Summary
New families of dual Padé iterations for computing matrixpth root and matrix p-sector function have beenintroduced.
Certain regions of convergence have been determined.
A certain property proved by Guo for iterates generatedby Newton’s and Halley’s methods for computing(I − B)1/p holds also for iterates generated by the dualPadé methods.
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Part I - memories Part II - talk Appendix - after the talk
References
References I
Cardoso, Loureiro, On the convergence of Schröderiteration function for the pth roots of complex numbers,Applied Math. Comput. (2011).Greco, Iannazzo, Poloni, The Padé iterations for thematrix sign function and their reciprocal are optimal,Lin. Alg. Appl. (2012).Guo, On Newton’s and Halley’s method for the principalpth root of a matrix, Lin. Alg. Appl. (2010).Higham, Stable iterations for the matrix square root,Numer. Alg. (1997).
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Part I - memories Part II - talk Appendix - after the talk
References
References II
Higham, Mackey, Mackey, Tisseur, Computing thepolar decomposition and the matrix sign decomposition inmatrix group, SIAM J. Matrix Anal. 25 (2004).Kenney, Laub, Rational iterative methods for the matrixsign function, SIAM J. Matrix Anal. Appl. 12 (1991).Minghua Lin, A residual recurrence for Halley’s methodfor the matrix pth root, Lin. Alg. Appl. (2010).
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Part I - memories Part II - talk Appendix - after the talk
References
References III
Gomilko, Greco, Zietak, A Padé family of iterationsfor the matrix sign function and related problems,Numer. Lin. Alg. Appl. (2011).Gomilko, Karp, Lin, Zietak, Regions of convergence ofa Padé family of iterations for the matrix sector function,J. Comput. Appl. Math. (2012).Laszkiewicz, Zietak, A Padé family of iterations for thematrix sector function and the matrix pth root,Numer. Lin. Alg. Appl. (2009).Zietak, The dual Padé families of iterations for thematrix pth root and the matrix p-sector function,J. Comput. Appl. Math., submitted.
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Part I - memories Part II - talk Appendix - after the talk
Appendix
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Part I - memories Part II - talk Appendix - after the talk
After the talk
Bhatia, Bini, Guo, Higham, Iannazzo, Ipsen, Meini, Poloni,Simoncini, Tisseur, Trefethen, Van Loan, and KZ
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Part I - memories Part II - talk Appendix - after the talk
7 days before my birthday
Manchester, April 11, 2013
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Part I - memories Part II - talk Appendix - after the talk
One of birthday presents
for more pictures see my home page (talks)
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Part I - memories Part II - talk Appendix - after the talk
Thank you!
Wrocław the city of DWARFS