rick@cs.arizona.edu (Rick Schlichting) (01/09/91)
[Dr. David Kahaner is a numerical analyst visiting Japan for two-years
under the auspices of the Office of Naval Research-Asia (ONR/Asia).
The following is the professional opinion of David Kahaner and in no
way has the blessing of the US Government or any agency of it. All
information is dated and of limited life time. This disclaimer should
be noted on ANY attribution.]
[Copies of previous reports written by Kahaner can be obtained from
host cs.arizona.edu using anonymous FTP.]
To: Distribution
From: David K. Kahaner ONR Asia [kahaner@xroads.cc.u-tokyo.ac.jp]
Re: Proceedings of Inst Stat Math, Vol 48, No. 1.
9 Jan 1991
The Institute of Statistical Mathematics was described in my report of
Apr 11, 1990, with file name "trp12-89".
Institute of Statistical Mathematics
National Inter-University Research Institute
4-6-7 Minamiazabu, Minako-ku
Tokyo, Japan 106
Tel: 81 3 3446-1501
At that time I commented that the Institute has been producing a unique
combination of exceptionally high quality basic research, and
applications oriented research; ISM's history dates to 1944, but in it's
present configuration only since 1985. Nevertheless it is not well known
in the West. Below is a set of six abstracts from a recent issue of
ISR's Proceedings, representing the best way I know to illustrate the
eclectic nature of ISR's work.
The quality of the research, combined with ISR's excellent location and
facilities in central Tokyo, make it very attractive for a sabbatical or
collaborative research project.
PROCEEDINGS OF THE INSTITUTE OF STATISTICS MATHEMATICS Vol. 48, no. 1
Contents
An Interpretation of Auto-Regression (AR) Model by Using Linear Algebra
Tomoyuki Higuchi (The Institute of Statistical Mathematics)
An auto-regressive (AR) model is expressed by using a matrix
formulation and revitalized through linear algebra. The
Levinson's recursion, which is an efficient recursive solution for
AR model parameters, is discussed within a framework of the linear
space. Moreover, several solutions for calculating a partial auto
correlation coefficients (PARCOR) are classified according to the
minimized prediction error.
Non-Commutative Analysis and Relative Entropy
Hiroaki Yoshida (The Institute of Statistical Mathematics)
In the latter of 1920's, the quantum theory was born. To describe
the quantization mathematically, von Neumann had investigated
operators acting on Hilbert spaces. From his researches, the
theory of operator algebras had appeared.
The following question will grow naturally. What is non-
commutative analysis? We should regard it as the analysis of
quantized objects. It is spread over many fields in mathematics
at present. However, we know that the theory of operator algebras
is the most essential part of non-commutative analyses.
Since commutative von Neumann algebras can be represented as
function spaces on measure spaces, we find that general von Neumann
algebras are the "quantum" analogue of measure spaces. Especially,
we see that von Neumann algebras called of finite type correspond
to "quantized" probability spaces.
In this note, we shall concentrate our interest on the relative
entropy in finite von Neumann algebras introduced by Primsner and
Popa. At first, we see that this relative entropy is one of the
extensions of the relative entropy in the probability theory.
Then, we give some technical formulas for evaluating the values of
the relative entropy. As an application of these results, we
investigate the relation between the actions of finite groups and
the values of the relative entropy for factors of type II using the
fixed point algebras. At last, we show the conjugate classes of
the actions of the symmetric group G3 graphically.
The technical terms and notions of the theory of operator algebras
are not so familiar. So, we shall begin with the explanation of
them comparing with those in the probability theory.
Fractal Growth of Bacterial Colonies
Hiroshi Fujikawa (Department of Microbiology, Tokyo Metropolitan Research
Laboratory of Public Health)
Mitsugu Matsushita (Department of Physics, Chuo University)
A Bacillus subtilis strain is inoculated on an agar plate
containing a low concentration (1 g/1) of peptone as nutrient and
incubated at 35xC. Colonies grow two-dimensionally with random
branches, similar to clusters of the diffusion-limited aggregation
(DLA) model. Colony pattern is found to be self-similar with the
fractal dimension of 1.73, in good agreement with the value of the
DLA model. During the growth of a colony the existence of a
screening effect is confirmed. These results clearly show that the
colony grows through a simple physical mechanism, that is, the DLA
process.
What makes a diffusion field for the bacterial DLA type growth? No
remarkable growth is observed on an agar plate without any peptone.
When the organism is spotted on an agar plate with the
unidirectional concentration gradient of peptone (0-2 g/1), colony
branches develop predominantly in the direction of higher peptone
concentration. These results strongly suggest that the diffusion
field of nutrient in an agar plate is essential for the bacterial
DLA growth.
