Letters to the Editor
The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.
Nonegenarian Fibonacci Devotee
Please let me take this opportunity to make one more obeisance to the Fi bonacci sequence.Fibonacci tended to take over my mathematical life from the time, many years ago, when I found that the occurrence of the numbers in leaf patterns needed more explaining. One thing led to another, decade after decade, paper after paper. 1 I lived com fortably among these numbers-until midnight of April 26, 2003. At that in stant, I ceased to be 89 years old; and there seems little prospect of my ever again having aFb i onacci number as my age. To be sure, my rural route address is now Box 532, Route 1, a concatena tion of Fibonacci numbers in reverse order, but that is small consolation. Something more is needed to re affirm my allegiance. Here is my offer ing. I will prove that theFb i onacci num bers with odd index can be generated iteratively from the quadratic equation x2+y2
(la)
=
3xy - 1
in the following way. Put x equal to any Fibonacci number with odd index;:::: 1, and solve (1a) for y; the larger root will be the Fb i onacci number with the next larger odd index. The Fibonacci num bers with even index are generated by
'For instance. 2V.
4
E.
my articles in J.
exactly the same procedure from the equation x2+y2=3xy+l.
(lb)
To prove these, I will use an imme diate consequence of the defining iter ation Fn+l = Fn+Fn-t: (2) I
Fn-2 + Fn+2
3F , .
will also use the identity
(3)
Fn-2Fn+2
=
Fn2+ (-1) n+l,
which is a special case of an identity in Hoggatt.2 Now I set x = Fn (n odd) in (la) (4)
Fr/ + y2
=
3FnY- 1,
and I am able to show that the larger root for y is F11+2 . Substituting (3) on the left and (2) on the right of (4) re duces it to
which does indeed have F,+2 as its larger root. Similarly for the assertion for even n. Irving Adler 297 Cold Spring Road North Bennington, VT 05257 USA e-mail:
[email protected] Theor. Bioi. 45 (1g74), 1-7g: and J. Algebra 205 (1ggs), 227-243.
Hoggatt, Jr. Fibonacci and Lucas Numbers (Houghton Mifflin, 1969). See p. 59.
THE MATHEMATICAL INTELLIGENCER © 2006 Springer Science+ Bus1ness Med1a, Inc
=
BARRY KOREN
Computationa F uid Dynamics· Science and Too The year 2003 marked the 1 OOth anniversary of both the birth of John von Neumann and the first manned flight with a powered plane-both events of great importance for computational fluid dynamics.
he science of flows of gases and liquids is fluid dynamics, a subdiscipline of physics. No courses in fluid dynamics are given in high school, as it requires too much mathematical background. Fluid dynamics is taught at university and at engineering colleges, for one cannot ignore fluid dynamics if one wants to design an aircraft, a rocket, a combustion engine, or an artificial heart.
Particularly for aircraft design, knowledge and under standing of fluid dynamics-aerodynamics in this case-is of major importance. Except for the dangerous gravity, all forces acting on a flying plane are forces exerted by air. To fly an aircraft safely (tanked up with fuel and with pas sengers on board), a precise knowledge, understanding, and control of these aerodynamic forces is a matter of life and death. Moreover, flying must not only be safe but also fuel-efficient and quiet. For aerospace engineering, aero dynamics is indispensable. Nowadays, both experimental and theoretical means are available for investigating fluid flows. Wind tunnels are the canonical tool for experimental aerodynamics. The Wright brothers, who made the first manned flight with a powered plane (Fig. 1), had at their disposal a wind tunnel, one they themselves had made. Wind-tunnel testing has many dis advantages, but it is deemed trustworthy because real air is used and not the virtual air of theoretical aerodynamics. A Brief History of Computational Fluid Dynamics
Nowadays, the technological relevance of theoretical aero dynamics, of theoretical fluid dynamics in general, is widely appreciated. However, in the past it was mainly an acade-
mic activity, with results that strongly differed from ex perimental observations. The technical applications of fluid dynamics developed independent of theory. Theoretical and technological breakthroughs have since closed the gap between theory and practice, and today we see a fruitful interaction between the two. The airplane has played a very stimulating role in this development. I proceed by highlighting some key developments from the history of theoretical fluid dynamics with an eye toward computational fluid dynamics. Revolutionary innovations
Theoretical fluid dynamics has an illustrious history [1, 2, 3]. In the course of centuries, many great names have con tributed to the understanding of fluid flow and have helped in building up theoretical fluid dynamics, step by step. The oretical fluid dynamics goes back to Aristotle (384-322 BC), who introduced the concept of a continuous medium. In my opinion, though, it actually began 2000 years later, when Leonhard Euler published his equations of motion for the flow of liquids and gases, on the basis of Newton's second law of motion [4, 5]. Euler's idea to describe the motion of liquids and gases
© 2006 Springer Science+Business Media, Inc., Volume 28. Number 1, 2006
5
Trail-blazing ideas from Princeton
Figure 1 . Glass-plate photo of the first manned flight with a powered plane (flight distance: 37 meters, flight time: 12 seconds), Kitty Hawk, North Carolina, 1 0.35 h., December 17, 1 903. Prone on the lower wing: Orville Wright. Running along with the plane to balance it by hand if necessary: Wilbur Wright. Visible in the foreground: the rail from which the plane took off and the bench on which the wing rested. The very same day, the Wright brothers made a flight of 260 meters and 59 seconds!
