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UNIVERSITY
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WROC£AW UNIVERSITY
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Kazimierz Urbanik (1930-2005)
Zbigniew J. Jurek1, Jan Rosiński2 and Wojbor A.
Woyczyński3
1)Instytut Matematyczny, Uniwersytet Wroc³awski
pl. Grunwaldzki 2/4, 50--384 Wroc³aw, Poland
2) Department of Mathematics, University of Tennessee,
Knoxville, TN 37996, U.S.A.
3)Department of Statistics and Center for Stochastic
and Chaotic Processes in Science and Technology,
Case Western Reserve University, Cleveland, OH 44106, U.S.A.
(adapted from Z. J. Jurek , J. Rosinski and W. A. Woyczynski cf. PMS 25.1 , (2005),
pp. 1-22.)
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Kazimierz Urbanik, the Founder and Editor-in-Chief of this journal, and Professor Emeritus of
Mathematics at Wroc³aw University, died of cancer Sunday,
May 29, 2005, at the age of 75. His research, teaching and administrative work was
decisive in creation of a major school in probability theory in Poland. Born in Krzemieniec in
Eastern Poland, following the end of World War II and transfer of the city to Western Ukraine,
he moved with his family to the western Polish territory of Lower Silesia, where he lived for the
next sixty years, almost all of them in the regional capital city of Wroc³aw.
Urbanik, a two-term Rector of Wroc³aw University and an Ordinary Member of the Polish Academy of
Sciences, led the Institute of Mathematics in Wroc³aw for several decades. His over 180
published scientific papers developed novel approaches to problems of probability
theory, the theory of stochastic processes, mathematical physics, and information theory. Today
they are well known in the global mathematics research community. His favored tools were
functional and analytic but he did not shrink from tackling difficult unsolved problems in
universal algebra either.
As an educator Urbanik was the principal advisor of seventeen doctoral students who continued
work on his ideas at academic institutions of five continents. His fairness, warmth, generosity
and devotion to them were legendary and they reciprocated in kind. He loved doing and teaching mathematics and despite his long and
incapacitating illness, about which he never complained, continued working with the students,
publishing and fulfilling his editorial duties almost to the last days of his life. He delivered
his final lecture on April 21, 2005, and his last published paper appeared in the Spring of this
year.
He is survived by his wife Stefania, son Witold, daughter Jadwiga, and a grandson, all of them
of Wroc³aw, Poland.
* * *
The town of Krzemieniec was for centuries,
until World War II, a part of the Commonwealth of Poland and
Lithuania. (This part of the article has been adapted, with
permission, from a paper that had been published in Demonstratio Mathematica 34(2001) on
the occasion of Professor Urbanik's 70th birthday.)
The town always played
a special role in the history of Polish culture, especially during the
nineteenth century when the Polish state ceased to exist as such and its
territories were partitioned among Russia, Prussia and Austria. The
pride of
Krzemieniec was the Lyceum, an educational institution of
considerable prestige and tradition in Eastern Europe. It counted
numerous
luminaries and statesmen among its graduates. Juliusz S³owacki
(1809--1849), one of the pantheon
of Polish Romantic poets, was raised and educated in Krzemieniec,
and so
was, a century later, Mark Kac
(1914-1984), a well known Polish-American mathematician.
It was in these environs that Kazimierz Urbanik was born on February
5,
1930. In due time he entered the Lyceum but his education at its
School of Exercises was interrupted by the war. First the Soviets,
then the Germans, and then, again the Soviets occupied the area, and by
1945 his family was forced to move to the town of Brzeg, 50 km
south-east
of Wroc\l aw, in Lower Silesia which was just reunited with Poland
as a result of the Yalta agreements. In 1948, he passed the final
matura high school examination and matriculated at the
University
of Wroc³aw. He majored both in mathematics and physics and showed
an
early interest in other areas
of natural sciences. At one point during his undergraduate studies he
was an
active
participant in nine different seminars. There he met his mentors,
Professors Hugo Steinhaus (1887-1972) and Edward Marczewski (Szpilrajn) (1907-1976) who
after World War II transplanted the traditions of the Lwów and
Warsaw
Schools
of Mathematics to the Polish Western Territories. The presence of those
two
distinguished mathematicians decisively influenced Urbanik's scholarly
interests. In October 1950, as a third-year undergraduate student, he
was
awarded a teaching assistantship at the University wherefrom he
graduated
in 1952, and where he was immediately employed as a junior faculty
member.
