NA Digest Sunday, August 9, 1993 Volume 93 : Issue 29

Today's Editor:

Cleve Moler
The MathWorks, Inc.

Submissions for NA Digest:

Mail to

Information about NA-NET:

Mail to


From: Daniel Okunbor <>
Date: Mon, 9 Aug 1993 14:53:36 -0500
Subject: Change of Address for Daniel Okunbor

I have taken up a teaching position at the Univesity of Missouri-Rolla
effective August 3rd. My current address is

Daniel Okunbor
University of Missouri-Rolla
Rolla, MO 65401

e-maill address:



From: Stephen Vavasis <>
Date: Fri, 6 Aug 93 11:31:54 -0400
Subject: Numerical Analysis Ideas in Science Magazine

This week's Science magazine (30 July 1993, pp 578-584) contains an
article about pseudospectra in fluid flows. "Science" and its British
counterpart "Nature" rarely publish articles about applied mathematics,
so this event may be of interest to NA digest readers.

The article, "Hydrodynamic Stability Without Eigenvalues", by L. N.
Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, proposes
a new way to analyze the transition from stable steady-state laminar
flow to unstable flow. Both Trefethens, as well as Driscoll, are at
Cornell, and Reddy just recently moved from NYU to Oregon State.

A key point of the article is that the Navier-Stokes equations, when
linearized around the steady-state flow, have highly nonorthogonal
eigenvectors. This means that the eigenvalues -- which are crucial to
traditional stability analyses -- are ill-conditioned and perhaps tell
the wrong story about local perturbations.

The connection between nonorthogonal eigenvectors and ill-conditioned
eigenvalues is well-known to numerical analysts but perhaps not to the
larger scientific community. This article is a great example of
spreading knowledge from within our community to the world at large.


From: L. M. Delves <>
Date: Tue, 3 Aug 93 9:31:34 BST
Subject: Formation of Institute of Advanced Scientific Computation

Liverpool University has recently set up the Institute of Advanced
Scientific Computation. This represents a merger of two previous
Research Centres: Center for Mathematical Software Research; and
North West Transputer Support Centre.

The Institute (IASC) has interests in all areas of technical computing, but
especial expertise and interest in MIMD parallelism. Current projects include:

1) Development of Distributed Data Library
2) Algorithms and software for oil reservoir simulation
3) Parallel database software for diagram retrieval
4) Temperature sensing for induction heating strip mill
5) Development of Fortran90 compiler
6) Development of HPF Fortran source-source translator
7) Parallelisation of codes for SAR processing, and Forging

All of these projects are collaborative with external sites/organisations.

The Institute has an active Visitor, and Postgraduate Teaching, program.

Here are some requests:

1) If you would like any info, mail me at
or the institute at

2) If your activities overlap ours, please put us on your email list for
circulation of whatever:

We are especially interested in collaboration with European and US sites.

3) Drop in and see us some time.


From: Robert I McLachlan <rxm@vortex.Colorado.EDU>
Date: Tue, 3 Aug 93 14:08:38 -0600
Subject: Two Preprints on the Numerical Integration of ODE's


Keywords: Lie-Poisson systems, Hamiltonian systems, symplectic integrators,
composition methods, operator splitting.

Postscript versions of the following papers are available by anonymous
ftp from (, directory pub/numerics/papers,
files and
Please direct comments to the author at

"Explicit Lie-Poisson integration and the Euler equations"

We give a wide class of Lie-Poisson systems for which explicit, Lie-Poisson
integrators, preserving all Casimirs, can be constructed. The methods are
extremely simple. Examples are the rigid body, a moment truncation,
and a new, fast algorithm for the sine-bracket truncation of the 2D Euler

"On the numerical integration of ordinary differential equations
by symmetric composition methods"

Differential equations of the form $\dot x=X=A+B$ are considered, where
the vector fields $A$ and $B$ can be integrated exactly, enabling
numerical integration of $X$ by composition of the flows of $A$ and
$B$. The relationships between various symmetric compositions
currently in use are investigated with regard to order, complexity, and
reversibility. Simple formulae are given for the number of determining
equations which must be solved for a method to have a particular order.
A new, more accurate way of applying the methods thus obtained to
compositions of an arbitrary first-order integrator is described and
tested. The determining equations are thoroughly explored, and new
methods up to 100 times more accurate (at constant work) than those
previously known are given.