in the form of partial differential equations was a revolu tionary innovation. However, his equations, known today as the Euler equations, were still unsuited for practical ap plications, because they neglect friction forces: only pres sure forces were taken into account. It was almost a cen tury later, in 1845, that George Stokes proposed fluid-flow equations which also consider friction [6]; equations which, for an incompressible flow, had already been found by Claude Navier [7] and are now known as the Navier-Stokes equations. With the introduction of the Navier-Stokes equa tions, the problem of understanding and controlling a large class of fluid flows seemed to be within reach, as it had been reduced to the integration of a handful of fundamen tal differential equations. Although formulating the Navier-Stokes equations con stituted great progress, the analytical solution of the com plete equations was not feasible. (It remains one of the out standing open mathematical problems of the 21st century.) One developed instead a large number of simplified equa tions, derived from the Navier-Stokes equations for special cases, equations that could be handled analytically. More over, a gap continued between experimental and theoreti cal fluid dynamics. The former developed greatly during the Industrial Revolution, independent of the latter. It was para doxical that the introduction of the Navier-Stokes equa tions led to a further fragmentation into different flow mod els, all of which described the flow of the same fluid (air in our case)-a theoretically highly undesirable situation. Theoretical fluid dynamics stagnated along a front of non linear problems. This barrier was finally broken in the sec ond half of the 20th century, with numerical mathematics, at the expense of much-often very much-computational work A key role was played in this by a Hungarian-born mathematician, John von Neumann.
6
THE MATHEMATICAL INTELLIGENCER
In the tens and twenties of the past century, Budapest was a fruitful breeding ground for scientific talent. It saw in 1903 the birth of John von Neumann (Fig. 2). In his early years von Neumann received a private education; at the age of 10 he went to school for the first time, directly to high school. There, his great talent for mathematics was discovered. He received extra lessons from mathematicians of the Univer sity of Budapest, among them Michael Fekete, with whom von Neumann wrote his first mathematics paper, at the age of 18. By then he was already a professional mathematician. Von Neumann studied at the ETH ZUrich and the University of Budapest; he obtained his PhD degree at the age of 22. Next he moved to Gem1any, where he lectured at universi ties in Berlin and Hamburg. There he was particularly active in pure mathematics: in set theory, algebra, measure theory, topology, and group theory. He contributed to existing theo ries: the sure way to quick recognition. From the mid 1930s, von Neumann chose a riskier way of working: breaking new ground. He turned to applied mathematics in the sense of mathematicians like Hilbert and Courant, i.e., not mathe matics applied to all kinds of ad hoc problems, but the sys tematic application of mathematics to other sciences, in par ticular to physics, with subdisciplines like aerodynamics. The rapidly deteriorating political situation in Europe, from which von Neun1ann had already emigrated to Princeton, 1 played a ·------- - -------
1Von Neumann was one of the many scientists who left Europe in the early 1 930s. For a description of the fall of Gottingen under Nazi pressures. see Richard Courant's biography [8].
Figure 2. John von Neumann, 1 903-1 957.
