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. ------------------------END OF REPORT------------------------------------