Although throughout his career he remained interested in a broad
spectrum of
scientific endeavors, his focus, after a brief flirtation with
topology, was
now firmly in probability theory. A research seminar on this
subject was then run at the Wroc³aw Branch of Mathematical Institute of
the Polish
Academy of Sciences by Marczewski and Steinhaus. Dubbed simply the
``Monday,
5 o'clock seminar'' by the insiders, it has continued its activities
for
more than fifty years and concentrated on analytic and functional
methods of
probability theory. Over the last several decades it was Urbanik who
directed it
and was its soul and spirit. All the three authors of this biographical
sketch
and numerous other probabilists got their initial training there, and
were
beneficiaries of Urbanik's patient and forgiving mentoring style.
Urbanik's academic career was swift. In 1956, he received his Ph.D.
for a dissertation on cascade processes, in 1957 obtained his
docentship and was appointed Associate Professor (Docent), and three
years later was promoted to the rank of Professor. In 1965, at the
age
of 35, he was elected to the Polish Academy of Sciences
as its youngest member ever.
Throughout the years he masterly combined a steady flow of research
(published in over 180 papers in a variety of areas), teaching and major
administrative responsibilities. The latter were not an afterthought in
his
academic life. For almost thirty years (1967-1978 and 1981-1996) he
guided
the Institute of Mathematics of the Wroc³aw University as its
Director,
and from 1975 to 1981 he served as the President (Rektor) of
Wroc³aw
University. He formally retired from the University in the Summer of
2000.
For a couple of terms he was also a Vice President of the Polish Academy
of
Sciences. He played key roles in developing several major projects
of
importance to Polish mathematics, such as the creation of the Stefan
Banach
Mathematical Center, which was initially an institution jointly funded
by
the Soviet Union, Poland, East Germany, Czechoslovakia, Hungary, Romania
and Bulgaria, but located in Warsaw, Poland. It should not be overlooked
that his effectiveness as a science administrator and a community leader
was greatly enhanced by his prominent position within the then ruling Polish
United Workers Party. However, he was never an ideological doctrinaire
and, within the mathematical community,
kept his political views private. With his
access to the communist establishment, he was able to protect the
mathematical community from political extremists, and many individual
mathematicians from the unpleasant consequences of their ``political
incorrectness''. Throughout the last half-century of Poland's political
trials and tribulations his integrity was above reproach as he kept the
respect and admiration of people from all parts of the political
spectrum.
It was a remarkable fact that in the 1990s, after Solidarity wrestled
power from the communist party, Urbanik was still elected by a popular
vote
to the directorship of the Institute.
As a teacher Urbanik developed a large and faithful following. His
delivery was
crisp and velvety, and we were all mesmerized by his lectures in which
deep
theories unfolded effortlessly in front of our eyes without any help
from notes
or textbooks. He had developed original approaches to almost every
subject he
lectured on and we regret that most of his course offerings were never
converted into published textbooks. His lectures attracted many
research
students to his seminars. Among the seventeen Ph.D. students who wrote
their
dissertations under his supervision are the first and the third authors
of
this
article, while the second author was a Ph.D. student of the third one.
Urbanik was also a popular speaker abroad, with invited visits to
Berkeley, Moscow, Paris, Cambridge, New Orleans, Beijing, G\"ottingen,
Hanoi, and Cleveland, among others. He spoke several times at the
Oberwolfach Institute in Germany. In 1966, during the World
Mathematical
Congress in Moscow, he delivered a major invited address.