role in this decision. War was looming and brought in creasing demands for answers to questions related to mil itary engineering. Whereas von Neumann had worked on a mathematical basis for the equations of quantum mechanics before the war, during the war he "lowered" himself to developing nu merical solution methods for the Euler equations. His idea to compute possible discontinuities in solutions of the Euler equations without explicitly imposing jump relations was very original. Instead, von Neumann proposed the in troduction of artificial (numerical) diffusion, in such a way that the discontinuities automatically appear in a physically correct way: shock capturing, nowadays a standard tech nique. He also came up with an original method for ana lyzing the stability of numerical calculations: a Fourier method, now a standard technique as well. In 1944, the urgent need arose to apply von Neumann's numerical methods on automatic calculators, computers, beyond the scope of the machines of that time. This moti vated von Neumann to start working also on the develop ment of the computer. In 1944 and 1945 he did trail-blazing work, writing his numerical methods for computing a fluid flow problem in a set of instructions for a still non-existent computer. These instructions were not to be put into the computer by changing its hardware or its wiring. Instead, von Neumann proposed to equip computers with hardware as general as possible, and to store the computing instruc tions in the computer, together with the other data involved (input data, intermediate results, and output data). In 1949,
the first computer was realized which completely fulfilled von Neumann's internal programming and memory princi ples: the EDSAC (Electronic Delay Storage Automatic Cal culator), by M. V. Wilkes, at Cambridge University. Today, the two principles are still generally applied. L. F. Richardson and Richard Courant and colleagues had combined theoretical fluid dynamics and numerical mathe matics before von Neumann [9, 10], but still without clear ideas about computers-without computer science. Compu tational fluid dynamics (abbreviated CFD) is a combination of three disciplines: theoretical fluid dynamics, numerical mathematics, and computer science. Because von Neumann brought in this last discipline, he can be considered the found ing father of CFD. A detailed description of von Neumann's contributions to scientific computing is given by Aspray [11). A good overview of his other pioneering work can be found in the scientific biography written by Ulam [12]. Traveling from place to place as an honored mathe matician with many social and political obligations, von Neumann must have had very little time to write down his scientific ideas. He published only one paper about both shock capturing and the aforementioned stability analysis, and that not until 1950 [13]. On his many travels, von Neumann visited the Nether lands. In 1954, he was an invited speaker during the Inter national Congress of Mathematicians held in Amsterdam. A tea party with Queen Juliana of the Netherlands was arranged for a select group of participants, among them John von Neumann (Fig. 3).
Figure 3. John von Neumann and colleagues at Soestdijk Palace. Above: all together, John von Neumann front row, far left. Queen Juliana, with white handbag, is flanked by the two new recipients of the Fields Medal: Jean-Pierre Serre (with Herman Weyl's hands on his shoulders) and Kunihiko Kodaira. At the right of von Neumann: Mary Cartwright.
© 2CXJ6 Springer Science+ Business Media, lnc., Volume 28, Number 1, 2006
7
Pioneering work in Amsterdam
In the third quarter of the 20th century, computer science was a new and growing discipline. Initially, the Netherlands played no significant role in the development of computer science, but the country was quickly moving forward. In 1946, the Mathematisch Centrum (MC) was founded in Am sterdam. The mission of this new institute was to do pure and applied mathematics research in order to increase "the level of prosperity and culture in the Netherlands and the contributions of the Netherlands to international culture." (Not at all the pure ivory tower.) The foundation of the MC did not proceed without struggle. The most prominent Dutch mathematician of the day, L. E. J. Brouwer of the University of Amsterdam, was of the opinion that mathe matics should be indifferent towards the physical sciences and even rejecting of technology; an odd point of view, con sidering the work of mathematicians like Hilbert, Courant, and von Neumann. With the MC, Brouwer wanted to tum Amsterdam into the new Gottingen of pure mathematics. It did not work out that way. His biographer feels that Brouwer was sacrificed to the foundation of the MC ([14], p. 479). The founders of the MC had heard about von Neumann's ideas about machines which should be able to perform a series of calculations as independently as possible. They wanted the MC to have a computing department in order to develop a computer and to execute advanced comput ing work. Aad van Wijngaarden (Fig. 4), former student of J. M. Burgers of the Delft University of Technology, was appointed as the first staff member of the MC, in 1947. That year he made a study tour to visit von Neumann in Prince ton. Van Wijngaarden and his co-workers designed and con structed the first Dutch computer: the ARRA I (Automa tische Relais Rekenmachine Amsterdam I, Fig. 5). New
Figure 4. Aad van Wijngaarden, 1 91 6- 1 987 (photo courtesy of CWI).
