![4 H. Neunzert
technology only recently? Mathematics became a technology when it re-
ceived a tool to evaluate complex, “near to reality” models, and that tool
was the computer. The model may be quite old. Navier–Stokes equations
describe flow behavior rather well, but to solve these equations for realistic
geometry and higher Reynolds numbers with sufficient precision, is even
for powerful parallel computing, a real challenge. Make the models as sim-
ple as possible, as complex as necessary and then evaluate them with the
help of efficient and reliable algorithms. These are genuine mathematical
tasks.
Science is designed to “understand” natural phenomena; scientific tech-
nology extends the domain of the validity of scientific theories to not yet
existing systems. We create a new, virtual world in which we may change
and optimize much easier and quicker than in the real world. Even that is
rather old. Some scholars of ancient science [9] and some philosophers [10]
consider this interplay of science and technology as crucial for the birth
of science during the Hellenistic period around 300 BC (with names like
Euclid or Archimedes on top). But now, since we may mathematically
optimize very complex virtual systems, we are able to use mathematics in
order to design better machines, to minimize the risk of financial actions,
and to plan optimal surgery.
This is the reason why mathematics has become a key technology. The
following technology fields emerged as crucial from several investigations
• Simulation of Processes and Products
• Optimization, Control, and Design
• Uncertainty and Risk
• Management and Exploitation of Data
• Virtual Material Design
• Biotechnology, Food, and Health
With the help of these road maps which contain examples and chal-
lenges for future mathematics gathered from all over Europe, European
mathematicians shall try to influence national and international research
policies in a way that may help mathematics get the weight in future pro-
grams which it has in reality already now. Mathematics was too long in
© 2006 by Taylor & Francis Group, LLC
in Europe (see [2, 6]).](https://image.slidesharecdn.com/90452-250227101950-91a84f3a/85/Mathematical-Models-and-Methods-for-Real-World-Systems-1st-Edition-K-M-Furati-21-320.jpg)
![Mathematics as a technology 5
an ivory tower, often used only as brain exercises for students. It needs
some time and a lot of effort to catch public awareness of its new role.
In this chapter I shall show examples from different technology fields
mentioned above, examples gained from our experience in the Fraun-
hofer Institute ITWM at Kaiserslautern. It was founded in 1996 and
became a member of the Fraunhofer-Gesellschaft in 2001; the Fraunhofer-
Gesellschaft is the leading German association for applied research with
altogether 12,000 employees in ca. 60 institutes, an annual turnover of ca.
1.2 billion euro and branches in the US and in some European countries.
Its decisive feature is that basic funding is given proportional to what is
earned in industry. To make a rather complicated story simple, a Fraun-
hofer Institute gets 40 cents from the federal government for each euro it
earns in industry. “No industrial project - no money at all and 40 % on top
in order to do fundamental research related to projects”–these are the two
rules which in my opinion are unique and uniquely successful worldwide.
ITWM has proved that mathematics as a technology is strong enough
to follow the Fraunhofer rules. Not only that, at present it is the most
successful institute of all the 15 Fraunhofer Institutes dealing with infor-
mation technologies. The reason is that it has a huge market, much wider
than any computer science institute. The disadvantage is that the market
doesn’t yet know it. The consequence is that there is a lot of space for all
other really applied mathematicians and for cooperation worldwide.
But now I want to become more substantial. Here are the technology
2 Simulation of Processes and the Behavior
of Products
Simulation means modelling-computing-visualizing. To find the right model
for the behavior of car components, as simple as possible and as compli-
cated as necessary, is, for example, a task for asymptotic analysis: identify
small parameters in very complex models, study the behavior for these
parameters tending to zero, and estimate the error using this “parameter
= 0 - model”. All this is tricky perturbation theory, sometimes advanced
functional analysis. But we should never “oversimplify” in order to get
an analytically treatable model; very often numerics will be necessary, and
very often advanced numerical ideas are necessary. Since a realistic geome-
try is sometimes very complex (think of a porous medium in a microscopic
© 2006 by Taylor & Francis Group, LLC
themes with examples and challenges, see references [1, 4].](https://image.slidesharecdn.com/90452-250227101950-91a84f3a/85/Mathematical-Models-and-Methods-for-Real-World-Systems-1st-Edition-K-M-Furati-22-320.jpg)
![6 H. Neunzert
view), we need, for example, new, gridfree algorithms efficiently imple-
mented for parallel systems. And finally, long lists of numbers as a result
of solving a PDE are completely useless-we have to interpret the results
in terms of the original questions, and quite often we have to visualize the
results as images or movies.
Simulation is now routinely used in many parts of industry all over the
world to support or to replace experimentation. “It can have a dramatic
effect on the design process, reducing the need for costly prototypes and
increasing the speed with which new products can be brought to market
[1].
There are industries where simulation has a long tradition, like aerospace
or automotive industries or in oil and gas prospection. In these areas,
commercial software is available and often easy to handle and efficient.
It is (at least for a Fraunhofer Institute) a very hard or even impossible
task to place a new algorithm to substitute this kind of software, even if
this algorithm is really better than the other one. What is possible for
mathematicians is to substitute some modules in software products, as,
for example, the second mathematical Fraunhofer Institute SCAI does in
offering an “Algebraic Multigrid Solver” for linear systems. Another pos-
sibility is postprocessing algorithms enabling the user to do an “optimal
experimental design” for virtual or “numerical experiments”. Industries
operating with more basic technologies such as textiles, glass, or even met-
als just begin to use simulation. The market for commercial software seems
too small, and tailor-made software is needed. How complicated this field
could be will now be shown by our experience with the glass industry.
ITWM has a 10-year close cooperation with Schott Glas at Mainz, where
cooperation may be taken literally. The enormous knowledge of Schott sci-
entists about materials and processes joins mathematical ideas in ITWM
to find innovative solutions. (The material was provided to me by Norbert
Siedow from ITWM; some parts and literature are described in the ITWM
annual report 2003, page 26 ff.)
from the glass tank with
molten glass of a temperature over 1000◦
C through a pipe to a kind of drop
called gob; in this process we identified 4 mathematical tasks which are
denoted by colors. Two are so-called “inverse problems” that measure the
temperature in the interior of the glass flow from radiation and optimize
the shape of the flanges carrying the pipe such that a given homogeneous
temperature is created through electrical currents. The shape of the gob,
a very viscous drop of liquid glass, has to be calculated by CFD codes able
© 2006 by Taylor & Francis Group, LLC
Figure 1 shows the glass making process,](https://image.slidesharecdn.com/90452-250227101950-91a84f3a/85/Mathematical-Models-and-Methods-for-Real-World-Systems-1st-Edition-K-M-Furati-23-320.jpg)
![Mathematics as a technology 7
to handle free surfaces very well.
Figure 1: Mathematical Problems in Glass Industry (Glassmaking)
panels ask for the simulation of radiation. In semi-transparent media,
this is a very elaborate task, since the radiation equation is a dimensional
integro-differential equation with enormous computational efforts.
Floatglass, an efficient
production process invented by Pilkington, shows sometimes wavy patterns
which have to be avoided. Whether these waves are instabilities created
in a modification of the Orr-Sommerfeld equations is the subject of an
ongoing PhD work. Glass fiber productions are extremely tricky processes
in which the fibers interact with the air around them. Turbulent flow-
fiber interaction is a topic where turbulence models are not enough, but
stochastic differential equations are crucial.
the cooling of glass. I would like to mention that already around 1800
Fraunhofer who gave the name to our society produced lenses and had
problems with the thermal tensions and the defects created by them.
Many of the problems here are “inverse problems” connected with heat
transfer, and they are very ill-posed. Inverse problems may be counted
under “optimization”; it is the combination of optimization and simulation
as in inverse problems, optimal shape design, etc. which creates many
mathematical challenges.
© 2006 by Taylor & Francis Group, LLC
Figures 2–5 show different kinds of glass processing, Pressing of TV
Figure 3 shows classical glass processing and problems connected with
One uses tricky scale asymptotics (see [8]).](https://image.slidesharecdn.com/90452-250227101950-91a84f3a/85/Mathematical-Models-and-Methods-for-Real-World-Systems-1st-Edition-K-M-Furati-24-320.jpg)
![8 H. Neunzert
Figure 2: Mathematical Problems in Glass Industry (Glassprocessing
I)
details of gob forming.
The hot glass leaves the feeder when the needle opens. A drop (gob) is
formed and cut off by a special cutter. J. Kuhnert (ITWM) has designed
a gridfree numerical method to calculate the glass flow. It is called the
“Finite Pointset Method” (FPM) and may be considered as an extension
of “Smoothed Particle Hydrodynamics” (SPH) [11]. Particles are moving
in the computational domain, carrying information about density, veloc-
ity, temperature, etc. This information has to be extrapolated to other
positions so that derivatives of these quantities as the Laplacian of the
velocity components, the temperature gradient, etc., can be calculated.
These extrapolations are denoted by a tilde, and the rest is Lagrangean
formalism.
The method is appropriate for fluids with free boundaries, changing
even the topology, as it happens, when the gob is cut off.
A more analytical task is the question of waves at floatglass surfaces.
Here is the industrial question: What is the origin of waves at the interface
of glass and molten tin (the glass flows over molten tin, a classical 2-phase
flow with quite different temperatures)? These waves are small defects
which should be removed. What are the causes?
Let us finish the glass field by describing a very nice, very ill-posed
© 2006 by Taylor & Francis Group, LLC
Let us have a closer look at a few of the problems. Figure 4 shows the](https://image.slidesharecdn.com/90452-250227101950-91a84f3a/85/Mathematical-Models-and-Methods-for-Real-World-Systems-1st-Edition-K-M-Furati-25-320.jpg)
![Mathematics as a technology 19
3 Optimization, Control, and Design
What we finally want to achieve in our man-made world are optimal so-
lutions: the process should be as cheap and as fast as possible, and the
product should at least behave better than the products of the competitors.
(Even nature seems to have a creator who is interested in optimality. That
is why we have so many variational principles, and that is why animals and
plants show us so many tricky solutions for their “technical” problems to
be as stable, as light, as smoothly moving as possible and necessary. This
is called “bionics” and there may be an interesting interplay between opti-
mization by mathematics and optimization by evolution.) “So rather then
asking how a product performs, the question is, how should the product
be designed in order to perform in a specified way. Scheduling, planning
and logistics also fall within that area of optimization. Optimal control
is used to provide real-time control of an industrial process or a product,
such as a plane or a car, in response to current operating conditions. A
related area is that of inverse problems, where the parameters (or even the
structure) of a model must be estimated from measurement of the system
output) [1].
We have mentioned inverse problems already in (1); they appear liter-
ally everywhere. We will show two examples from our projects at ITWM;
however they are very short.
There is the wide field of topological shape optimization; “topological”
means that one may change the topology of a structure, for example, by
admitting holes. One has to minimize an objective function (maximal
stress, mean compliance, etc.) with respect to the shape.
As an example for a multicriteria optimization, we consider a project
of [5].
How should we optimally control the radiation in cancer therapy such
that the cancer cells are destroyed as much as possible, but at the same
time organs or important healthy parts of the body remain undamaged.