Robert McLachlan


From: Alfred Gautschy <puls@cvxastro.MPA-Garching.MPG.DE>
Date: Thu, 5 Aug 93 19:09:49 +0200
Subject: A Difficult BEVP in Astrophysics

Numerical Methods for a Difficult BEVP in Astrophysics

In the following I describe shortly a problem in stellar astrophysics
I am working with for years now. I wonder if there is anything better
we can do for its solution than what we are applying presently.
I am sure the na-community would know best.

In stellar stability theory we usually have to solve a system
of four or six first-order linear differential equations, that,
together with suitable boundary conditions, constitutes the
boundary-eigenvalue problem. The domain of the independent
variable extends from 0 to 1, with the system having a regular
singularity at 0. Formally we can bring the system of
equations into the form:

d Y
x --- = A * Y (1)
d x

where Y is a complex vector with 4 or 6 components and A is a
4X4 or 6X6 matrix with complex coefficients. The eigenvalues
(normal modes of the system) enter the matrix A in two components
nonlinearly. The components of the matrix A are strongly varying between
x = 0 and x = 1 and the values of the components are known only
at a few hundred discrete points between 0 and 1. Furthermore, matrix A
has no patricular symmetry properties.

We are typically interested in the lowest few dozen eigen-solutions.
For particular stellar problems we have to extract high overtones,
but again only a few dozen of them.

For some time the BEVP was solved with finite difference methods
which, however, are pretty cumbersome due to the approximate solutions
which have to be guessed. Often the eigenfunctions are rapidly oscillating
in space, and that is something one cannot know ahead.

Another method we try for a few years now is the transformation of the
BEVP into an initial-value problem of Riccati type, which is then
solved by means of a shooting method and fitting the solutions by
adapting the eigenvalue. Due to the singularities of the Riccati equation
on the path of integration one has to transform to the inverse Riccati
form rather frequently which makes the calculations rather expensive.

My question is now, are there methods around that allow for the calculation
of eigenfrequencies and eigenvectors of a BEVP of the type (1) that
do not need any a priori guesses of the approximate solution and
do (hopefully) not need supercomputers for their application.
Is there anything like a generalized matrix eigenvalue method that
works for systems?

For any informtation concerning improvements a e-mail note would be
highly appreciated.

Kind regards

Alfred Gautschy
Max-Planck-Institut fuer Astrophysik
85748 Garching/Germany


From: Gustavo Montero Garcia <>
Date: Fri, 6 Aug 93 11:54:06 +0100
Subject: Pollution & preconditioners

I'll be so grateful if you send me some information about the
following subjets:

Modelling the pollution phenomena by finite element method in:
- Atmosphera
- Groundwater
- Seawater

Fast preconditioners for solving linear and non linear systems
of equations.

Thank you.

Gustavo Montero Garcia
Prof. Tit. U.
Centro de Aplicaciones Numericas en Ingenieria CEANI
Universidad de Las Palmas de Gran Canaria
Campus de Tafira Baja
35017 Tafira ( Las Palmas de Gran Canaria)
Islas Canarias -Espa~na-

Tf.: + 34 28 451 917
FAX.: + 34 28 451 921


From: Nick Higham <>
Date: Fri, 6 Aug 93 14:44:16 +0100 (BST)
Subject: New Book ``How To Teach Mathematics''

The American Mathematical Society has just published a book called
``How To Teach Mathematics: A Personal Perspective'' by
Steven G. Krantz. While the book is not directly concerned with teaching
numerical mathematics, I think it will be of interest to many NA-Digest
readers. In 76 pages Krantz gives much valuable advice based on his
twenty years experience of teaching. Among the topics he covers are

- preparation of lectures
- blackboard technique
- how to deal with student questions
- how to deal with student complaints
- handling large classes
- setting homework and exams, and grading
- use of computers in teaching (he prefers traditional pen and paper

As Krantz says, ``this is a book about the obvious'', but the obvious
can be suprisingly easy to overlook and much of the advice proffered
here is often learned the hard way, by making mistakes.

The book contains many informative examples and entertaining
anecdotes, a quote from Bereseford Parlett, and advice from Paul
Halmos on how to teach the Fundamental Theorem of Algebra.