8
THE MATHEMATICAL INTELLIGENCER
computers were designed and built (one per design only), in 1955 exclusively for the Fokker aircraft industries: the FERTA (Fokker Elektronische Rekenmachine Type ARRA). Much human labor was required to perform com putations on these early computers. At the MC, this was done by young women (Fig. 6), schooled in mathematics by Van Wijngaarden. A highlight was the project for the de velopment of the Fokker Friendship airplane, a numerical project on which Van Wijngaarden and his "computing girls" worked from 1949 until 1951. The computations con cerned oscillations of the airplane's wing in subsonic flow: flutter. The first computing work was still very "external" and machine-dependent; for each computation, cables had to be plugged into the computer. With the accomplishment of internal programming as proposed by von Neumann, at tention shifted entirely to the invention of algorithms and their coding as computer programs. Edsger Dijkstra, a later Turing Award recipient, was appointed at the MC as the first Dutch computer programmer. Van Wijngaarden and Dijkstra left an international mark on computer science with their contributions to the development of the pro gramming language Algol 60 [15], and Van Wijngaarden later added to that reputation with Algol 68 [16]. In 1979, Van Wijngaarden was awarded an honorary doctorate for his pioneering work by the Delft University of Technology, and the MC grew into the present CWI (Centrum voor Wiskunde en Informatica), which celebrates its 60th an niversary in 2006. Computational fluid dynamics research on the basis of the Euler or Navier-Stokes equations, of the same funda mental character as that established by von Neumann, was not done in the Netherlands of 1945-1960. For this funda mental work, we have to go to the United States and the Soviet Union of the 1950s.
Figure 5. The first computer at the MC and in the Netherlands, the ARRA I in its final set-up. From left to right: power frame and the three arithmetic registers. On the table in the middle: the punch-tape reader. Some 1 200 relays are at the back of the machine {photo courtesy of CWI).
A continuous flow of CFD from New York
In early December 1941, a passenger ship carried a 15-year old Hungarian boy from Europe to the United States. The boy, along with his parents, was escaping the tragic fate threatening European Jews. (It was to be the last passen ger ship from Europe's mainland to the United States for years to come. During the voyage, the United States was drawn into the Second World War by the attack on Pearl
Harbor.) The young ship passenger carried with him two letters of recommendation from his teachers. It seems likely that von Neumann saw those letters brought by his young fellow-countryman, for the boy, Peter Lax (Fig. 7), had a meteroric rise to success. In 1945, while still a teenager, he became involved in the Manhattan Project. In 1949, he received his PhD degree from New York Univer sity, with Richard Courant as his thesis advisor, and in 1951,
Figure 6. Female arithmeticians at the MC, the "girls of Van Wijngaarden." In the foreground: Ria Debets, later the spouse of Edsger Dijk stra. {photo courtesy of CWI).
© 2CX>6 Springer Science+Business Media, Inc., Volume 28, Number 1, 2006
9
from Lax: "The impact of computers on mathematics (both applied and pure) is comparable to the roles of telescopes in astronomy and microscopes in biology." Despite the Second World War and the Cold War, Lax has always had very good connections with scientists worldwide. One such relation is with a famous Russian mathematician, mentioned in the next section. A brilliant idea from Moscow
Figure 7. Peter D. Lax.
he became assistant professor there. His work in mathe matics continues to this day and has led to many honors and awards, among them the 2005 Abel Prize in mathe matics [17]. Like von Neumann, Lax is a homo universalis in math ematics. He has performed ground-breaking research, and has been a productive and versatile author of mathematics books. His books deal with such diverse topics as partial differential equations, scattering theory, linear algebra, and functional analysis. Above all, he is known for his research on numerical methods for partial differential equations, in particular for hyperbolic systems of conservation laws, such as those arising in fluid dynamics. Lax's name has been given to several mathematical discoveries of impor tance to CFD: •
•
•
•
•
the Lax equivalence theorem [18], stating that consis tency and stability of a finite-difference discretization of a well-posed initial-boundary-value problem are neces sary and sufficient for the convergence of that dis cretization, the Lax-Friedrichs scheme [ 19], a stabilized central finite difference scheme for hyperbolic partial differential equations, the Lax-Wendroff scheme [20], a more accurate but equally stable version of the Lax-Friedrichs scheme, the Lax entropy condition [21], a principle for selecting the unique physically correct shock-wave solution of nonlinear hyperbolic partial differential equations that al low multiple shock-wave solutions, and the Harten-Lax-Van Leer scheme [22], a very efficient nu merical method for solving the Riemann problem.