There are, besides optimization, a lot of simulation problems? f. e. to
simulate how radiation penetrates the body, but let’s concentrate on opti-
mization assuming that the transmission of the radiation to different parts
of the body given the external source, which can be controlled, is known.
The goal is that a medical doctor can operate with the optimization tool,
allowing more or less radiation to certain organs by “pulling” in the cor-
responding direction of a navigation scheme; the program then computes
the different doses of different sources and different directions, getting at
© 2006 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/90452-250227101950-91a84f3a/85/Mathematical-Models-and-Methods-for-Real-World-Systems-1st-Edition-K-M-Furati-36-320.jpg)
![24 H. Neunzert
4 Uncertainty and Risk
Many processes in nature, in economy, and even in daily life are or seem
to be strongly accidental; we therefore need a stochastic theory in order
to model these processes. Randomness creates uncertainty, and uncer-
tainty creates risk, for example, in decisions about investments, about
medication, and about security of technical systems like planes or power
plants. Whether this randomness is genuine or just a consequence of high
complexity is a philosophical question which does not influence stochastic
modeling. You will find very complex systems in catastrophes like earth-
quakes or floods; biological systems, for example are extremely complex
the human body. Experiments are not possible, and simulation is therefore
highly necessary, but very difficult, too.
Also in economy, experiments are impossible, but one needs help for
decisions which minimize the risk.
The law of large numbers leads often to models which are deterministic
PDEs and very similar to deterministic models in natural sciences. But at a
closer look they are even more complex, for example, very high dimensional
(the independent variables are not geometric, but may be the values of
different stocks). Therefore, even if we get at the end a treatable PDE,
we have to use Monte-Carlo methods to solve them approximately, and we
are back to stochastic differential equations. Now quite often derivatives of
these solutions with respect to variables and parameters are needed, and
to differentiate a function given by a Monte-Carlo method is not always
successful.
invented for practical problems, is a great help [3]. Here is an example
from option prizing.
Of course, there are other uncertainties and risks such as in floods and
earthquakes. In technical systems, very different methods are involved.
5 Management and Exploitation of Data
We are flooded by data which, if structured, create information and finally
knowledge. The extraction of this information or knowledge from data
is called “data mining”. Data may be given as signals or images; if we
want to discover patterns, and if we want to “understand” these signals
or images, we need image processing and pattern recognition methods. If
we want to study and predict input-output systems for which we do not
have enough theory (simple models) but many observations from the past,
© 2006 by Taylor & Francis Group, LLC
The Malliavin calculus shown in Figures 28–31, initially not](https://image.slidesharecdn.com/90452-250227101950-91a84f3a/85/Mathematical-Models-and-Methods-for-Real-World-Systems-1st-Edition-K-M-Furati-41-320.jpg)
![Mathematics as a technology 29
Figure 34: Simulation of 3-D flow through oil filters
tem which is a combination of incompressible, steady Navier-Stokes with
a Darcy term.
The interface condition describing the behavior of the fluid on the sur-
face of the filter material is a rather delicate issue, but in this model (with
© 2006 by Taylor & Francis Group, LLC
Brinkman homogenization) it is easier to handle (see the Ph.D. thesis by
Laptev [7]). The flow field is given below.](https://image.slidesharecdn.com/90452-250227101950-91a84f3a/85/Mathematical-Models-and-Methods-for-Real-World-Systems-1st-Edition-K-M-Furati-46-320.jpg)
![Mathematics as a technology 31
Figure 37: Simulation of 3-D flow through oil filters
The correspondence with measurements (where the pressure loss for
different Reynold numbers at different temperatures with correspondingly
different permeabilities) is remarkable.
I call this correspondence sometimes “prestabilized harmony”: a rather
crude model which is numerically approximated and gives results which
correspond with nature to an extent which one really might not expect.
But, of course, care is necessary. Models have their range of applicability,
and their limitations should be carefully respected.
To compute the flow field of a filter is not enough to understand its
efficiency. The transport of the particles, which have to be filtered out,
must be simulated. Therefore, we have to model their absorption by the
fibers of the filter and the motion of the particles by the fluid velocity, its
friction, and the influence of diffusion. Finally, the absorption is, of course,
filter and particle dependent. This is an area of exciting modelling (see,
7 Biotechnology, Food, and Health
This field has created new research areas which are rather interdisciplinary,
for example, bio-informatics or system biology. Statistics, discrete math-
ematics, computer science and system and control theory, data mining,
© 2006 by Taylor & Francis Group, LLC
for example [8]).](https://image.slidesharecdn.com/90452-250227101950-91a84f3a/85/Mathematical-Models-and-Methods-for-Real-World-Systems-1st-Edition-K-M-Furati-48-320.jpg)
![Mathematics as a technology 37
References
[1] A. Cliff, R. Matheij, and H. Neunzert, Mathematics: Key to the european
knowledge based economy, in MACSI-Net Roadmap for Mathematics in
European Industry, Edited by A. Cliffe, B. Matheij, and H. Neunzert, A
project of European commission, Mark 2004.
[2] A. Cliffe, B. Matheij, and H. Neunzert (Eds), A project of European
commission, MACSI-Net Roadmap for Mathematics in European Indus-
try, Mark 2004,
[3] Fournier et al., Application of Malliavin calculus to Monte-Carlo methods
in finance, Finance and Stochastics 3(4), 1999.
[4] Fraunhofer Institute for Industrial Mathematics, Kaiserslauntern, Ger-
[5] H. Neunzert, N. Siedow, and F. Zingsheion, Simulation temperature be-
havior of hot glass during cooling, In, Mathematical Modeling, Edited by
E. Cumberbatch and A. Fitt, Cambridge University Press, 2001.
[6] H. Neunzert and U. Trottenberg (Eds), Mathematik als Technologie, Die
Fraunhofer–Institute ITWM und SCAI, to appear.
[7] V. Laptev, Numerical Solution of Complex Flow in Plain and Porous Me-
dia, Dissertation, Department of Mathematics, Technical University of
Kaiserlauntern, Germany, 2004.
[8] A. Latz and A. Wiegmann, Simulation of fluid particle simulation in realis-
tic 3-dimensional fiber structures, in Proceedings Filtech Europa, I-353-360,
2003.
[9] L. Russo, The Forgotten Revolution, Springer-Verlag, Heidelberg, 2004.
[10] M. Scheler, Soziologie des Wissens, in Die Wis-sensformen und die
Gesellschaft, Francke-Verlag, 1960.
[11] S. Tiwari and J. Kuhnert, A numerical scheme for solving incompress-
ible and law mach number flows by finite pointset methods, in, Meshfree
Methods for Partial Differential Equations, Edited by M. Griebel and M.
Schwertzer, Volume 2, Springer-Verlag, Berlin, 2004.
© 2006 by Taylor & Francis Group, LLC
many, Annual report 2003. (info@itwm.fraunhofer.de).](https://image.slidesharecdn.com/90452-250227101950-91a84f3a/85/Mathematical-Models-and-Methods-for-Real-World-Systems-1st-Edition-K-M-Furati-54-320.jpg)
![hands and knees and remain so until he should return. He never
returned, and to this day the poor doomed grizzlies go on all fours.
A wonderful feat of jugglery, but a greater was that of the Olympian
goddess who changed the beautiful maiden Callisto into a bear,
which Jupiter set in the heavens, and where she is to be seen every
night, beside her son the Little Bear.
The angry Manitou turned his strange grandchildren out of doors,
fastened the door and carried his daughter away to his own
wigwam.
The Indians to this day believe that a bear can talk if you will only sit
still and listen to him. The Indians will not harm a bear. Now for the
meaning of those queer little piles of stones one sees so frequently
in the Shasta mountains. If an Indian is killed by a bear he is burned
on the spot where he fell. Every Indian who passes that way will
fling a stone at the fated place to dispel the charm that hangs over
it.
“All that wide and savage water-shed of the Sacramento tributaries
to the south and west of Mt. Shasta affords good bear hunting at
almost any season of the year—if you care to take the risks. But he
is a velvet-footed fellow, and often when and where you expect
peace you will find a grizzly. Quite often when and where you think
that you are alone, just when you begin to be certain that there is
not a single grizzly bear in the mountains, when you begin to
breathe the musky perfume of Mother Nature as she shapes out the
twilight stars in her hair, and you start homeward, there stands your
long lost bear in your path! And your bear stands up! And your hair
stands up! And you wish you had not lost him! And you wish you
had not found him! And you start for home! And you go the other
way glad, glad to the heart if he does not come tearing after you.”[1]
Downward from Mt. Shasta flows the Sacramento river. For thirty
miles it goes tumbling over bowlders and granite ledges on its way
to the sea. In mid-summer the Sacramento cañon is a paradise of
umbrageous beauty, a region of forest and groves, of leafy shrubs,](https://image.slidesharecdn.com/90452-250227101950-91a84f3a/85/Mathematical-Models-and-Methods-for-Real-World-Systems-1st-Edition-K-M-Furati-77-320.jpg)
Mathematical Models and Methods for Real World Systems 1st Edition K.M. Furati
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- 6. M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology S. Kobayashi University of California, Berkeley Marvin Marcus University of California, Santa Barbara W. S. Massey Yale University Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University David L. Russell Virginia Polytechnic Institute and State University Walter Schempp Universität Siegen Mark Teply University of Wisconsin, Milwaukee PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EDITORIAL BOARD EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey Zuhair Nashed University of Central Florida Orlando, Florida © 2006 by Taylor & Francis Group, LLC
- 7. MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS Recent Titles J. R. Weeks, The Shape of Space, Second Edition (2002) M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces (2002) V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Second Edition (2002) T. Albu, Cogalois Theory (2003) A. Bezdek, Discrete Geometry (2003) M. J. Corless and A. E. Frazho, Linear Systems and Control: An Operator Perspective (2003) I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions (2003) G. V. Demidenko and S. V. Uspenskii, Partial Differential Equations and Systems Not Solvable with Respect to the Highest-Order Derivative (2003) A. Kelarev, Graph Algebras and Automata (2003) A. H. Siddiqi, Applied Functional Analysis: Numerical Methods, Wavelet Methods, and Image Processing (2004) F. W. Steutel and K. van Harn, Infinite Divisibility of Probability Distributions on the Real Line (2004) G. S. Ladde and M. Sambandham, Stochastic versus Deterministic Systems of Differential Equations (2004) B. J. Gardner and R. Wiegandt, Radical Theory of Rings (2004) J. Haluska, The Mathematical Theory of Tone Systems (2004) C. Menini and F. Van Oystaeyen, Abstract Algebra: A Comprehensive Treatment (2004) E. Hansen and G. W. Walster, Global Optimization Using Interval Analysis, Second Edition, Revised and Expanded (2004) M. M. Rao, Measure Theory and Integration, Second Edition, Revised and Expanded (2004) W. J. Wickless, A First Graduate Course in Abstract Algebra (2004) R. P. Agarwal, M. Bohner, and W-T Li, Nonoscillation and Oscillation Theory for Functional Differential Equations (2004) J. Galambos and I. Simonelli, Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions (2004) Walter Ferrer and Alvaro Rittatore, Actions and Invariants of Algebraic Groups (2005) Christof Eck, Jiri Jarusek, and Miroslav Krbec, Unilateral Contact Problems:Variational Methods and Existence Theorems (2005) M. M. Rao, Conditional Measures and Applications, Second Edition (2005) K. M. Furati, Zuhair Nashed, and Abul Hasan Siddiqi, Mathematical Models and Methods for Real World Systems (2005) © 2006 by Taylor & Francis Group, LLC