Every beginning mathematics lecturer will benefit from reading this
book. I recommend it to anyone who wants to improve their teaching

author = "Steven G. Krantz",
title = "How To Teach Mathematics: {A} Personal Perspective",
publisher = "American Mathematical Society",
address = "Providence, RI",
year = 1993,
isbn = "0-8218-0197-X"

Nick Higham
Department of Mathematics
University of Manchester


From: I. G. Graham <>
Date: Fri, 6 Aug 93 20:07:40 +0100 (BST)
Subject: Finite Element Error Analysis

Dear Colleagues,

Can anyone help me with the following technical question about finite element
error analysis?

Let Omega be a bounded convex polygonal domain in the plane.
Triangulate Omega with a sequence of meshes which are regular in the sense of
Ciarlet's book (1978), and which are parametrised by the mesh diameter h.
For each h, let a_h be a continuous function on Omega which is
smooth in the interior of each triangle of the mesh.
Moreover assume that a_h is bounded above and below on Omega by
positive constants which are independent of h.

Consider (in weak form) the linear elliptic problem

- div [(a_h) grad u] = f,

which is to be solved for the scalar function u subject to mixed Dirichlet-
Neumann boundary conditions, with the Dirichlet part of the boundary having
non-trivial measure . The given function f and the given Dirichlet and
Neumann data are smooth and independent of h. Let u_h be the solution of
this problem by the standard finite element method.

Standard arguments then show that u_h is bounded independently of h in
the W^1_p norm when p = 2.
I am interested in proving the same property for some p > 2 (*)
(I wouldn't mind if p had to be arbitrarily close to 2.)

This property of finite element approximation was proved for the Dirichlet
problem for Laplace's equation by Rannacher and Scott (Math Comp 38, 1982).
By the Sobolev Embedding theorem the result (*) implies the slightly weaker
result that the uniform norm of the finite element solution is bounded
independently of h. There are many results around along these lines
e.g. Schatz and Wahlbin (Math Comp 38, 1982) or Suzuki and Fujita
(Numer Math 49, 1986). Again Schatz and Wahlbin (and many other authors)
considered only Laplace's equation, but Suzuki and Fujita proved their
result for quite general coefficients a which could be discontinuous but
which were independent of h. Both Schatz and Wahlbin and Suzuki and Fujita
considered only the Dirichlet problem.

These are the closest results I have found to (*). I have looked through
all issues of Math Comp, Numer Math and SIAM JNA since 1985 but have not
found anything else. I may of course have missed it. Can anyone help with
a reference or an opinion please?

Note that I do not care about convergence of finite element solutions, only
boundedness independent of h. I am also interested in the analogous result
but with a_h discontinuous across triangle boundaries.

Many thanks in advance,

Ivan Graham
School of Mathematical Sciences
University of Bath
Bath BA2 7AY
United Kingdom

fax: (+44 225) 826492


From: Jorge More <>
Date: Mon, 2 Aug 93 17:18:22 CDT
Subject: Postdoctoral Research at Argonne National Laboratory

Postdoctoral Research
Mathematics and Computer Science Division
Argonne National Laboratory

The Mathematics and Computer Science Division of Argonne National
Laboratory invites applications for a postdoctoral research position.

The successful candidate will help develop a parallel environment for
solving large-scale numerical optimization problems.
A strong background in numerical optimization and
tool development for scientific computing applications is desirable.
Nominal requirements include a Ph.D. in applied mathematics,
computer science, applied science, or engineering.

This project is interdisciplinary in nature, interfacing with efforts
in linear algebra, computational differentiation, parallel computing
tools, and large-scale simulation of physical processes.
Project members have access to state-of-the art computing facilities,
including the IBM SP1 and the Intel Touchstone DELTA.

Argonne is located in the southwestern Chicago suburbs, offering the
advantages of affordable housing and good schools, as well as easy
access to the cultural attractions of the city.

Applicants must have received their Ph.D. not more than three
years prior to the beginning of the appointment.
Applications must be addressed to
Walter McFall, Box mcs-postdoc, Employment and Placement,
Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439,
and must include a resume' and the names and addresses of three
references. For further information, contact
Jorge More' (708-252-7238;

Argonne is an affirmative action/equal opportunity employer.