Like von Neumann, Lax was (and still is) a strong pro ponent of the use of computers in mathematics. A quote
10
A substantial part of the Euler and Navier-Stokes software used worldwide is based on a single journal paper [23], dis tilled by the then-young Russian mathematician Sergei Kon stantinovich Godunov (Fig. 8) from his PhD thesis. Godunov proposed the following. Suppose one has a tube and in it a membrane separating a gas on the left with uni formly constant pressure, from a gas on the right with a like wise uniformly constant but lower pressure (Fig. 9, top). If the membrane is instantaneously removed-the traffic light changes from red to green-then the yellow gas will push the blue gas to the right; the interface between the two gases, the contact discontinuity, runs to the right. At the same time, two pressure waves start running through the tube: a compression wave running ahead of the contact dis continuity and an expansion wave running to the left (Fig. 9, bottom). In the 19th century, the Euler flow in this tube, a shock tube, had already been computed by Riemann, with "pencil and paper" [24]. (For this old work of Riemann, Duivesteijn has written a nice, interactive Java applet [25].) For the computation of the flow in a tube in the case of an initial condition which has more spatial variation, Godunov proposed to decompose the tube into virtual cells (Fig. 10, above), with a uniformly constant gas state in each cell, and with each individual cell wall to be considered as the afore mentioned membrane (traffic light). To know the interac-
THE MATHEMATICAL INTELLIGENCER
Figure 8. Sergei Konstantinovich Godunov.
rarefaction wave
contact discontinuity .....
shockwave
Figure 9. Shock tube. Top: condition of rest in left and right part: high and low pressure, respectively. Bottom: condition of motion with shock wave and contact discontinuity running to the right and rarefaction wave running to the left. (drawing: Tobias Baanders, CWI).
tion between the gas states in two neighboring cells, one in stantaneously 'removes' the cell wall separating the two cells, and computes the Riemann solution locally there, and hence the local mass, momentum, and energy flux (Fig. 10, bottom). This is done at all cell faces. With this, the net transport for each cell is known and a time step can be made. A plain method and a very simple flow problem, so it seems. If one can do this well, the flow around a com plete aircraft or spacecraft can be computed. The remark able property of the method is that at the lowest discrete level, that of cell faces, a lot of physics has been built into it, not just numerical mathematics. The more cells, the better the accuracy, yet also the more expensive the computation. Godunov did not have ac cess to computers, but to "computing girls," who called Godunov and his fellow PhD students "that science," and who received payment on the basis of the number of com putations they performed, right or wrong. No real CFD there either! In 1997, Godunov received an honorary doctorate from the University of Michigan, and a symposium was organized
for him at the university's Department of Aerospace Engi neering. At that symposium, in a one-and-a-half-hour lec ture, Godunov gave insight into his earlier research, whose strategic importance was not appreciated in the Soviet Union at the time. This historic lecture has since been pub lished [26, 27]. A second important result in Godunov's classical paper from 1959 [23] is his proof that it is impossible to devise a linear method which is more than first-order accurate, with out being plagued by physically incorrect oscillations in the solution: wiggles (Fig. 11). With a first-order-accurate method, the solution becomes twice as accurate and re mains free of wiggles when the cells are taken twice as small. With a second-order-accurate method, the solution becomes four times more accurate then, but-unfortu nately-possibly wiggle-ridden. Wiggles can be very troublesome in practice. For ex ample, a simple speed-of-sound calculation in a single cell only may break down the entire flow computation, because of a possibly negative pressure. The wiggle problem does not occur only with Godunov's method; it is a general prob-
Figure 10. Shock tube divided into small cells. Top: cells. Bottom: wave propagation over all cell faces. (drawing: Tobias Baanders, CWI).
© 2006 Springer Science + Bus1ness Media, Inc., Volume 28, Number 1, 2006
11
pressure
pressure
1
0
Figure 1 1 . Right and wrong pressure distribution. Left: without wiggles. Right: with wiggles. (drawing: Tobias Baanders, CWI).
lem. A drawback of Godunov's method is that it is com puting-intensive; at each cell face, the intricate Riemann problem is solved exactly. Technology pushes from Lelden
It took about two decades before good remedies were found for the wiggles of higher-order methods and the high cost of the Godunov algorithm. The aid came from an as tronomer. In space, large clouds of hydrogen are found. Simulation of the flow of this hydrogen provides models of the development of galaxies. The literally astronomical
Figure 12. Bram van Leer (photo: Michigan Engineering).