- 8. Boca Raton London New York Singapore K. M. Furati King Fahd University of Petroleum & Minerals Dhahran, Saudi Arabia Zuhair Nashed University of Central Florida Orlando, Florida, USA Abul Hasan Siddiqi King Fahd University of Petroleum & Minerals Dhahran, Saudi Arabia Mathematical Models and Methods for Real World Systems © 2006 by Taylor & Francis Group, LLC
- 9. Published in 2006 by Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8493-3743-7 (Hardcover) International Standard Book Number-13: 978-0-8493-3743-7 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Catalog record is available from the Library of Congress Visit the Taylor & Francis Web site at and the CRC Press Web site at Taylor & Francis Group is the Academic Division of T&F Informa plc. DK6028_Discl.fm Page 1 Thursday, June 2, 2005 11:54 AM © 2006 by Taylor & Francis Group, LLC (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, http://www.taylorandfrancis.com http://www.crcpress.com For permission to photocopy or use material electronically from this work, please access www.copyright.com
- 10. CONTENTS Preface xi Contributing Authors xiii Part I Mathematics for Technology Chapter 1 3 Mathematics as a Technology – Challenges for the Next Ten Years H. Neunzert Chapter 2 39 Industrial Mathematics – What Is It? N. G. Barton Chapter 3 47 Mathematical Models and Algorithms for Type-II Superconductors K. M. Furati and A. H. Siddiqi Part II Wavelet Methods for Real-World Problems Chapter 4 73 Wavelet Frames and Multiresolution Analysis O. Christensen Chapter 5 107 Comparison of a Wavelet-Galerkin Procedure with a Crank-Nicolson-Galerkin Procedure for the Diffusion Equation Subject to the Specification of Mass S. H. Behiry, J. R. Cannon, H. Hashish, and A. I. Zayed vii © 2006 by Taylor & Francis Group, LLC
- 11. Chapter 6 125 Trends in Wavelet Applications K. M. Furati, P. Manchanda, M. K. Ahmad, and A. H. Siddiqi Chapter 7 179 Wavelet Methods for Indian Rainfall Data J. Kumar, P. Manchanda, and N. A. Sontakke Chapter 8 211 Wavelet Analysis of Tropospheric and Lower Stratospheric Gravity Waves O. Oğuz, Z. Can, Z. Aslan, and A. H. Siddiqi Chapter 9 225 Advanced Data Processes of Some Meteorological Parameters A. Tokgozlu and Z. Aslan Chapter 10 245 Wavelet Methods for Seismic Data Analysis and Processing F. M. Khène Part III Classical and Fractal Methods for Physical Problems Chapter 11 273 Gradient Catastrophe in Heat Propagation with Second Sound S. A. Messaoudi and A. S. Al Shehri Chapter 12 283 Acoustic Waves in a Perturbed Layered Ocean F. D. Zaman and A. M. Al-Marzoug Chapter 13 301 Non-Linear Planar Oscillation of a Satellite Leading to Chaos under the Influence of Third-Body Torque R. Bhardwaj and R. Tuli viii © 2006 by Taylor & Francis Group, LLC
- 12. Chapter 14 337 Chaos Using MATLAB in the Motion of a Satellite under the Influence of Magnetic Torque R. Bhardwaj and P. Kaur Chapter 15 373 A New Analysis Approach to Porous Media Texture – Mathematical Tools for Signal Analysis in a Context of Increasing Complexity F. Nekka and J. Li Part IV Trends in Variational Methods Chapter 16 389 A Convex Objective Functional for Elliptic Inverse Problems M. S. Gockenbach and A. A. Khan Chapter 17 421 The Solutions of BBGKY Hierarchy of Quantum Kinetic Equations for Dense Systems M. Yu. Rasulova, A. H. Siddiqi, U. Avazov, and M. Rahmatullaev Chapter 18 429 Convergence and the Optimal Choice of the Relation Parameter for a Class of Iterative Methods M. A. El-Gebeily and M. B. M. Elgindi Chapter 19 443 On a Special Class of Sweeping Process M. Brokate and P. Manchanda ix © 2006 by Taylor & Francis Group, LLC
- 14. PREFACE The International Congress of Industrial and Applied Mathematics is organized at 4-year intervals under the auspices of the International Coun- cil of Industrial and Applied Mathematics (ICIAM). The ICIAM com- prises 16 national societies: ANIAM (Australian and New Zealand Indus- trial and Applied Mathematics), CAIMS (Canada Applied and Industrial Mathematics Society), CSIA (Chinese Society for Industrial and Applied Mathematics), ECMI (European Consortium for Mathematics in Indus- try), ESMTB (Eupropean Society for Mathematics and Theoretical Biol- ogy), GAMM (Gescllschaft fur Angewandte Mathematik und Mechanike), IMA (Institute for Mathematics and Applications), ISIAM (Indian Soci- ety for Industrial and Applied Mathematics) JSIAM (Japan Society for Industrial and Applied Mathematics), Nortim (Nordiska Foreningen for Tillampad och Industriell Mathematik), SBMAC (Sociedade Brasiliera de Matematika Aplicade Computacional), SEMA (Sociedal Espanola de Matematica Applicada), SIMAI (Societa’ Italiana di’ Matematica, Appli- cata e Industiale), SMAI (Societa de Mathematiques Appliquees et In- dustrielles), SIAM (Society for Industrial and Applied Mathematics), and VSAM (Vietnamese Society for Applications of Mathematics). The objec- tive of the national societies of ICIAM is similar. EMS (European Math- ematical Society), LMS (London Mathematical Society), and SMS (Swiss Mathematical Society) are its associate members. The First Congress of Industrial and Applied Mathematics was held in Paris (1987), the second in Washington (1991), the third in Hamburg (1995), and the fourth in Ed- inburgh (1999). The sixth is scheduled to be held in Zurich (2007). It is the premier organization in the world for promoting teaching and research of applications of mathematics in diverse fields. Mini-symposiums are very important activities of such congresses. The member societies and distin- guished workers of different areas are requested to submit proposals which are accepted after an appropriate reviewing process. In recent years, all knowledgeable and responsible mathematicians are arguing vehemently for establishing linkage between mathematics and the physical world (besides many, we refer to professor Phillipe A. Griffiths’ ad- dress “Trends for Science and Mathematics in 21st Century” (the inaugural function of an event of the WMY2000 in Cairo), and Professor Tony F. xi © 2006 by Taylor & Francis Group, LLC
- 15. Chan’s article “The Mathematics Doctorate: A Time for Change” (Notices AMS, Sept. 2003)). Now, it is the general belief that mathematics cannot prosper in isolation. This book is an attempt to strengthen the linkages between mathematical sciences and other disciplines such as superconduc- tors (an emerging area of science, technology, and industry), data analysis of environmental studies, and chaos. It also contains some valuable results concerning variational methods, fractal analysis, heat propagation, and multiresolution analysis having potentiality of applications. The first two chapters are written by two distinguished industrial and applied mathematicians, Professor Dr. Helmut Neunzert, a distinguished industrial mathematician and the founding director of the prestigious In- stitute of Industrial Mathematics in Germany, and Dr. Noel G. Barton, Director of the Sydney Congress. This book comprises chapters by those who were invited to the mini- symposium in three parts on Mathematics of Real-World Problems. It is divided into four parts: Mathematics for Technology, Wavelet Meth- ods for Real-World Problems, Classical and Fractal Methods for Physical Problems, and Trends in Variational Methods. S.H. Behiry et al., K.M. Furati et al., J. Kumar et al., O. Oğuz et al., A. Tokgozlu and Z. Aslan, and F.M. Khène. chapters by M.A. Messaoudi and A.S. Al Shehri, F.D. Zaman and A.M. Al-Marzoug, R. Bhardwaj and R. Tuli, R. Bhardwaj and P. Kaur, and A.A. Khan, M.Yu. Rasulova et al., M.A. El-Gebeily and M.B.M. Elgindi, and M. Brokate and P. Manchanda. This book will be welcomed by all those having interest in acquiring knowledge of contemporary applicable analysis and its application to real-world problems. The class of specialists who may have keen interest in the subject mat- ter of this book is quite large as it includes mathematicians, meteorologists, engineers, and physicists. Khaled M. Furati and A.H. Siddiqi would like to thank the King Fahd University of Petroleum & Minerals for providing financial assistance to attend the 5th ICIAM at Sydney. The help of Dr. P. Manchanda and Dr. Q. H. Ansari is acknowledged. K. M. Furati, M. Z. Nashed, and A. H. Siddiqi xii © 2006 by Taylor & Francis Group, LLC Part I contains chapters by H. Neunzert, N.G. Barton, and K.M. Furati and A.H. Siddiqi. Part II is based on the contributions of O. Christensen, Part III is devoted to the F. Nekka and J. Li. Part IV comprises chapters of M.S. Gockenbach and
- 16. CONTRIBUTING AUTHORS 1. M. K. Ahmad, Department of Mathematics, Aligarh Muslim Uni- versity, Aligarh 202002, India 2. Z. Aslan, Department of Mathematics and Computing, Beykent University, Faculty of Science and Letters, İstanbul, Turkey; and Faculty of Engineering and Design, İstanbul Commerce University, Istanbul 34672, Turkey 3. U. Avazov, The Institute of Nuclear Physics, Ulughbek, Tashkent 702132, Uzbekistan 4. N. G. Barton, Sunoba Renewable Energy Systems, P.O. Box 1295, North Ryde BC, NSW 1670, Australia 5. S. H. Behiry, Department of Mathematics and Physics, Faculty of Engineering, Mansoura University, Mansoura, Egypt 6. R. Bhardwaj, Department of Mathematics, School of Basic and Ap- plied Sciences, Guru Gobind Singh Indraprastha University, Kash- mere Gate, Delhi 110006, India 7. M. Brokate, Institute of Applied Mathematics, Technical Univer- sity of Munich, Munich, Germany 8. Z. Can, Department of Physics, Yildiz Technical University, Faculty of Science and Letters, İstanbul, Turkey 9. J. R. Cannon, Department of Mathematics, University of Central Florida, Orlando, FL 32816 10. O. Christensen, Department of Mathematics, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark 11. M. B. M. Elgindi, Department of Mathematics, University of Wisc- onsin–Eau Claire, Eau Claire, WI 54702-4004 12. K. M. Furati, Mathematical Sciences Department, King Fahd Uni- versity of Petroleum & Minerals, Dhahran 31261, Saudi Arabia xiii © 2006 by Taylor & Francis Group, LLC
- 17. 13. M. A. El-Gebeily, Mathematical Sciences Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia 14. M. S. Gockenbach, Department of Mathematical Sciences, 319 Fisher Hall, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931-1295 15. H. Hashish, Department of Mathematics and Physics, Faculty of Engineering, Mansoura University, Mansoura, Egypt 16. P. Kaur, Department of Mathematics, School of Basic and Ap- plied Sciences, Guru Gobind Singh Indraprastha University, Kash- mere Gate, Delhi 110006, India 17. A. A. Khan, Department of Mathematical Sciences, 319 Fisher Hall, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931-1295 18. F. M. Khène, Research Institute, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia 19. J. Kumar, Department of Mathematics, Gurunanak Dev University, Amritsar 143005, India 20. J. Li, 1 - Faculté de Pharmacie, 2 - Centre de Recherches Mathémati- ques, Université de Montréal, C.P. 6128, Succ. Centre-ville, Montréal, Québec, Canada H3C 3J7 21. P. Manchanda, Department of Mathematics, Gurunanak Dev Uni- versity, Amritsar 143005, India 22. A. M. Al-Marzoug, Saudi Aramco, Dhahran 31311, Saudi Arabia 23. S. A. Messaoudi, Mathematical Sciences Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia 24. F. Nekka, 1 - Faculté de Pharmacie, 2 - Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128, Succ. Centre- ville, Montréal, Québec, Canada H3C 3J7 25. H. Neunzert, Fraunhofer Institute for Industrial Mathematics, Kai- serslautern, Germany 26. O. Oğuz, İstanbul Commerce University, Faculty of Engineering and Design, İstanbul, Turkey 27. M. Rahmatullaev, The Institute of Nuclear Physics, Ulughbek 702132, Tashkent xiv © 2006 by Taylor & Francis Group, LLC