Despite his retirement
he continued to direct his Monday Seminar, teach graduate
courses and
serve as the Editor-in-Chief of the journal Probability and
Mathematical Statistics which was founded by him in 1980. Numerous
awards
and honors bestowed on him are listed in a separate appendix below.
Kazimierz Urbanik's most substantial research contributions, already
acknowledged in Jean
Dieudonn\'e's historical analysis A Panorama of Pure
Mathematics as seen by N. Bourbaki (cf. [a],
Section B VII, pp. 223-228), were in
probability and stochastic processes.
He also made, however, major discoveries in other areas that included
information theory,
mathematical physics (including foundations of quantum mechanics),
theory
of universal algebras, mathematical analysis, functional analysis and
topology.
In this broad scope of research he was a faithful follower of his
mentor,
Hugo Steinhaus.
* * *
In the remainder of this sketch we will attempt to
describe Kazimierz Urbanik's
principal lines of research dividing
them into several topics as was done in the 1974 Nauka Polska
article on
Urbanik's work (cf. [g]). The numbered references refer to the
complete
bibliography of Urbanik's papers which is enclosed as an appendix
below;
letters denote other references.
(i) Probability theory. In the years 1956-1960 Urbanik was one
of
the first who investigated limit theorems for sequences of independent
random
elements with values in compact groups, and introduced the notion of a
Gaussian
measure on a locally compact abelian group . One of his fundamental and
strikingly elegant results was that the existence of a Gaussian measure
on a
group is equivalent to the connectedness of the group. His results
in this
area are now a standard fare in monographs on probability theory on
groups
(cf.
e.g., Heyer's monograph [c] and Chapters 3 and 6 in Grenander's book
[b]).
While visiting Aarhus University in 1962 Urbanik learned about
Kingman's work [f]
on random walks with spherical symmetry which lead him to consider a
new
type
of convolution.
In the fundamental paper [79] he introduced
a formal
notion of a
generalized
convolution as a binary operation on
probability measures on the positive half-line satisfying five axioms, one
of
them being the weak law of large numbers for $\delta_1$ measures.
These
axioms permit a study of generalized characteristic functions, Laplace
transforms, infinitely divisible laws, stable laws, Linnik class $I_0$, moments, domains of attractions, and other concepts
hitherto
studied only for classical convolutions. In his most recent work
those fundamental results are used to introduce and investigate some
``generalized'' special functions. Over the years Urbanik wrote
almost
twenty papers on that subject; in literature,
generalized convolutions are now commonly referred to as ``Urbanik
systems".
Some of the generalized convolutions are related to the
theory of hypergroups. Urbanik's pioneering work in this area was
followed
up by
numerous contributions to the subject from other mathematicians such
as D.
Kendall, N. Bingham, V.E. Volkovich, N. van Thu, H.Heyer, R. Jajte and
Z. J.
Jurek.
In 1968, Urbanik has ingeniously applied the analytical method of
extreme
points, and the Choquet's Theorem in particular, to find
characteristic
functions of many limit probability laws, including a description
of the
L\'evy class $L$ of selfdecomposable distributions. He also used that
tool
to characterize Feller's class, autoregressive systems and limit laws
in
non-commutative probability theory, cf. [119], [122], [141].
Four years later, in [109], he described limit laws of partial sums of
random vectors normalized by matrices (linear operators), which pushed forward the subject originated independently in V. Sakovic's and M. Sharpe's dissertations. For this purpose he
introduced
a completely new notion of decomposability semigroups. These are
matrix (linear operator) semigroups associated with probability
measures.
In numerous papers Urbanik has shown how topological and algebraic
properties
of those semigroups can be used to describe probability distributions;
cf. [120], [123], [128], [134]. This research had many followers, including
M. Klass, M.G. Hahn, V. Semovskii, J. Kucharczak, R. Jajte, W. Krakowiak,
B. Mincer, W.N. Hudson, Z.J. Jurek, J.A. Veeh, W. Hazod, M.M.
Meerschaert, H.P. Schaefler; and again in this context, in [109] and [134], Urbanik
masterfully applied Choquet's Theorem to find an analogue of the
L\'evy-Khintchine formula. A historical sketch of the operator-limit
laws theory can be found in [e]. Chapter 3 of the latter monograph consists
mostly of Urbanik's results and the book also provides a new
random integral representation method which permits to circumvent the extreme
points technique.