12
THE MATHEMATICAL INTELUGENCER
speeds and pressures which may arise in these computa tions impose high demands on the accuracy, and particu larly the robustness, of the computational methods to be applied. While still in Leiden, in the 1970s, the astronomer Bram van Leer (Fig. 12) published a series of papers in which he proposed methods which are second-order ac curate and do not allow wiggles. The fifth and last paper in this series is [28). Furthermore, Van Leer introduced a computationally efficient alternative to the Godunov algo rithm [29): two technology pushes, not only for astronomy but also for aerospace engineering, as well as for other dis-
ciplines.In 1990, Van Leer was awarded an honorary doc torate for this work by the Free University of Brussels. Efficient solution algorithms from Rehovot and other places
Broadly speaking, how does an Euler- or Navier-Stokes flow computation around an aircraft work? The airspace out to a large distance from the aircraft, may be divided into (say) small hexahedra, small 3D cells. Just as in the 1D shock tube example, one can then compute for each cell the net inflow of mass, momentum, and energy, using at each cell face the Godunov alternative ala Van Leer or other alternatives, like the Roe scheme [30] or the Osher scheme [31]. The finer the mesh of cells around the aircraft, the grid (Fig. 13), the more accurate the solution, but also the higher the computing cost. A grid of one million cells for an Euler- or Navier-Stokes-flow computation is not un usual. Suppose that we want to simulate a steady flow. We then have to solve, per cell, five coupled nonlinear partial differential equations. The cells themselves are coupled as well: what flows out of a cell flows into a neighboring cell (or across a boundary of the computational domain). In Navier-Stokes-flow computations, the flow solution in a sin gle cell may influence the flow solutions in all other cells. In our modest example, we may have to solve a system of five million coupled nonlinear algebraic equations. Effi cient solution of these millions of equations is an art in it self.Many efficient solution algorithms have been devel-
oped, the most efficient of which are the multigrid algo rithms.Multigrid methods were invented at different loca tions and by several people. A leading role has been played by Achi Brandt from the Weizmann Institute of Science in Rehovot, Israel [32].Multigrid algorithms have a linear in crease of the computing time with the number of cells. This may seem expensive-2, 3, or 4 times higher computing cost for a grid with 2, 3, or 4 times more cells, respectively but it is not. In numerical mathematics, no bulk discount is given. For many solution algorithms the rule is 22 ,32,42, . ..times higher computing cost for a grid with 2,3,4, . .. times more cells! For the interested reader, a book on multi grid methods is [33]. Present State of the Art in CFD An example
A quick impression will now be given of what can be done with CFD by looking at a standard flow problem. It con cerns the recent MSc work of Jeroen Wackers. From scratch, he developed 2D Euler software in which the grid is automatically adapted to the flow, and what follows de scribes one of his results. Consider the channel depicted in Figure 14, and in it a uniformly constant supersonic air flow (from left to right) at three times the speed of sound. One may consider the channel to be a stylized engine inlet of a supersonic aircraft. In fact it is just a benchmark geometry [34, 35]. The red vertical valve at the bottom of the chan nel instantaneously snaps up, so that, together with the red
Figure 1 3. Cross sections of a hexahedral grid around the Space SHuttle.
© 2006 Spnnger Science+ Business Med1a, Inc .. Volume 28, Number 1. 2006
13
'
' \_)
\ 7
flow speed
speed of sound
' ..J\
I
Figure 1 4. 20, parallel channel. In it, a parallel plate and a vertical valve which is still open.
horizontal plate, it forms a step which suddenly chokes part of the channel. Figure 15 shows a computational result. We see how the uniformly constant initial solution and the grid have de veloped after some time. The computational method highly satisfies the often conflicting requirements of numerical stability, accuracy, and monotonicity on the one hand, and computing and memory efficiency on the other [36].
also scientific journals dedicated to CFD. Moreover, off the-shelf CFD software can be purchased these days. Each issue of, e.g., the monthly Aerospace America contains col orful, full-page advertisements for CFD software. A practi cal overview of the CFD literature, software, and also va cancies can be found on the Web site of CFD Online [41]. Today, CFD's role is about as important as that of ex perimental fluid dynamics. And CFD continues to grow. It is fed by improvements in both computer science and nu merical mathematics. In addition, CFD itself stimulates re search in computer science and numerical mathematics: a fruitful interaction.
Books, journals, and software
Twenty years ago, textbooks on CFD were rare, but sev eral are available now (see, e.g., [37, 38, 39, 40]). There are
>
X
>
X Figure 15. Computational result some time after instanteously closing the lower part of the channel. Top: iso-lines of density. When the ver tical valve is still in the open position, the density in the entire channel is constant (everywhere the same blue color as at the inlet). Bottom: computational grid automatically adapted to flow solution.
14
THE MATHEMATICAL INTELLIGENCER
At present, CFD enters into full cooperation with other disciplines, such as structural mechanics (computational fluid-structure interactions) and electromagnetism (com putational magnetohydrodynamics).