- 18. 28. M. Yu. Rasulova, The Institute of Nuclear Physics, Ulughbek 702132, Tashkent 29. A. S. Al Shehri, Mathematics Department, School of Sciences, Girl’s College, Dammam, Saudi Arabia 30. A. H. Siddiqi, Mathematical Sciences Department, King Fahd Uni- versity of Petroleum & Minerals, Dhahran 31261, Saudi Arabia 31. N. A. Sontakke, Indian Institute of Tropical Meteorology, Dr. Homi Bhabha Road, Pashan, Pune 411008, India 32. A. Tokgozlu, Department of Geography, Faculty of Science and Letters, Süleyman Demirel University, Isparta 32260, Turkey 33. R. Tuli, Department of Mathematics, School of Basic and Applied Sciences, Guru Gobind Singh Indraprastha University, Kashmere Gate, Delhi 110006, India 34. F. D. Zaman, Mathematical Sciences Department, King Fahd Uni- versity of Petroleum & Minerals, Dhahran 31261, Saudi Arabia 35. A. I. Zayed, Department of Mathematical Sciences, DePaul Uni- versity, Chicago, IL 60614 xv © 2006 by Taylor & Francis Group, LLC
- 19. Part I Mathematics for Technology © 2006 by Taylor & Francis Group, LLC
- 20. Chapter 1 MATHEMATICS AS A TECHNOLOGY– CHALLENGES FOR THE NEXT TEN YEARS H. Neunzert Fraunhofer Institute for Industrial Mathematics Abstract The main focus of this chapter is the interlinking of mathemat- ical models and methods to real-world systems. Six areas of technological themes which have emerged as crucial from inten- sive investigation in Europe, namely, Simulation of Processes and Products; Optimization, Control, and Design; Uncertainty and Risk; Management and Exploitation of Data; Virtual Ma- terial Design; and Biotechnology, Food, and Health, are elabo- rated. Contributions of the Fraunhofer Institute for Industrial Mathematics, Kaiserslautern, Germany in this field are high- lighted. 1 Introduction There is no doubt that mathematics has become a technology in its own right, maybe even a key technology. Technology may be defined as the application of science to the problems of commerce and industry. And science? Science may be defined as developing, testing, and improving models for the prediction of system behavior; the language used to de- scribe these models is mathematics, and mathematics provides methods to evaluate these models. Here we are! Why has mathematics become a 3 © 2006 by Taylor & Francis Group, LLC
- 21. 4 H. Neunzert technology only recently? Mathematics became a technology when it re- ceived a tool to evaluate complex, “near to reality” models, and that tool was the computer. The model may be quite old. Navier–Stokes equations describe flow behavior rather well, but to solve these equations for realistic geometry and higher Reynolds numbers with sufficient precision, is even for powerful parallel computing, a real challenge. Make the models as sim- ple as possible, as complex as necessary and then evaluate them with the help of efficient and reliable algorithms. These are genuine mathematical tasks. Science is designed to “understand” natural phenomena; scientific tech- nology extends the domain of the validity of scientific theories to not yet existing systems. We create a new, virtual world in which we may change and optimize much easier and quicker than in the real world. Even that is rather old. Some scholars of ancient science [9] and some philosophers [10] consider this interplay of science and technology as crucial for the birth of science during the Hellenistic period around 300 BC (with names like Euclid or Archimedes on top). But now, since we may mathematically optimize very complex virtual systems, we are able to use mathematics in order to design better machines, to minimize the risk of financial actions, and to plan optimal surgery. This is the reason why mathematics has become a key technology. The following technology fields emerged as crucial from several investigations • Simulation of Processes and Products • Optimization, Control, and Design • Uncertainty and Risk • Management and Exploitation of Data • Virtual Material Design • Biotechnology, Food, and Health With the help of these road maps which contain examples and chal- lenges for future mathematics gathered from all over Europe, European mathematicians shall try to influence national and international research policies in a way that may help mathematics get the weight in future pro- grams which it has in reality already now. Mathematics was too long in © 2006 by Taylor & Francis Group, LLC in Europe (see [2, 6]).
- 22. Mathematics as a technology 5 an ivory tower, often used only as brain exercises for students. It needs some time and a lot of effort to catch public awareness of its new role. In this chapter I shall show examples from different technology fields mentioned above, examples gained from our experience in the Fraun- hofer Institute ITWM at Kaiserslautern. It was founded in 1996 and became a member of the Fraunhofer-Gesellschaft in 2001; the Fraunhofer- Gesellschaft is the leading German association for applied research with altogether 12,000 employees in ca. 60 institutes, an annual turnover of ca. 1.2 billion euro and branches in the US and in some European countries. Its decisive feature is that basic funding is given proportional to what is earned in industry. To make a rather complicated story simple, a Fraun- hofer Institute gets 40 cents from the federal government for each euro it earns in industry. “No industrial project - no money at all and 40 % on top in order to do fundamental research related to projects”–these are the two rules which in my opinion are unique and uniquely successful worldwide. ITWM has proved that mathematics as a technology is strong enough to follow the Fraunhofer rules. Not only that, at present it is the most successful institute of all the 15 Fraunhofer Institutes dealing with infor- mation technologies. The reason is that it has a huge market, much wider than any computer science institute. The disadvantage is that the market doesn’t yet know it. The consequence is that there is a lot of space for all other really applied mathematicians and for cooperation worldwide. But now I want to become more substantial. Here are the technology 2 Simulation of Processes and the Behavior of Products Simulation means modelling-computing-visualizing. To find the right model for the behavior of car components, as simple as possible and as compli- cated as necessary, is, for example, a task for asymptotic analysis: identify small parameters in very complex models, study the behavior for these parameters tending to zero, and estimate the error using this “parameter = 0 - model”. All this is tricky perturbation theory, sometimes advanced functional analysis. But we should never “oversimplify” in order to get an analytically treatable model; very often numerics will be necessary, and very often advanced numerical ideas are necessary. Since a realistic geome- try is sometimes very complex (think of a porous medium in a microscopic © 2006 by Taylor & Francis Group, LLC themes with examples and challenges, see references [1, 4].
- 23. 6 H. Neunzert view), we need, for example, new, gridfree algorithms efficiently imple- mented for parallel systems. And finally, long lists of numbers as a result of solving a PDE are completely useless-we have to interpret the results in terms of the original questions, and quite often we have to visualize the results as images or movies. Simulation is now routinely used in many parts of industry all over the world to support or to replace experimentation. “It can have a dramatic effect on the design process, reducing the need for costly prototypes and increasing the speed with which new products can be brought to market [1]. There are industries where simulation has a long tradition, like aerospace or automotive industries or in oil and gas prospection. In these areas, commercial software is available and often easy to handle and efficient. It is (at least for a Fraunhofer Institute) a very hard or even impossible task to place a new algorithm to substitute this kind of software, even if this algorithm is really better than the other one. What is possible for mathematicians is to substitute some modules in software products, as, for example, the second mathematical Fraunhofer Institute SCAI does in offering an “Algebraic Multigrid Solver” for linear systems. Another pos- sibility is postprocessing algorithms enabling the user to do an “optimal experimental design” for virtual or “numerical experiments”. Industries operating with more basic technologies such as textiles, glass, or even met- als just begin to use simulation. The market for commercial software seems too small, and tailor-made software is needed. How complicated this field could be will now be shown by our experience with the glass industry. ITWM has a 10-year close cooperation with Schott Glas at Mainz, where cooperation may be taken literally. The enormous knowledge of Schott sci- entists about materials and processes joins mathematical ideas in ITWM to find innovative solutions. (The material was provided to me by Norbert Siedow from ITWM; some parts and literature are described in the ITWM annual report 2003, page 26 ff.) from the glass tank with molten glass of a temperature over 1000◦ C through a pipe to a kind of drop called gob; in this process we identified 4 mathematical tasks which are denoted by colors. Two are so-called “inverse problems” that measure the temperature in the interior of the glass flow from radiation and optimize the shape of the flanges carrying the pipe such that a given homogeneous temperature is created through electrical currents. The shape of the gob, a very viscous drop of liquid glass, has to be calculated by CFD codes able © 2006 by Taylor & Francis Group, LLC Figure 1 shows the glass making process,
- 24. Mathematics as a technology 7 to handle free surfaces very well. Figure 1: Mathematical Problems in Glass Industry (Glassmaking) panels ask for the simulation of radiation. In semi-transparent media, this is a very elaborate task, since the radiation equation is a dimensional integro-differential equation with enormous computational efforts. Floatglass, an efficient production process invented by Pilkington, shows sometimes wavy patterns which have to be avoided. Whether these waves are instabilities created in a modification of the Orr-Sommerfeld equations is the subject of an ongoing PhD work. Glass fiber productions are extremely tricky processes in which the fibers interact with the air around them. Turbulent flow- fiber interaction is a topic where turbulence models are not enough, but stochastic differential equations are crucial. the cooling of glass. I would like to mention that already around 1800 Fraunhofer who gave the name to our society produced lenses and had problems with the thermal tensions and the defects created by them. Many of the problems here are “inverse problems” connected with heat transfer, and they are very ill-posed. Inverse problems may be counted under “optimization”; it is the combination of optimization and simulation as in inverse problems, optimal shape design, etc. which creates many mathematical challenges. © 2006 by Taylor & Francis Group, LLC Figures 2–5 show different kinds of glass processing, Pressing of TV Figure 3 shows classical glass processing and problems connected with One uses tricky scale asymptotics (see [8]).