From: Tony Skjellum <tony@Aurora.CS.MsState.Edu>
Date: Mon, 2 Aug 93 22:09:09 CDT
Subject: Scalable Parallel Libraries Conference

Scalable Parallel Libraries Conference
(including Multicomputer Toolbox Developers' & Users' Meeting)
October 6-8, 1993
National Science Foundation Engineering Research Center
for Computational Field Simulation,
Mississippi State, Mississippi

More information: Request from OR
anonymous ftp announcement:
directory: pub/SCL, files: SCL.2, Registration_Form

Deadline for posters extended to August 15 (please see files on anonymous ftp).


* Charles L. Seitz, Caltech, [KEYNOTE ADDRESS]
Title: High-Performance Workstations + High-Speed LANs >= Multicomputers

* Milo R. Dorr, LLNL
Title: "A Concurrent, Multigroup, Discrete Ordinates Model of Neutron

* David Walker, ORNL
Title: "The design of scalable linear algebra libraries for concurrent

* David Womble, Sandia ALBQ
Title: "Out of core, out of mind: making parallel I/O practical".

* Dan Quinlan, LANL
Title: "Run-time Recognition of Task Parallelism Within the P++
Parallel Array Class Library
* Steven Smith, LLNL
Title: "High-Level Message-Passing Constructs for Zipcode 1.0:
Design and Implementation"

* Robert Falgout, LLNL
Title: "Modeling Groundwater Flow on Massively Parallel Computers"

* Anthony Skjellum, MSU/NSF ERC
Title: "The Multicomputer Toolbox: Current and Future Directions,"

* William Gropp, Argonne National Laboratory
Title: TBD

* Ewing Lusk, Argonne National Laboratory
Title: "The MPI Communication Library: Its Design and a Portable

* Charles H. Still, LLNL
Title: "The Multicomputer Toolbox: Experiences with the Meiko CS-2".

* S. Lennart Johnsson, Harvard University and Thinking Machines Corp.
Title: "Scientific Libraries on Scalable Architectures"

* Linda Petzold, AHPCRC & UMN
Title: Solving Large-Scale Differential-Algebraic Systems via DASPK
on the CM5

* Dan Reed, UIUC
Title: TBD

* Dan Meiron, Caltech
Title: "Using Archetypes to Develop Scientific Parallel Applications"

* Eric F. Van de Velde, Caltech
Title: Stepwise Program Refinement in Concurrent Scientific Computing

* Padma Raghavan, UIUC/NCSA
Title: Parallel Solution of Linear Systems using Cholesky Factorization

* Anna Tsao, SRC
Title: "Performance of a Parallel Eigensolver Based on the Invariant
Subspace Decomposition for Dense Symmetric Matrices."

* Steve Lederman, SRC
Title: "Comparison of Scalable Parallel Matrix Multiply Libraries"

* Chuck Baldwin, UIUC
Title: "Dense and Iterative Concurrent Linear Algebra in the
Multicomputer Toolbox"

* Nikos Chrisochoides, NPAC Syracuse
Title: "An alternative to data-mapping for scalable iterative PDE
solvers : Parallel Grid Generation"

* Sanjay Ranka, NPAC Syracuse
Title: "Scalable Libraries for High Performance Fortran"

* Anthony Skjellum, MSU/NSF ERC
Title: "Building Parallel Libraries using MPI"


From: John Gregory <>
Date: Tue, 3 Aug 93 09:05:08 CDT
Subject: Call For Votes: sci.op-research

Perhaps most na-net readers have seen this announcement, but for
those few who have not, I send this copy of what appeared recently
on the Usenet group sci.math.num-analysis.

regarding the proposed creation of the unmoderated newsgroup sci.op-research

Votes must be received by 2359 GMT on Sunday 22nd August 1993.

This Call for Votes will also be sent to the following mailing lists:

The main purpose of this group is to act as the umbrella group from
which different operations research interest groups will branch off in
the future, as envisioned by the technology committee of the ORMS

In the interim, the newsgroup will support the RESEARCH, APPLICATION
and TEACHING of operations research through the exchange of information
through various activities including:
-- Posting information about accepted papers
-- Asking questions and posting summaries of replies
-- Posting Frequently Asked Questions (FAQ) and other lists such as
-Ajay Shah's list of Free C/C++ programs for numerical methods
-Arthur Geoffrion's list of mail reflectors relevant to O.R.
-John Gregory's FAQ on LP
Those interested in a particular area could bring out regular FAQs
answering questions or likely questions from those new to their
-- Posting information about ARCHIVES (e.g. those at Rutgers,
-- Sharing teaching approaches
-- Announcement of new textbooks; Discussion on existing textbooks
-- New product announcements
-- Users' impressions of commercial software (No advertisements.)
-- JOB announcements in universities and industry

The group will be UNMODERATED.