[3] J. D . Anderson, A History of Aerodynamics, Cambridge University Press, Cambridge, 1997. [4] L. Euler, Principes gf!meraux du mouvement des fluides, Memoires de I'Academie des Sciences de Berlin, 11 (1755), pp. 274-31 5 . [5] M . D . Salas, Leonhard Euler and his contributions t o fluid me
Outlook
The fact that commercial CFD software is a success is proof of the practical importance of the theoretical fluid-dynam ics work since Euler. The growing availability of CFD soft ware may seem to be a threat for CFD research; CFD re searchers seem to make themselves redundant by their own success. Yet, this growing software availability may also be considered a good development. Not everyone has to write his/her own Euler or Navier-Stokes code. Coding such soft ware from scratch gives the best insight and is pleasing work, but it may easily take too much time.
chanics , AIM-paper 88-3564, AIM, Reston, VA, 1988. [6] G. G . Stokes, On the theories of the internal friction of fluids in mo
tion, and of the equilibrium and motion of elastic solids, Transac tions of the Cambridge Philosophical Society, 8 (1845), pp. 287. [7] C. L. M . H. Navier, Memoire sur les lois du mouvernent des f/uides , Memoires de I'Academie des Sciences, 6 (1822), pp. 389-440. [8] C. Reid, Courant in Gottingen and New York. The Story of an Im
probable Mathematician, Springer-Verlag, New York, 1976. [9] L. F. Richardson , Weather Prediction by Numencal Process, Cam bridge University Press, Cambridge, 1922. [10] R. Courant, K. 0. Friedrichs, and H . Lewy, Ober die partie/len Dif
ferenzgleichungen der mathematischen Physik, Mathematische Education
A new question arises: How to teach CFD, now that it has become more and more important as an easily available, au tomatic tool? Not just factual knowledge but also under standing of the mathematical and physical principles of CFD remains indispensable, not only when practicing it as a sci ence, but also when using it as a tool. The CFD-tool user must know and understand these principles well in (1) pos ing the computational problem, (2) choosing the numerical method to solve that problem, and (3) interpreting the com putational results. The user must know the possibilities and limitations of computational methods and should be able to assess whether the computational results obtained fulfill the expectations or not. If not, it should be found out why. Thus CFD is not solving flow problems by blind numerical force. On the contrary, stimulated by the growing potential of CFD, still more complicated flow problems will be considered, problems which will require even more knowledge and un derstanding of flow physics and numerical mathematics.
Annalen, 1 00 (1 928), pp. 32-74. [11] W. Aspray, John von Neumann and the Origins of Modern Com
puting, MIT Press, Cambridge, Massachusetts, 1990. [1 2] S. Ulam, John von Neumann, 1903-1957, Bulletin of the Ameri
can Mathematical Society, 64 (1958), pp. 1-49. [13] J. von Neumann and R. D. Richtmyer, A method for the numeri cal calculation of shocks, Journal of Applied Physics, 21 (1950), pp. 232-237. [14] D . van Dalen, L. E. J . Brouwer, 1 881-1966, Het Heldere Licht van
de Wiskunde, Bert Bakker, Amsterdam, 2002. [15] P. Nauer (ed .), Revised Report on the Algorithmic Language Algol
60
(available
for
download
from
http://www. masswerk.at/
algol60/report.htm). [16] A. van Wijngaarden et al., Revised Report on the Algorithmic Lan
guage Algol 68, Springer-Verlag , Berlin, 1 976. [17] http://www.abelprisen.no/en/. [18] P. D. Lax and R. D. Richtmyer, Survey of the stability of linear fi nite difference equations, Communications on Pure and Applied
Mathematics, 9 (1 956), pp. 267-293. [19] P. D . Lax, Weak solutions of nonlinear hyperbolic equations and
Research
their numerical computation, Communications on Pure and Ap
As
plied Mathematics, 7 (1954), pp. 159-1 93.
CFD becomes more and more mature, it also becomes more difficult to contribute fundamental research to it. In re cent decades a PhD student can hardly do such fundamental work as Godunov did. Students will have to acquire an ever growing knowledge and understanding of CFD before they can start working in it themselves. On the other hand, thanks to the availability of CFD tools, the possibilities for applica tion of CFD are far greater now than in Godunov's era. Just how CFD will develop remains unpredictable, and this is part of what makes it an exciting and attractive discipline. In CFD plenty of research questions remain. New fluid flow problems will continue to arise, and there will cer tainly be times when we may say with Orville Wright, "Isn't it astonishing that all these secrets have been preserved for so many years just so that we could discover them!"