- 25. 8 H. Neunzert Figure 2: Mathematical Problems in Glass Industry (Glassprocessing I) details of gob forming. The hot glass leaves the feeder when the needle opens. A drop (gob) is formed and cut off by a special cutter. J. Kuhnert (ITWM) has designed a gridfree numerical method to calculate the glass flow. It is called the “Finite Pointset Method” (FPM) and may be considered as an extension of “Smoothed Particle Hydrodynamics” (SPH) [11]. Particles are moving in the computational domain, carrying information about density, veloc- ity, temperature, etc. This information has to be extrapolated to other positions so that derivatives of these quantities as the Laplacian of the velocity components, the temperature gradient, etc., can be calculated. These extrapolations are denoted by a tilde, and the rest is Lagrangean formalism. The method is appropriate for fluids with free boundaries, changing even the topology, as it happens, when the gob is cut off. A more analytical task is the question of waves at floatglass surfaces. Here is the industrial question: What is the origin of waves at the interface of glass and molten tin (the glass flows over molten tin, a classical 2-phase flow with quite different temperatures)? These waves are small defects which should be removed. What are the causes? Let us finish the glass field by describing a very nice, very ill-posed © 2006 by Taylor & Francis Group, LLC Let us have a closer look at a few of the problems. Figure 4 shows the
- 26. Mathematics as a technology 9 Figure 3: Mathematical Problems in Glass Industry (Glassprocessing II) problem which deals with temperature measurements. The high temper- ature of the glass melt asks for remote measurements or at least only measurements at the boundary. Here is the problem. We measure the temperature at parts of the boundary. Assuming that the heat transport is given by conduction and radiation and assuming that the heat flux at the boundary is known ev- erywhere, what is the temperature inside? The problem was solved without radiation in a very nice master’s the- sis by L. Justen and is with radiation the subject of a Ph.D. thesis by Pereverzyev jun. For one dimension it works, but the real world is three dimensional. The situation is similar for melt spinning processes in textile industries; there is an intersection with the previous field when we talk about glass fibers. But, in general, we have polymer fibers, leaving nozzles as a liquid, but crystallizing when an air flow is cooling and pulling the fibers. Here are some mathematical problems connected with the process. Of course, there are curtains of fibers in a real process. The industrial question belongs to “reverse engineering”: these are the properties of the product we want to have (even to describe these properties is a mathematical problem). How can we create them? The crystallization is a mathematical problem too and the subject of © 2006 by Taylor & Francis Group, LLC
- 27. 10 H. Neunzert Figure 4: Mathematical Problems in Glass Industry (Gob forming) a Ph.D. thesis by Renu Dhadwal . Let us have a closer look at the inter- action of fibers with a turbulent flow. The main question is, How does the stochastic behavior of the turbulent air flow influence the (stochastically described) properties of the fabric? Markeinkewho just finished her Ph.D. Things may even be more complicated – see for example a quickly rotating spinneret for producing glass fibers: © 2006 by Taylor & Francis Group, LLC Figures 6–20 describe the work of N.
- 28. Mathematics as a technology 11 Figure 5: Mathematical Problems in Glass Industry (Gob forming) Figure 6: Mathematical Problems in Glass Industry (Gob forming) © 2006 by Taylor & Francis Group, LLC
- 29. 12 H. Neunzert Figure 7: Mathematical Problems in Glass Industry (Floatglass) Figure 8: Mathematical Problems in Glass Industry (Reconstruciton of initial temperature) © 2006 by Taylor & Francis Group, LLC
- 30. Mathematics as a technology 13 Figure 9: Mathematical Problems in Spinning Processes (Production of nonwovens) Figure 10: Mathematical Problems in Spinning Processes (Fiber-fluid interaction: Fiber Dynamics) © 2006 by Taylor & Francis Group, LLC
- 31. 14 H. Neunzert Figure 11: Mathematical Problems in Spinning Processes (Foner-fluid interaction: Nonwoven Materials) Figure 12: Mathematical Problems in Spinning Processes (Turbulence Effects) © 2006 by Taylor & Francis Group, LLC
- 32. Mathematics as a technology 15 Figure 13: Mathematical Problems in Spinning Processes (Turbulence Effects) Figure 14: Mathematical Problems in Spinning Processes (Turbulence Effects) © 2006 by Taylor & Francis Group, LLC
- 33. 16 H. Neunzert Figure 15: Mathematical Problems in Spinning Processes (Turbulence Effects) Figure 16: Mathematical Problems in Spinning Processes (Turbulence Effects) © 2006 by Taylor & Francis Group, LLC
- 34. Mathematics as a technology 17 Figure 17: Mathematical Problems in Spinning Processes (Turbulence Effects) Figure 18: Mathematical Problems in Spinning Processes (Turbulence Effects) © 2006 by Taylor & Francis Group, LLC
- 35. 18 H. Neunzert Figure 19: Mathematical Problems in Spinning Processes (Deposition with Turbulence Effects) Figure 20: Mathematical Problems in Spinning Processes (Melt- Spinning of Glass Fibers) © 2006 by Taylor & Francis Group, LLC
- 36. Mathematics as a technology 19 3 Optimization, Control, and Design What we finally want to achieve in our man-made world are optimal so- lutions: the process should be as cheap and as fast as possible, and the product should at least behave better than the products of the competitors. (Even nature seems to have a creator who is interested in optimality. That is why we have so many variational principles, and that is why animals and plants show us so many tricky solutions for their “technical” problems to be as stable, as light, as smoothly moving as possible and necessary. This is called “bionics” and there may be an interesting interplay between opti- mization by mathematics and optimization by evolution.) “So rather then asking how a product performs, the question is, how should the product be designed in order to perform in a specified way. Scheduling, planning and logistics also fall within that area of optimization. Optimal control is used to provide real-time control of an industrial process or a product, such as a plane or a car, in response to current operating conditions. A related area is that of inverse problems, where the parameters (or even the structure) of a model must be estimated from measurement of the system output) [1]. We have mentioned inverse problems already in (1); they appear liter- ally everywhere. We will show two examples from our projects at ITWM; however they are very short. There is the wide field of topological shape optimization; “topological” means that one may change the topology of a structure, for example, by admitting holes. One has to minimize an objective function (maximal stress, mean compliance, etc.) with respect to the shape. As an example for a multicriteria optimization, we consider a project of [5]. How should we optimally control the radiation in cancer therapy such that the cancer cells are destroyed as much as possible, but at the same time organs or important healthy parts of the body remain undamaged. There are, besides optimization, a lot of simulation problems? f. e. to simulate how radiation penetrates the body, but let’s concentrate on opti- mization assuming that the transmission of the radiation to different parts of the body given the external source, which can be controlled, is known. The goal is that a medical doctor can operate with the optimization tool, allowing more or less radiation to certain organs by “pulling” in the cor- responding direction of a navigation scheme; the program then computes the different doses of different sources and different directions, getting at © 2006 by Taylor & Francis Group, LLC
- 37. 20 H. Neunzert Figure 21: Topological Optimization the end corresponding isodose levels. To be more detailed: we have a target, the tumor and we have “risks”, which should get as little as possible, but at most at given thresholds for the radiation. To do this so fast, that it is finally online, and to do it so, that the doc- tors can easily handle it, are interesting and highly relevant mathematical tasks. © 2006 by Taylor & Francis Group, LLC One uses Pareto solutions, which are defined in the next figure:
- 38. Mathematics as a technology 21 Figure 22: Mathematical ideas Figure 23: Cube with pointwise load: 10 % volume reduction per iteration (1) © 2006 by Taylor & Francis Group, LLC
- 39. 22 H. Neunzert Figure 24: Cube with pointwise load: 10 % volume reduction per iteration (2) Figure 25: Optimization and Control (Multicriteria optimization of intensity modulated radiotherapy) © 2006 by Taylor & Francis Group, LLC
- 40. Mathematics as a technology 23 Figure 26: Ideal planning goals-not achievable Figure 27: Multicriteria approximation problem © 2006 by Taylor & Francis Group, LLC
- 41. 24 H. Neunzert 4 Uncertainty and Risk Many processes in nature, in economy, and even in daily life are or seem to be strongly accidental; we therefore need a stochastic theory in order to model these processes. Randomness creates uncertainty, and uncer- tainty creates risk, for example, in decisions about investments, about medication, and about security of technical systems like planes or power plants. Whether this randomness is genuine or just a consequence of high complexity is a philosophical question which does not influence stochastic modeling. You will find very complex systems in catastrophes like earth- quakes or floods; biological systems, for example are extremely complex the human body. Experiments are not possible, and simulation is therefore highly necessary, but very difficult, too. Also in economy, experiments are impossible, but one needs help for decisions which minimize the risk. The law of large numbers leads often to models which are deterministic PDEs and very similar to deterministic models in natural sciences. But at a closer look they are even more complex, for example, very high dimensional (the independent variables are not geometric, but may be the values of different stocks). Therefore, even if we get at the end a treatable PDE, we have to use Monte-Carlo methods to solve them approximately, and we are back to stochastic differential equations. Now quite often derivatives of these solutions with respect to variables and parameters are needed, and to differentiate a function given by a Monte-Carlo method is not always successful. invented for practical problems, is a great help [3]. Here is an example from option prizing. Of course, there are other uncertainties and risks such as in floods and earthquakes. In technical systems, very different methods are involved. 5 Management and Exploitation of Data We are flooded by data which, if structured, create information and finally knowledge. The extraction of this information or knowledge from data is called “data mining”. Data may be given as signals or images; if we want to discover patterns, and if we want to “understand” these signals or images, we need image processing and pattern recognition methods. If we want to study and predict input-output systems for which we do not have enough theory (simple models) but many observations from the past, © 2006 by Taylor & Francis Group, LLC The Malliavin calculus shown in Figures 28–31, initially not
- 42. Mathematics as a technology 25 Figure 28: Malliavin calculus for Monte-Carlo methods (1) we may develop “black-box” models like linear control models or neural networks. If for parts of the system a theory is available, we may talk of “grey-box” models. Data mining, signal or image processing, and black- or grey-box models are the mathematical disciplines involved here. Some of them are not as mature as PDE, optimization, or stochastics, but are certainly a field, where new ideas are needed. (There are many, especially in the field of pattern recognition: look, for example, at the articles of David Munford or Yves Meyer from the last 10 years.) A typical input output system, where we do not have much theory, is–the human body; medicine is therefore a main application area, and we want to show only one example from our experience, the interpretation of long-term electrocardiograms. If we register only the heart beats, we get quite long sequences, (ti)i=1,...N with N ∼ 100, 000, and have to find the in- formation about the risk for sudden cardiac death. To do so we use Lorenz plots, sets consisting of points {(ti, ti+1, ti+2), (ti+1, ti+2, ti+3), . . .}i=1,...N , and try to understand the structure of these sets. Of course, the beat is rather regular, if the Lorenz plot is a slim club (but too slim is again dangerous). The picture shows the clearly visible influence of drugs; to estimate the risk, one needs very tricky data mining techniques. © 2006 by Taylor & Francis Group, LLC
- 43. 26 H. Neunzert Figure 29: Malliavin calculus for Monte-Carlo methods (2) Figure 30: Malliavin calculus for Monte-Carlo methods (3) © 2006 by Taylor & Francis Group, LLC
- 44. Mathematics as a technology 27 Figure 31: Malliavin calculus for Monte-Carlo methods (4) Figure 32: Comparison of computations of delta for a call © 2006 by Taylor & Francis Group, LLC
- 45. 28 H. Neunzert Figure 33: Risk parameters in the case of arrhythmic heartbeat 6 Virtual Material Design One of the objectives of material science is to design new materials which have desirable properties; to do so by using simulation is called virtual material design”. Mathematics is used to relate the large-scale (macro- scopic) properties of materials such as stiffness, fatigue, permeability, and impedance to the small-scale (microscopic) structure of the material. The microscopic structure has to be optimized in order to guarantee the re- quired macroscopic properties. This is an application of multi scale anal- ysis, where we use averaging and homogenization procedures to pass from micro to macro. The scales may reach from nano to the size of constituents of composite materials. Typical materials are textiles, paper, food, drugs, and alloys. At ITWM we try to design appropriate filter material. This is a very wide field, since filters are used everywhere: they serve different purposes and require therefore different properties. The example here deals with oil filters. The research work in its first part was done by Iliev and Laptev from ITWM. We use a system which we get through homogenization from Navier- Stokes through a “very porous ” medium: a Navier-Stokes-Brinkman sys- © 2006 by Taylor & Francis Group, LLC
- 46. Mathematics as a technology 29 Figure 34: Simulation of 3-D flow through oil filters tem which is a combination of incompressible, steady Navier-Stokes with a Darcy term. The interface condition describing the behavior of the fluid on the sur- face of the filter material is a rather delicate issue, but in this model (with © 2006 by Taylor & Francis Group, LLC Brinkman homogenization) it is easier to handle (see the Ph.D. thesis by Laptev [7]). The flow field is given below.