In papers [110] and [114] Urbanik introduced a classification of limit
laws by introducing a countable decreasing family
$L_m, m= 0,1,2,...,$ which begins with the L\'evy class $L$ of
selfdecomposable distributions. This circle of ideas was picked up,
extended and generalized, among others, by J. Bunge, Z. J. Jurek,
K. Sato, M. Maejima, M. Yamazato, B. Schreiber, N. van Thu.
A novel identification theorem for probability distributions
via moments of sums of independent random variables was obtained by
Urbanik [171] in
1993. The proof borrowed techniques from the theory of Banach algebras.
(ii) Stochastic processes. In one of his first papers, published
in
1954,
Urbanik investigated asymptotic behavior of homogenous Markov
processes and,
in particular, the distribution of their extreme values. He proposed a
Markovian
model of cosmic ray cascades, and the physical problem of forecasting
the
sun's
activity led him to the prediction theory for stationary processes
without
the moment condition. He proved that in this context Orlicz spaces play
a
role
analogous to the role of Hilbert spaces in the Wiener-Kolmogorov theory
based on
the covariance function. In a 1967 article [97], which was quickly
followed by
a joint
paper with W.A. Woyczynski [98] on a similar topic, the stochastic
integrability with respect to general processes with independent
increments
was
characterized in terms of Orlicz spaces. This approach was later
extended to
Bartle-type stochastic integrals by J. Rosi\'nski, and to
semimartingale
integrals by Kwapie\'n and Woyczy\'nski; cf.[g]. We should also mention a beautiful exposition
of the classical linear prediction theory for second-order stationary sequences published by
Urbanik in Springer's Lecture Notes in Mathematics series [99]. The book was a written version of
a series of lectures delivered at Erlangen University, Germany.
In 1956, Urbanik begun his systematic study of generalized stochastic
processes
and random fields whose sample functions are (Schwartz) distributions,
and introduced local characteristics for such processes; cf.
[21],[27], [30] [33-38],[49], [70]. This work, of great importance for
physics
and,
in particular, quantum field theory, was done contemporaneously but
independently of I.M. Gelfand's investigations in the same area, and
used
different techniques.
More recently, in 1988 papers [157] and [165], Urbanik introduced the
concept of an analytic stochastic process which was based on the
Wiener-It\^o decomposition of chaos. His fundamental theorem provides
an
isomorphism between the class of analytic processes and the space of
entire
functions. This permits an application of tools from analytic function
theory
to random special functions.
In 1992, Urbanik introduced a new analytic method for
studying random functionals of transient stochastic processes, which
include functionals of geometric Brownian motion,
cf. [168] and
[169]. The latter found applications in foundations of modern
mathematical
finance theory.
(iii) Information theory and theoretical physics. In 1957,
Urbanik
working together with
G.S. Rubinstein, solved a problem posed by A.N. Kolmogorov, concerning
the
maximum value of information, [26].
His further investigations in this field were closely related to
statistical
physics and done in collaboration with physicist Roman S. Ingarden. In
particular,
using ideas of E.T. Jaynes, they proposed an original foundation for
informational thermodynamics. The law of entropy increase was proved
rigorously; cf. [65], [67-69]. In foundations of quantum mechanics,
Urbanik proved a remarkable fact that commutativity of observables is
equivalent
to the existence of their joint distribution; cf. [66], [94].
Since 1961, Urbanik made several attempts to define information without
probability theory. These efforts finally bore fruit in 1972, when he
proposed
new axioms for information theory based on four postulates: (1) the law
of the
broken choice; (2) the local character of information; (3) the
indistinguishability of equivalent systems of information; (4) the law
of
increase of information; cf. [111], [113], [116].