The colony morphology varies a great deal with the nutrient
concentration and the surface moisture of agar plates, including a
DLA type, dense branching morphology, a round type, and fast
spreading without any openings. Repulsion between two neighboring
colonies is observed in the DLA and dense branching morphologies
only, suggesting that these growths may be strongly influenced by
the concentration field of nutrient. Local growth mechanism
characterized by an organism itself is also considered to affect
its colony formation. When the agar plate is covered with a thin
layer of glycerol, the colony morphology of our strain becomes
thoroughly round. This may come from some physical factor such as
surface tension of glycerol.
Dynamical Structure Factor of Percolating Networks
Tsuneyoshi Nakayama, Kousuke Yakubo, Hiroyuki Ohta (Faculty of
Engineering, Hokkaido University)
Computer simulations are essential to develop the insight into
dynamics of percolating nets. We have succeeded in treating very
large percolation clusters of more than 10**5 particles recently,
revealing a wealth of detailed quantitative information. These
have become possible with the advent of array-processing
supercomputers, and with the use of a numerical method that does
not require diagonalization. In this article, we have extended our
computer simulations to calculate the dynamic structure factor
S(q,w) for d=2 percolating networks. It is found that S(q,w) has
universal behavior scaled by a single wave number. It is shown
that this wave number behavior cannot be explained by the effective
medium theory or using average wave functions.
Analysis of Water Flow of the Kusu River in an Interconnected Multi-Reservoir
Power System
Emiko Arahata, Kunio Tanabe, Yoshiyasu-Hamada Tamura, Genshiro Kitagawa,
Tohru Ozaki (The Institute of Statistical Mathematics)
Ryuichi Seki, Katsuhiro Urayama (Kyushu Electric manufacturing Co., Ltd.)
Hiroyuki Tamura (Faculty of Engineering, Osaka University)
We are concerned with the problem of predicting water flows of the
Kusu river system in Kyushu, in the southern part of Japan, which
interconnects several multi-reservoir power systems. In
particular, we are interested in predicting the residual inflows
into the reservoirs. The difficulty with this problem arises from
the lack of a sufficient number of observation points as well as a
large error in measurements. Due to this problem, the conventional
methods which depend on an ad hoc technique were unable to give
meaningful information for controlling the river flow system.
The purpose of this paper is to explore the possibility of
estimating the unknown residual river flows by introducing a prior
model which reflects the mass balance of water and gradual change
of water flows. A state space model for this system is introduced.
We introduce smoothing prior to unknown variables to specify the
transition equations, and add the second-order difference equations
for unmeasurable variables to the observation equations. Using the
Kalman filtering technique, we get the state estimates. The
structural parameters which are estimated for low water level cases
are successfully applied to high water level ones. While
conventional estimation of residual river flows often gives
negative values which do not reflect reality, our method has no
such problem.
Generalization of Fokker-Planck Equations By Means of a Projection
Operator Technique - A Study Of Stochastic Processes Subject To Additive
Noises
Takashi Okasaki (The Institute of Statistical Mathematics)
An attempt is presented to generalize Fokker-Planck (FP) equations
to be applicable to stochastic processes driven by arbitrary
external noises. Except for Gaussian white noises, the ordinary FP
equation fails to determine the probability density function (PDF)
of the stochastic variables rigorously, because it has no term
incorporating the detailed information of the noise process but a
parameter related with the noise-correlation function. To lift
this limitation, an approach is developed based on a projection
operator technique devised originally in statistical mechanics.
The present approach starts with a specification of the equations
that describe the evolutions of the main variable U and of the
noise variable W; the equation of U may additively contain
functions nonlinear in W, and that of W is required to have driving
forces specified by a Gaussian white process. The joint
probability density function, obeying an exact whole-system FP
equation associated with these equations, gives the PDF of the main
variable U as the expectation of a delta function in U, which is
here referred to as a density creator (DCR).
Through the PDF of the noise variable, an operator p is constructed
to project every phase function of (U, W) out onto a space of
linear combinations of DCR's. An elaboration on the calculi
involving the projection operator p and the whole-system FP
operator enables one to express the equation of motion of the DCR
in terms of the DCR itself, thereby to obtain an accurate equation,
e.g., "a generalized FP (GFP) equation", which includes a memory
integral over the past history of the PDF to recover informations
originally owned by the joint PDF. The GFP equation properly
describes the interaction effects between the main variable and the
noises.
While the GFP equation thus derived has a compact form, it may, in
applications, show some difficulties due to the presence of
complicated projection operators in the expression of the diffusion
coefficient. To attain a tractable form of the GFP equation, the
diffusion coefficient, and the PDF as well, are expanded with
respect to the interaction terms in the whole-system FP operator.
The diffusion coefficient then turns out to be an integral of the
noise-correlation function and of the Jacobian related with a non-
stochastic process which obeys an equation the same as for the main
variable, but with no noise term.
The GFP equation is applied, to confirm its validity, to a simple
linear process driven by coloured noises. The resulting PDF
indicates a precise coincidence with the exact one, in contrast
with the poor result given by the ordinary FP equation.
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