- 47. 30 H. Neunzert Figure 35: Simulation of Flow through a Filter Flow Rate Figure 36: Simulation of 3-D flow through oil filters © 2006 by Taylor & Francis Group, LLC
- 48. Mathematics as a technology 31 Figure 37: Simulation of 3-D flow through oil filters The correspondence with measurements (where the pressure loss for different Reynold numbers at different temperatures with correspondingly different permeabilities) is remarkable. I call this correspondence sometimes “prestabilized harmony”: a rather crude model which is numerically approximated and gives results which correspond with nature to an extent which one really might not expect. But, of course, care is necessary. Models have their range of applicability, and their limitations should be carefully respected. To compute the flow field of a filter is not enough to understand its efficiency. The transport of the particles, which have to be filtered out, must be simulated. Therefore, we have to model their absorption by the fibers of the filter and the motion of the particles by the fluid velocity, its friction, and the influence of diffusion. Finally, the absorption is, of course, filter and particle dependent. This is an area of exciting modelling (see, 7 Biotechnology, Food, and Health This field has created new research areas which are rather interdisciplinary, for example, bio-informatics or system biology. Statistics, discrete math- ematics, computer science and system and control theory, data mining, © 2006 by Taylor & Francis Group, LLC for example [8]).
- 49. 32 H. Neunzert differential-algebraic systems, and parameter and structure identification are involved, together with all kinds of life sciences. Biological systems are extremely complex, involving huge molecules which interact in poorly understood ways. It is a long way to get a full understanding in terms of fundamental chemistry and physics. Moreover, it is a mathematical task to gain as much information as possible from the data we have; the classical idea to use a linear control system and to identify the coefficients does not work. We therefore need grey models, complex enough to allow prediction, but simple enough that parameters may be identified from the measurements. Health is very much related to deterministic models for biophysical processes, a better image understanding, and efficient data mining. Food is one of the emerging application fields of science, especially simulation. To simulate a process preparing food, for example, cooking of an omelette or frying a piece of meat in order to optimize the quality or the energy consumption, is a mathematical task of extremely high diffi- culty. However, the economic value is enormous for companies which offer food worldwide and for companies which produce, for example, household appliances. The ITWM has not yet many projects in this field; however, its joint venture with Chalmers University of Technology, the Fraunhofer Chalmers Research Centre (FCC) at Gothenburg deals with bio-informatics and system biology. Jirstrand, FCC. By metabolism we mean the processes inside living cells. These are complicated biochemical processes; even a “simple” process as glycoly- sis is not at all simple. We have to model biochemical pathways, i.e. chains of reactions, happening in collisions change the concentration of molecules of different types. Even simple enzymatic reactions lead to non- linear systems. Finally, one does what every modeler has to do: we non- dimensionalize and look for small parameters to apply perturbation meth- ods. This leads to rational expression, called Michaelis-Menten dynamics in biology. At the end we get very large, rational right-hand sides for the system of ODEs. The problem is that we do not know the parameters of the system, even the structure (which reactions should be included; do we need to include hysteresis, etc.) is not clear. Can we deduce from the behavior which structural elements the model should include? And how many parameters are we able to identify? How can we adopt the model to © 2006 by Taylor & Francis Group, LLC Figures 38–43 are taken from a presentation by Mats
- 50. Mathematics as a technology 33 the knowledge we have? Some steps are done, but there is still a long way to go. Figure 38: Metabolism Figure 39: Modeling of Biochemical Reactions © 2006 by Taylor & Francis Group, LLC
- 51. 34 H. Neunzert Figure 40: Modeling of Biochemical Reactions Figure 41: Modeling of Biochemical Reactions © 2006 by Taylor & Francis Group, LLC
- 52. Mathematics as a technology 35 Figure 42: Modeling of Biochemical Reactions 8 Conclusions As mentioned in the beginning and shown during the description of the technology fields, one of the major drivers behind the dramatic change towards a knowledge-based economy is the advent of powerful and afford- able digital computers. The rate of progress in hardware follows Moore’s Law, telling us that computer power doubles every two years. Equally important, but not so widely appreciated, is the fact that there has been a similar improvement in the algorithms used to evaluate complex math- ematical models. The improvement in speed, due to better algorithms, has been as significant as the improvements in hardware. All this has made computer simulation an accepted tool; in science, Computational X is dominating. Industry is already feeling the benefits of these advances, resulting in an increase in efficiency and competitiveness. This in turn makes mathematics, being at the core of all simulation, poised to become a key technology. Mathematics by its abstraction allows the transfer of ideas from one application field to another. Mathematicians are “cross thinkers”. This kind of cross thinking creates creativity and leads to in- novation. To give mathematics its power, the classical “engineering mathematics” © 2006 by Taylor & Francis Group, LLC
- 53. 36 H. Neunzert Figure 43: Modeling of Biochemical Reactions is not sufficient. I hope I have made clear that new ideas, some from pure mathematics too, are needed in order to get good results: new function spaces, new ideas in non-linear analysis or in stochastic calculus, new ideas to deal with inverse problems and to deal with pattern recognition, etc. It is not a question of “pure or applied”, there is a need for ”pure and applied.” Both should be in balance and they should work together; the fact that there is a widespread separation weakens both parts. There is a need for properly educated mathematicians all over the world, too. What a proper education means for an “industrial mathe- matician” would be a subject in its own. The European Consortium for Mathematics in Industry (ECMI) has put a lot of effort into that issue. However, what we have to strive after is creativity and flexibility in finding proper models and more efficient algorithms. “Industrial Mathematics” or, as it is called in Europe, Technomathematics, Economathematics, or Fi- nance Mathematics, is not a subject in its own like algebra or topology. It is more a new attitude towards the world it is the curiosity in order to understand and the drive to improve. If we mathematicians work together, if we are courageous enough to leave the ivory tower of our science and act in the real world, I am sure we shall see a bright future for our science and for our students, too. © 2006 by Taylor & Francis Group, LLC
- 54. Mathematics as a technology 37 References [1] A. Cliff, R. Matheij, and H. Neunzert, Mathematics: Key to the european knowledge based economy, in MACSI-Net Roadmap for Mathematics in European Industry, Edited by A. Cliffe, B. Matheij, and H. Neunzert, A project of European commission, Mark 2004. [2] A. Cliffe, B. Matheij, and H. Neunzert (Eds), A project of European commission, MACSI-Net Roadmap for Mathematics in European Indus- try, Mark 2004, [3] Fournier et al., Application of Malliavin calculus to Monte-Carlo methods in finance, Finance and Stochastics 3(4), 1999. [4] Fraunhofer Institute for Industrial Mathematics, Kaiserslauntern, Ger- [5] H. Neunzert, N. Siedow, and F. Zingsheion, Simulation temperature be- havior of hot glass during cooling, In, Mathematical Modeling, Edited by E. Cumberbatch and A. Fitt, Cambridge University Press, 2001. [6] H. Neunzert and U. Trottenberg (Eds), Mathematik als Technologie, Die Fraunhofer–Institute ITWM und SCAI, to appear. [7] V. Laptev, Numerical Solution of Complex Flow in Plain and Porous Me- dia, Dissertation, Department of Mathematics, Technical University of Kaiserlauntern, Germany, 2004. [8] A. Latz and A. Wiegmann, Simulation of fluid particle simulation in realis- tic 3-dimensional fiber structures, in Proceedings Filtech Europa, I-353-360, 2003. [9] L. Russo, The Forgotten Revolution, Springer-Verlag, Heidelberg, 2004. [10] M. Scheler, Soziologie des Wissens, in Die Wis-sensformen und die Gesellschaft, Francke-Verlag, 1960. [11] S. Tiwari and J. Kuhnert, A numerical scheme for solving incompress- ible and law mach number flows by finite pointset methods, in, Meshfree Methods for Partial Differential Equations, Edited by M. Griebel and M. Schwertzer, Volume 2, Springer-Verlag, Berlin, 2004. © 2006 by Taylor & Francis Group, LLC many, Annual report 2003. (info@itwm.fraunhofer.de).
- 55. Other documents randomly have different content
- 56. stoicism of his race, but he went away alone into the wood, returning at the dawn of day. When he returned Black Eagle was dying. RAPIDS, COLUMBIA RIVER.
- 57. Slowly the pale lids closed over the sunken eyes, a breath and the brave lad had trusted his soul to the white man’s God. The broken-hearted old chief sat the long night through by the corpse of his son. When morning came he called the tribe together and told them he wished to follow his last child to the grave, but he wanted them to promise him that they would cease to war with the white man and seek his friendship. At first many of the warriors refused, but Umatilla had been a good chief, and always had given them fine presents at the potlatches. Consulting among themselves they finally consented. When the grave was ready, the braves laid the body of Black Eagle to rest. Then said the old chief: “My heart is in the grave with my son. Be always kind to the white man as you have promised me, and bury us together. One last look into the grave of him I loved and Umatilla too shall die.” The next instant the gentle, kind hearted old chief dropped to the ground dead. Peace to his ashes. They buried him as he had requested and a little later sought the teacher’s friendship, asking him to guide them. That year saw the end of the trouble between the Indians and the white race at the Dalles. The old chief still lives in the history of his country. Umatilla is a familiar name in Dalles City. The principal hotel bears the name of Umatilla. On either side of the river farm houses, orchards and wheat fields dot the landscape. Salmon fishing is the great industry on the river. The wheels along both sides of the river have been having a hard time of it this season from the drift wood, the high water and the big sturgeon, which sometimes get into the wheels. A big sturgeon got into a wheel belonging to the Dodon Company and slipped into the bucket, but was too large to be thrown out. It was carried around and around until it was cut to pieces, badly damaging the wheel. Now the law expressly states, as this is the close season for sturgeon, that when caught they must be thrown back in the water. “But what is the use,” inquires the Daily News, “if they are dead?”