Newsgroups line:
sci.op-research Operational Research

Cut this out this ballot form, fill in your name and delete the
choices that aren't applicable. Send it to
If your software honours Reply-To lines you can just reply to this
message (please trim the message down, though, if you do it by
quoting; I don't need a copy of the whole CFV!).

My name is:
My vote on sci.op-research: yes/no/abstain
End of ballot

The line with your name is mandatory. Do not insert any blank lines.

Votes posted to newsgroups or mailed to anyone except the vote taker
will not be counted. Votes for a group with a different name to that
shown above will likewise not be counted.

You will receive an acknowledgement of your vote by email; A mass
acknowledgement will be posted during the voting period; at the end of
the voting period a complete list of votes will be published.

You may only vote once and only from one address, if you have several.
The end of the voting period is specified at the top of this CFV.

This vote is being conducted by Ian Jackson <>, a
neutral third party, with possible assistance from Ron Dippold

Please contact if you have any questions about the
voting procedure or about particular votes.

Please contact the proposer, Mohan Sodhi <>, if
you have questions about the proposed group.

Mohan Sodhi is grateful to the following, and to others, too numerous
to mention, for their constructive comments and encouragement: (In
alphabetical order by 1st name):
Arthur Geoffrion, Chris Bullen, Craig Willits, David Lawrence, James
Bean, Lester Ingber, Mark Moraes, Matthew Saltzmann, Ramesh Sharda,
Ron Dippold, Scott Huddleston, Taner Bilgic.

Ian Jackson ...!uknet!cam-orl!iwj
acting as vote taker for sci.op-research under the auspices of UVV.
These opinions are not necessarily those of Olivetti Research Ltd.


From: Bob Geller <>
Date: Wed, 4 Aug 93 22:14:23 JST
Subject: Postdoc Opportunity in Japan and Preprint Offer

Greetings from Japan!

The main purpose of this email is to inform you about the opportunities
available to be a post-doc in Japan, and to ask you to call this to
the attention of anyone who's interested. Naturally, I'd especially
like to have inquiries from people interested in working with our
group at Tokyo University.

The following papers give an idea of what we're now working on. If
you'd like a preprint let me know, and I'll be happy to mail one
to you. The following are our recent papers that are now in the

(1) Hara, T., S. Tsuboi and R. J. Geller (1993). Inversion for laterally
heterogeneous upper mantle S-wave velocity structure using iterative waveform
inversion, Geophys. J. Int., in press.

(2) Geller, R. J. and T. Hara (1993). Two efficient algorithms for iterative
linearized inversion of seismic waveform data,
Geophys. J. Int., in press.

(3) Geller, R. J. and T. Ohminato (1993). Computation of
synthetic seismograms and their partial derivatives for heterogeneous media
with arbitrary natural boundary conditions using the Direct Solution
Method, Geophys. J. Int., in press.

(4) Hara, T. and R. J. Geller (1993). Anamolously large near-field
Rayleigh waves excited by the 1992 Landers, California, earthquake,
Bull. Seism. Soc. Am., submitted.

Obviously I'd be most interested in hearing from people who'd like
to work on the above topics, or topics closely related thereto.
But this is by no means an absolute requirement.


The Japan Society for the Promotion of Science (JSPS) Postdocs pay Y270,000
per month (tax free) with an extra housing stipend of up to Y100,000 per
month. One round trip air ticket is also provided. There is a one-time
settling-in allowance of Y200,000, as well as other gooodies (family allowance,
health insurance, Japanese language training, research funds, etc.).

The current exchange rate is Y104= $1 US (approx).

Tenure of appointment is 12 months, with a possibility of renewal for up to
12 additional months.


Citizens of almost any country are eligible. In many cases you can apply
through your country's own agency (e.g., NSF in the case of the US)
rather than through me to the JSPS. But in any case you need to have
an agreement with the host researcher in Japan before proceeding further.


For additional information contact:
Bob Geller ( FAX +81-3-3818-3247
TEL +81-3-5800-6973
Address: Dept. of Earth&Planetary Physics, Faculty of Science,
Tokyo University, Bunkyo-ku, Tokyo 113 JAPAN

Please send publication list, c.v., names of three references (letters
are not necessary at this point), and reprints or preprints of three
recent papers.