(iv) General algebras. It was Edward Marczewski, one of
Urbanik's
mentors who, in 1958, initiated studies of the notion of independence in
universal algebras. One of the deepest problems in that field was the
characterization of those algebras whose independence has the properties
of
linear independence in linear spaces. During the following eight years
Urbanik completely solved that problem, proving that those algebras are
linear
or affine spaces over appropriate fields; cf. [48], [51], [57], [71], [75],
[89].
Also, during his visit at Tulane University, New Orleans, Louisiana, in
the
academic year 1959-60, he made fundamental contributions
to the theory of algebras with absolute values. The work contained in
almost
twenty papers written by Urbanik in the field of general algebras forms
an
essential part of George Gr\"atzer's 1979 monograph [d]. The journal
Algebra Universalis is now one of the main outlets for research
in the
area where Urbanik's work was of such fundamental importance.
(v) Topology, measure theory and analysis. Urbanik's first
paper,
written in 1953 jointly with B. Knaster, characterized zero-dimensional
$G_\delta$ sets. But he did not stay in the area although,
occasionally, he
returned to topological issues. In [5] he proved the non-topological
structure
of the field of Mikusi\'nski operators. Jointly with Paul Erd\"os, in
[41], he
proved a theorem about sets measured by multiples of irrational
numbers. A
collaboration with H. Fast, resulted in an extension of
Titchmarsh convolution theorem, while in [64] he developed
Fourier analysis on Marcinkiewicz spaces. Urbanik also solved the
Hartman's
problem on the existence of common extension of isomorphic images of
Haar
measures induced by different topologies on a given group.
The above paragraphs provide only a rough and imprecise description of
Kazimierz Urbanik's opus of research. A complete listing of his
publications
is enclosed as an appendix.
There are several characteristic features
of
Urbanik's style of doing mathematics in particular, and science in
general.
The
first and foremost in our minds is the elegance of his theories and
sheer power of his deductive reasoning combined with the crispness and
clarity of
their presentation. He never shrunk from frontal attacks on the
problems he
was working on and was capable to marshal considerable artillery to
support his offensives. But now and then you see a totally unexpected
tack in
his proofs and analytic ingenuity that we, his students, all tried to
emulate.
In his work on probability, he employed powerful abstract
tools,
ranging from functional analysis to abstract algebras and topology, with
great mastery even in situations that seemingly, at first sight, were
unlikely to benefit from them. There was also a persistent physical thinking behind a lot of
his
abstract
arguments. His great intuition and insight in
finding
the most appropriate and often eye-opening formal framework for
theories
he was working on has always been remarkable.
It is our considered opinion that the recognition and importance of
Urbanik's
multifaceted work will grow as time goes on and that the popularity
of his pioneering ideas in research programs of other mathematicians
and
theoretical physicists will expand.
The probability school he has
created in Wroc³aw, continuing Hugo Steinhaus' traditions, has by
now
radiated its ideas and its style of doing mathematics to many other
international research centers, and his former students spread their
scholarly activities to
five
continents.
References
[a] J. Dieudonn\'e, A Panorama of Pure Mathematics, as Seen by N.
Bourbaki,
Academic Press, New York, 1982.
[b] U. Grenanader, Probabilities on Algebraic Structures,
J.Wiley,
New York, 1963.
[c] H. Heyer, Probability Measures on Locally Compact Groups,
Springer-Verlag, Berlin, New York, 1977.
[d] G. Gratzer, Universal Algebra, 2nd ed. Springer-Verlag,
New York, 1979.
[e] Z.J.Jurek and J.D. Mason, Operator-limit distributions in
probability theory, J.Wiley, New York, 1993.
[f] J.F.C. Kingman, Random walks with spherical symmetry, Acta
Mathematica,
Vol. 63(1963), pp. 11-53.
[g] S. Kwapie\'n and W.A. Woyczy\'nski, Random Series and
Stochastic Integrals: Single and Multiple, Birkh\"auser-Boston, 1992.
[h] E. Marczewski, C.Ryll-Nardzewski and W. Woyczy\'nski, Kazimierz
Urbanik, Nauka Polska (1974), No. 1, pp.
101-105.
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