- 58. FARM ON THE BANK OF THE COLUMBIA RIVER, BELOW THE DALLES, OREGON. A visit to a salmon cannery is full of interest. As the open season for salmon is from April first to August first, the buildings though large
- 59. are mere sheds. The work is all done by Chinamen. The fish are tossed onto the wharf, where they are seized by the men, who carry them in and throw them on to long tables, chop off their heads, dress them and hold them, one fish at a time, under a stream of pure mountain water, which pours through a faucet over the long sink. Next they are thrown onto another table, where other Chinamen cut them up ready for the cans, all in much less time than it takes to tell about it. The tin is shipped in the sheet to the canneries and the cans are made on the ground. Astoria, the Venus of America, is headquarters for the salmon fishing on the Columbia River. Joaquin Miller described it as a town which “clings helplessly to a humid hill side, that seems to want to glide into the great bay-like river.” Much of it has long ago glided into the river. Usually the salmon canneries are built on the shores, but down here and on toward the sea, where the river is some seven miles wide, they are built on piles in mid stream. Nets are used quite as much as wheels in salmon fishing. Sometimes a hungry seal gets into the nets, eating an entire “catch,” and playing havoc with the net. Up toward the Dalles on the Washington side of the river, are three springs. These springs have long been considered by the Indians a veritable fountain of youth. Long before the coming of the white man they carried their sick and aged to these springs, across the “Bridge of the Gods.” Just above Dalles City lies the dalles which obstruct navigation for twelve miles. Beyond this point the river is navigable two hundred miles. Here, too, legends play an important part. When the volcanoes of the northwest were blazing forth their storm of fire, ashes and lava, a tribe known as the Fire Fiends walked the earth and held high revelry in this wild country. When Mount Rainier had ceased to burn the Devil called the leaders of the tribe together one day and proposed that they follow nature’s mood and live more peaceably, and that they quit killing and eating each other. A howl met this proposal. The Devil deemed it wise just at this moment to move on, so off he set, a thousand Fire Fiends after him. Now his majesty could easily whip a score of Fiends, but he was no match for
- 60. a thousand. He lashed his wondrous tail about and broke a great chasm in the ground. Many of the Fiends fell in, but the greater part leaped the rent and came on. A second time the ponderous tail came down with such force that a large ravine was cracked out of the rocks, the earth breaking away into an inland sea. The flood engulfed the Fiends to a man. The bed of the sea is now a prairie and the three strokes of the Devil’s tail are plainly visible in the bed of the Columbia at the dalles. Just across the river from Dalles City on a high bluff, stands a four story building, the tower in the center running two stories higher. The building stands out there alone, a monument to the enterprise of one American. He called it a shoe factory, but no machinery was ever put in position. After the pseudo shoe factory was completed false fronts of other buildings were set up and the rugged bluffs laid out in streets. An imaginary bridge spanned the broad river. Electric lights, also imaginary, light up this imaginary city. The pictures which this genius drew of his town showed street cars running on the principal streets and a busy throng of people passing to and fro. As to the shoe factory, it was turning out thousands of imaginary shoes every day. Now this rogue, when all was ready, carried the maps and cuts of his town to the east, where he sold the factory and any number of lots at a high figure, making a fortune out of his paper town. From Dalles City across the country to Prineville in the Bunch Grass country, a distance of a hundred miles, the country is principally basalt, massive and columnar, presenting many interesting geological features. Deep gorges separate the rolling hills which are covered with a soil that produces bunch grass in abundance. This same ground produces fine wheat and rye. This is a good sheep country and wool is one of the principal products. Crater Lake is haunted by witches and wizards. Ghosts, with seven leagued boots, hold high revelry on its shores on moonlight nights, catching any living thing that comes their way and tossing it into the deep waters of the lake, where the water devils drag it under.
- 61. SCENE ON AN OREGON FARM IN THE WILLAMETTE VALLEY. We spent two delightful days on an Oregon farm near Hubbard, thirty miles south of Portland.
- 62. We drove from Hubbard in the morning to Puddin river. The bridge was being repaired, so we walked across, our man carrying our traps. We had just passed Whisky hill when we met our friend Mr. Kauffman and his daughter, driving down the road. We were warmly welcomed and after an exchange of greetings we drove back with them to their home, where we partook of such a dinner as only true hospitality can offer. Mr. Kauffman owns three hundred acres of fine farming land. There is no better land anywhere on the Pacific coast than in this beautiful valley of the Willamette river. Beautiful flowers and shrubs of all sorts in fine contrast to the green lawn surround the house, which is painted white, as Ruskin says all houses should be when set among green trees. Near by is a spring of pure mountain water. In the woods pasture beyond the spring pheasants fly up and away at your approach. Tall ferns nod and sway in the wind, while giant firs beautiful enough for the home of a hamadryad lend an enticing shade at noontime. If any part of an Oregon farm can be more interesting than another it is the orchard, where apple, peach, plum, pear and cherry trees vie with each other in producing perfect fruit. Grapes, too, reach perfection in this delightful climate. One vine in Mr. Kauffman’s vineyard measures eighteen inches in circumference. The dryhouse where the prunes are dried for market is situated on the south side of the orchard. No little care and skill is required to dry this fruit properly. Wednesday morning we reluctantly bade good-by to our kind hostess and departed with Mr. Kauffman for Woodburn, where we took the train for Portland. The drive of ten miles took us through a fine farming district. Here farms may be seen in all stages of advancement from the “slashing” process, which is the first step in making a farm in this wooded country, to the perfect field of wheat, rye, barley or hops. Arriving at Woodburn we lunched at a tidy little restaurant. The train came all too soon and we regretfully bade our host farewell.
- 63. The memory of that delightful visit will linger with us as long as life shall last. ROADWAY IN OREGON.
- 64. There are few regions in the West to-day where game is as abundant as in times past. Yet there are a few spots where sport of the old time sort may be had, and the lake district of Southern Oregon is one of these. Here, deer and bear abound as in days of yore, while grouse, squirrel, mallard duck and partridge are most plentiful. Fort Klamath lake is a beautiful sheet of water, sixty miles long by thirty wide. Among the tules in the marshes the mallard is at home, while grouse and nut brown partridge by the thousands glide through the grass. Fish lake speaks for itself, while the very name, Lake of the Woods, carries with it an enticing invitation to partake of its hospitality and royal sport. Travel is an educator. It gives one a broader view of life and one soon comes to realize that this great world swinging in space is a vast field where millions and millions of souls are traveling each his own road, all doing different things, all good, all interesting. In our journeyings we have met many interesting people, but none more interesting than Miss McFarland, whom we met on our voyage up the Columbia river. Miss McFarland was the first American child born in Juneau, Alaska. Her only playmates were Indian children. She speaks the language like a native and was for years her father’s interpreter in his mission work. She has lived the greater part of her life on the Hoonah islands. The Hoonah Indians are the wealthiest Indians in America. Having all become Christians they removed the last totem pole two years ago. Reminiscences of Miss McFarland’s childhood days among the Indians of Alaska would make interesting reading. The old people as well as the children attend the mission schools. One day an old chief came in asking to be taught to read. He came quite regularly until the close of the school for the summer vacation. The opening of the school in the autumn saw the old man in his place, but his eyes had failed. He could not see to read and was in
- 65. despair. Being advised to consult an optician he did so and triumphantly returned with a pair of “white man’s eyes.” Upon one occasion Miss McFarland’s mother gave a Christmas dinner to the old people of her mission. It is a custom of the Indians to carry away from the feast all of the food which has not been eaten. One old man had forgotten his basket, but what matter, Indian ingenuity came to his aid. Stepping outside the door he removed his coat and taking off his dress shirt triumphantly presented it as a substitute in which to carry home his share of the good things of the feast. These Indians believe that earthquakes are caused by an old man who shakes the earth. Compare this with Norse Mythology. When the gods had made the unfortunate Loke fast with strong cords, a serpent was suspended over him in such a manner that the venom fell into his face causing him to writhe and twist so violently that the whole earth shook. When Miss McFarland left her home in Hoonah last fall to attend Mill’s college every Indian child in the neighborhood came to say good-by. They brought all sorts of presents and with many tears bade her a long farewell. “Edna go away?” “Ah! Oh! Me so sorry.” “Edna no more come back?” “We no more happy now Edna gone,” “No more happy, Oh! Oh!” “Edna no more come back.” “Oh, good-by, Edna, good-by.” Every Christmas brings Miss McFarland many tokens of affection from her former playmates. Pin cushions, beaded slippers, baskets, rugs, beaded portemonnaies. Always something made with their own hands. Miss McFarland’s name, through that of her parents, is indissolubly connected with Indian advancement in Alaska. One meets curious people, too, in traveling. In the parlor at the hotel one evening a party of tourists were discussing the point of extending their trip to Alaska. The yeas and nays were about equal when up spoke a flashily dressed little woman, “Well,” said she,
- 66. “what is there to see when you get there?” That woman belongs to the class with some of our fellow passengers, both men and women who sat wrapped in furs and rugs from breakfast to luncheon and from luncheon to dinner reading “A Woman’s Revenge,” “Blind Love,” and “Maude Percy’s Secret,” perfectly oblivious to the grandest scenery on the American Continent, scenery which every year numbers of foreigners cross continents and seas to behold. One of our fellow travelers is a German physician who is spending the summer on the coast. He is deeply interested in the woman question in America. He is quite sure that American women have too much liberty. “Why,” said he, “they manage everything. They rule the home, the children and their husbands, too. Why, madam, it is outrageous. Now surely the man ought to be the head of the house and manage the children and the wife too, she belongs to him, doesn’t she?” “Not in America,” we replied, “the men are too busy, and besides they enjoy having their homes managed for them. Then, too, the women are too independent.” “That is just what I say, madam, they have too much liberty, they are too independent. They go everywhere they like, do everything they like and ask no man nothings at all.” My German friend evidently thinks that unless this wholesale independence of women is checked our country will go to destruction. The war with Spain does not compare with it. I am wondering yet if our critic’s wife is one of those independent American women. Just below Portland on the banks of the Willamette river and connected with Portland by an electric street railway stands the first capital of Oregon, Oregon City, the stronghold of the Hudson Bay Company, which aided England in so nearly wrenching that vast territory from the United States. This quaint old town is rapidly taking on the marks of age. The warehouse of that mighty fur company stands at the wharf, weather
- 67. beaten and silent. No busy throng of trappers, traders and Indians awaken its echoes with barter and jest. No fur loaded canoe glides down the river. No camp fire smoke curls up over the dark pine tops. The Indian with his blanket, the trapper with his snares and the trader with his wares have all disappeared before the march of a newer civilization. The camp fire has given place to the chimney; the blanket to the overcoat; the trader to the merchant and the game preserves to fields of waving grain. The lonely old warehouse looks down in dignified silence on the busy scenes of a city full of American push and go. All the forenoon the drowsy porter sat on his stool at the door of the sleeper, ever and anon peering down the aisle or scanning the features of the passengers. What could be the cause of his anxiety? Was he a detective in disguise? Had some one been robbed the night before? Had some one forgotten to pay for services rendered? Had that handsome man run away with the beautiful fair haired woman at his side? Visions of the meeting with an irate father at the next station dawned on the horizon. The train whirled on and still the porter kept up his vigilance. It was nearly noon when I stepped across to my own section and picked up my shoes. The sleepy porter was wide awake now. His face was a study. For one brief moment I was sure that he was a detective and that he thought he had caught the rogue for whom he was looking. “Them your shoes, Madam?” said he approaching me. “Yes.” “Why, Madam, I’ve been waitin’ here all mornin’ for the owner to come and get ’em.” Ah, now I understood. He was responsible for the shoes and he thought that they belonged to a man. Fifty cents passed into the
- 68. faithful black hands and my porter disappeared with just a hint of a smile on his face.