Please hurry if you're interested. The deadline may be coming up soon.
(It varies from country to country).


Phil Cummins was a JSPS fellow in our lab from Feb. '92 to April '93. He's
kindly agreed to describe his experiences, and answer questions, for anyone
who's interested. He's now at ANU in Canberra. You can contact him at:


From: Karen Hahn <>
Date: Mon, 9 Aug 93 15:16:38 EDT
Subject: IMACS Workshop on Turbulence

First Announcement/ Call for Participation
IMACS WORKSHOP ON (theoretical and computational aspects of) TURBULENCE
Rutgers University, New Brunswick, NJ, USA / February 10-11, 1994

(organized by the IMACS Technical Committee
on Computational Fluid Dynamics and Rutgers University)

In order for CFD to get to new levels of usefulness to industry, there
is a need for better turbulence models for high Re flows. An honest
appraisal by industry on needs, and a similar word by academia and
researchers at national labs on what can be done now is probably
necessary. The workshop will deal with the numerical analysis and CFD
algorithms that could bridge the gap, and will include those aspects
of turbulence theory that have relevance to modeling used in
simulation. Topics to include: Modeling, Non-equilibrium
Turbulence, Sprectral and High Order Methods, Adaptive Grids,
Lagrangian Methods, Ocean Dynamics. It is currently planned to have
proceedings published after the workshop; preparing a written paper
for these proceedings is, however, optional.

Organizing Committee: Prof. R. Pelz (Rutgers University)
Dr. R. Agarwal (McDonnell Douglas Aerospace)

CALL FOR PAPERS/CALL FOR SESSIONS: Those wishing to contribute a paper,
or to organize a session are invited to make themselves known to the
Workshop Committee as soon as possible.

Mail to: IMACS Turbulence Workshop
c/o IMACS Secretariat
Department of Computer Science
Rutgers University e-mail:
New Brunswick, NJ 08903, USA fax: 908-932-0537


From: Richard Brualdi <>
Date: Wed, 4 Aug 1993 14:52:13 -0500 (CDT)
Subject: Contents, Linear Algebra and its Applications

Contents Volume 191

Song Xu (Beijing, People's Republic of China)
Notes on Sufficient Matrices 1

Irving S. Reed (Los Angeles, California)
Generalized de Moivre's Theorem, Quaternions, and Lorentz
Transformations on a Minkowski Space 15

David E. Stewart (St. Lucia, Queensland, Australia)
An Index Formula for Degenerate LCPs 41

Robert E. Hartwig (Raleigh, North Carolina)
The Pyramid Decomposition and Rank Minimization 53

Ali H. Sayed, Hanoch Lev-Ari, and Thomas Kailath
(Stanford, California)
Fast Triangular Factorization of the Sum of Quasi-Toeplitz
and Quasi-Hankel Matrices 77

Bernd Fritzsche, Stefan Fuchs, and Bernd Kirstein
(Leipzig, Bundesrepublik Deutschland)
Schur-Sequence Parametrizations of Potapov-Normalized
Full-Rank jpq-Elementary Factors 107

Han H. Cho (Seoul, Korea)
Regular Matrices in the Semigroup of Hall Matrices 151

Miroslav Fiedler (Prague, Czech Republic) and Thomas L. Markham
(Columbia, South Carolina)
Quasidirect Addition of Matrices and Generalized Inverses 165

B. Najman (Zagreb, Bijenicka, Croatia) and Q. Ye (Winnipeg,
Manitoba, Canada)
A Minimax Characterization for Eigenvalues of Hermitian Pencils. II 183

Jiu Ding (Hattiesburg, Mississippi)
Perturbation Analysis for the Projection of a Point to an Affine Set 199

M. T. Alcalde, C. Burgueno (Casilla, Temuco, Chile) and
C. Mallol (Montpellier, France)
Les Pol(n, m)-Algebres: Indentites Polynomiales Symetriques
dans des Algebres 213

S. Gonzalez and C. Martinez (Zaragoza, Spain)
Bernstein Algebras With Zero Derivation Algebra 235

Eugene Seneta (Sydney, Australia)
Explicit Forms for Ergodicity Coefficients of Stochastic Matrices 245

Author Index 253


End of NA Digest