- 69. CHAPTER XII OFF FOR CALIFORNIA We left Portland on the night train for San Francisco. I took my gull, the Captain we called him, into the sleeper with me. He was asleep when I placed his basket under my berth, but about midnight he awoke and squawked frightfully. I rang for the porter but before he arrived the Captain had awakened nearly every one in the car. Angry voices were heard inquiring what that “screeching, screaming thing,” was. An old gentleman thrust his red night capped head out of his berth next to mine and angrily demanded of me where that nasty beast came from. When I politely told him he said he wished that I had had the good sense to leave it there. Then he said something that sounded dreadfully like swear words, but being such an old gentleman I’ve no doubt that my ears deceived me. At any rate it was something about sea gulls in general and my own in particular. His red flannel cap disappeared and presently I heard him snoring away up in G. Now my poor gull only squawked on low C. After that the Captain traveled in the baggage car with the trunks and packages. Traveling south from Portland one passes farms and orchards until the foot of the Sierra Nevada range is reached. Most of the farms are well improved. Many of the orchards are bearing, while others are young. Here and there in the mountains are cattle ranches. These mountains are not barren, rugged rocks like the Selkirks of Alaska. Here there is plenty of pasture to the very summit of the mountains. Wolf Creek valley is one vast hay field. Up we go until the far-famed Rogue River valley is reached. This noble valley lying in the heart of
- 70. the Sierras reminds one of the great Mohawk valley of New York. Ashland is the center of this prosperous district. The Southern State Normal School is located here. The seventh annual assembly of the Southern Oregon Chautauqua will convene in Ashland in July. This assembly is always well attended. Farmers bring their families and camp on the grounds. The program contains the names of musicians prominent on the coast. Among the lecturers are the names of men and women prominent in their special fields. Frank Beard, the noted chalk talk lecturer, will be present. So you see that the wild and woolly west is not here, but has moved on to the Philippines. When the passenger train stops at the station of Ashland a score of young fruit venders swarm on the platform, crying plums, cherries, peaches and raspberries at fifteen cents a box. When the train-bell rings fruit suddenly falls to ten cents and when the conductor cries “All aboard” fruit takes a downward plunge to five cents a box, but the fruit is all so delicious that you do not feel in the least cheated in having paid the first price. “Look here, you young rascal,” said a newspaper man, who travels over the road frequently to one of the young fruit dealers, “I bought raspberries of you yesterday at five cents a box.” “O no you didn’t, mister, never sold raspberries at five cents a box in my life sir, pon honor.” In less than three minutes this young westerner was crying “Nice ripe raspberries here, five cents a box.” “Why,” said I, “I thought you told the gentleman that you never sold berries at five cents a box.” “No, Madam, I didn’t, pon honor,” and the little rogue really looked innocent.
- 71. CLIMBING THE SHASTA RANGE. Leaving Ashland with three big engines we climb steadily up four thousand one hundred and thirty feet to the summit of the range.
- 72. The Rogue River valley spreads out below us in a grand panorama of wheat, oats, barley fields and orchards. Down the southern slope the commercial interest centers in large saw-mills and cattle ranches. Off to the east lie the lava beds where Gen. Canby and his companions were so treacherously assassinated by the Modoc Indians under the leadership of Captain Jack and Scar Faced Charley. Crossing the Klatmath River valley the dwelling place in early days of the Klatmath Indians, the engines make merry music as they puff, puff, puff in a sort of Rhunic rhyme to the whir of the wheels as they groan and climb three thousand nine hundred feet to the summit of the Shasta range. There is something wonderfully fascinating about mountain climbing. Whether by rail over a route laid out by a skilled engineer; on the back of a donkey over a trail just wide enough for the feet of the little beast, or staff in hand you go slowly up over rocks and bowlders, or around them, clinging to trees and shrubs for support. The very fact that the train may without a moment’s notice plunge through a trestle or go plowing its way down the mountain side; the donkey lose his head and take a false step; the shrub break or a bowlder come tearing down the rock-ribbed mountain and crush your life out, thrills the blood and holds the mind enthralled as a bird is held enchanted by the charm of the pitiless snake. Throughout the mountains mistletoe, that mystic plant of the Druids, hangs from the limbs and trunks of tall trees. It was with an arrow made from mistletoe that Hoder slew the fair Baldur. All day long snow-covered Mt. Shasta has been in sight and toward evening we pass near it on the southern side of the range and stop at the Shasta Soda Springs. The principal spring is natural soda water. This is the fashionable summer resort of San Francisco people, who come here to get warm, the climate of that city being so disagreeable during July and August that people are glad to leave town for the more genial air of the mountains.
- 73. THE HIGHEST TRESTLE IN THE WORLD, NEAR MUIR’S PEAK, SHASTA RANGE. It certainly is odd to have people living in the heart of a great city ask you during these two months if it is hot out in the country. “Out
- 74. in the country” means forty or fifty miles out, where there is plenty of heat and sunshine. At Shasta Springs, however, the weather is cooler. The climate is delightful, the water refreshing and the strawberries beyond compare. Boteler, known as a lover of strawberries, once said of his favorite fruit: “Doubtless God could have made a better berry, but doubtless God never did.” Just beyond the springs stand the wonderful Castle Crags. Hidden in the very depths of these lofty Crags lies a beautiful lake. This strange old castle of solid granite, its towers and minarets casting long shadows in the moonlight for centuries, is not without its historic interest, though feudal baron nor chatelaine dainty ever ruled over it. Joaquin Miller, in the “Battle of Castle Crag,” tells the tale of its border history. Not far away at the base of Battle Rock a bloody battle was once fought between a few whites and the Shasta Indians on one side and the Modoc Indians on the other. MOUNT SHASTA. By permission of F. Laroche, Photographer, Seattle, Washington.
- 75. The Indians of California say that Mt. Shasta was the first part of the earth created. Surely it is grand enough and beautiful enough to lay claim to this pre-eminence. When the waters receded the earth became green with vegetation and joyous with the song of birds, the Great Manitou hollowed out Mt. Shasta for a wigwam. The smoke of his lodge fires (Shasta is an extinct volcano) was often seen pouring from the cone before the white man came. Kmukamtchiksh is the evil spirit of the world. He punishes the wicked by turning them into rocks on the mountain side or putting them down into the fires of Shasta. Many thousands of snows ago a terrible storm swept Mt. Shasta. Fearing that his wigwam would be turned over, the Great Spirit sent his youngest and fairest daughter to the crater at the top of the mountain to speak to the storm and command it to cease lest it blow the mountain away. She was told to make haste and not to put her head out lest the Wind catch her in his powerful arms and carry her away. The beautiful daughter hastened to the summit of the peak, but never having seen the ocean when it was lashed into a fury by the storm wind, she thought to take just one peep, a fatal peep it proved. The Wind caught her by her long red hair and dragged her down the mountain side to the timber below. At this time the grizzly bears held in fee all the surrounding country, even down to the sea. In those magic days of long ago they walked erect, talked like men and carried clubs with which to slay their enemies. At the time of the great storm a family of grizzlies was living in the edge of the forest just below the snow line. When the father grizzly returned one day from hunting he saw a strange little creature sitting under a fir tree shivering with cold. The snow gleamed and glowed where her beautiful hair trailed over it. He took her to his wife who was very wise in the lore of the mountains. She knew who the strange child was but she said nothing about it to old father
- 76. grizzly, but kept the little creature and reared her with her own children. When the oldest grizzly son had quite grown up his mother proposed to him that he marry her foster daughter who had now grown to be a beautiful woman. Many deer were slain by the old father grizzly and his sons for the marriage feast. All the grizzly families throughout the mountains were bidden to the feast. When the guests had eaten of the deer and drank of the wine distilled from bear berries and elder berries in moonlight at the foot of Mt. Shasta, when the feast was over, they all united and built for their princess a magnificent wigwam near that of her father. This is “Little Mt. Shasta.” The children of this strange pair were a new race,—the first Indians. Now, all this time the great spirit was ignorant of the fate of his beloved daughter, but when the old mother grizzly came to die she felt that she could not lie peacefully in her grave until she had restored the princess to her father. Inviting all the grizzlies in the forest to be present at the lodge of the princess, she sent her oldest grandson wrapt in a great white cloud to the summit of Mt. Shasta to tell the Great Spirit where his daughter lived. Now when the great Manitou heard this he was so happy he ran down the mountain side so fast that the snow melted away under his feet. To this day you can see his footprints in the lava among the rocks on the side of the mountain. The grizzlies by thousands met him and standing with clubs at “attention” greeted him as he passed to the lodge of his daughter. But when he saw the strange children and learned that this was a new race he was angry and looked so savagely at the old mother grizzly that she died instantly. The grizzlies now set up a dreadful wail, but he ordered them to keep quiet and to get down on their
- 77. hands and knees and remain so until he should return. He never returned, and to this day the poor doomed grizzlies go on all fours. A wonderful feat of jugglery, but a greater was that of the Olympian goddess who changed the beautiful maiden Callisto into a bear, which Jupiter set in the heavens, and where she is to be seen every night, beside her son the Little Bear. The angry Manitou turned his strange grandchildren out of doors, fastened the door and carried his daughter away to his own wigwam. The Indians to this day believe that a bear can talk if you will only sit still and listen to him. The Indians will not harm a bear. Now for the meaning of those queer little piles of stones one sees so frequently in the Shasta mountains. If an Indian is killed by a bear he is burned on the spot where he fell. Every Indian who passes that way will fling a stone at the fated place to dispel the charm that hangs over it. “All that wide and savage water-shed of the Sacramento tributaries to the south and west of Mt. Shasta affords good bear hunting at almost any season of the year—if you care to take the risks. But he is a velvet-footed fellow, and often when and where you expect peace you will find a grizzly. Quite often when and where you think that you are alone, just when you begin to be certain that there is not a single grizzly bear in the mountains, when you begin to breathe the musky perfume of Mother Nature as she shapes out the twilight stars in her hair, and you start homeward, there stands your long lost bear in your path! And your bear stands up! And your hair stands up! And you wish you had not lost him! And you wish you had not found him! And you start for home! And you go the other way glad, glad to the heart if he does not come tearing after you.”[1] Downward from Mt. Shasta flows the Sacramento river. For thirty miles it goes tumbling over bowlders and granite ledges on its way to the sea. In mid-summer the Sacramento cañon is a paradise of umbrageous beauty, a region of forest and groves, of leafy shrubs,
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