[20] P. D. Lax and B. Wendroff, Systems of conservation laws, Commu
nications on Pure and Applied Mathematics, 13 (1960) , pp. 217-237. [21] P. D. Lax, Hyperbolic Systems of Conservation Laws and the Math
ematical Theory of Shock Waves, SIAM, Philadelphia, 1973. [22] A. Harten, P. D. Lax, and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws,
SIAM Review, 25 (1 983), pp. 35-61. [23] S. K. Godunov, Finite difference method for the numerical com putation of discontinuous solutions of the equations of fluid dy namics, Mathemat1cheskfi"Sborn1k, 47 (1959), pp. 271-306. Trans lated from Russian at the Cornell Aeronautical Laboratory. [24] G. F. B. Riemann, O ber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite,
i n : Gesammelte Werke,
Leipzig,
1876. Reprint: Dover, New York, 1953. [25] G . F. Duivesteij n , Visual shock tube solver (to be downloaded from
REFERENCES
[1] T. von Karman, Aerodynamics, McGraw-Hill, New York, 1963. [2] H . Rouse and S. lnce, History of Hydraulics, Dover, New York, 1963.
http://www. piteon .ni/cfd/) . [26] B. van Leer, An Introduction to the article "Reminiscences about difference schemes", by S. K. Godunov, Journal of Computational
Physics, 153 (1999), pp. 1-5.
© 2006 Spn nger Science+Business Media,
Inc., Volume 28, Number
1, 2006
15
[27] S. K. Gudunov, Reminiscences about difference schemes, Jour nal of Computational Physics, 1 53 (1 999), pp. 6-25.
AUTHOR
[28] B. van Leer, Towards the ultimate conseNative difference scheme. V. A second-order sequel to Godunov's method, Journal of Com
putational Physics, 32 (1 979), pp. 1 0 1 -1 36. Reprint: Journal of Computational Physics, 135 (1997}, pp. 229-248. [29] B. van Leer, Flux-vector splitting for the Euler equations, in Lec ture Notes in Physics, Vol. 170, Springer-Verlag , Berli n , 1 982, pp. 507-5 1 2 . [30] P . L. Roe, Approximate Riemann solvers, parameter vectors, and differences schemes, Journal of Computational Physics, 43 (1981) . pp. 357-372. BARRY KOREN
[31] S. Osher and F. Solomon, Upwind difference schemes for hyper
CWI
bolic systems of conseNation laws, Mathematics of Computation ,
P.O. Box 94079
38 (1 982), pp. 339-374
1 090 GB Amsterdam
[32] A Brandt Multi-level adaptive solutions to boundary-value prob ,
lems, Mathematics of Computation , 31 (1 977), pp. 333-390.
The Netherlands
[33] U. Trottenberg, C. W Oosterlee, and A Schuller, Multigrid, Aca
e-mail:
[email protected] demic Press, New York, 2001 .
[34] A F. Emery, An evaluation of several differencing methods tor in
Barry Koren studied Aerospace Engineering at the Delft Insti
viscid fluid flow problems, Journal of Computational Physics, 2
tute of Technology, and Computational Fluid Dynamics at the
(1 968), pp. 306--331 .
Von Karman Institute for Fluid Dynamics in Belgium. He is now
[35] P. R . Woodward and P. Colella, The numerical simulation of two dimensional fluid flow with strong shocks, Journal of Computa
leader of the research group in Computing and Control at the Dutch Centre for Mathematics and Computer Science (CWI) in Amsterdam, and also professor of Computational Fluid Dy
tional Physics, 54 (1 984), pp. 1 1 5-1 73. [36] J. Wackers and B. Koren, A simple and efficient space-time adap tive grid technique for unsteady compressible flows, in Proceed
ings 1 6th AIM CFD Conference (CD-ROM), AIM-paper 2003-
namics at the Delft Institute of Techno l ogy . More information
can be found at http://homepages.cwi.nV-barry/. He is married and the father of three children.
3825, American Institute of Aeronautics and Astronautics. Reston , VA, 2003.
[37] P.
Wesseling,
Principles of Computational Fluid Dynamics.
Springer-Verlag, Berlin, 2001
tiona/ Methods for lnviscid and Viscous Flows, Wiley, Chichester, 1 988-1 990.
[38] P. J. Roache, Fundamentals of Computational Fluid Dynamics, Hermosa, Albuquerque. NM, 1 998.
[40] C. A J. Fletcher, Computational Techniques for Fluid Dynamics, Vol. 1 Fundamental and General Techniques, Vol. 2 Specific Tech
[39] Ch. Hirsch, Numerical Computation of Internal and External Flows. Vol. 1 Fundamentals of Numerical Discretization, Vol. 2 Computa-
niques for Different Flow Categories, Springer-Verlag, Berlin, 1 988. [41] http://www cfd-online.com
ROCKY MOUNTAIN MATHEMATICA
Mathematica workshop in the scenic and cool mountain environment of 9100 feet above sea level. From July 9-14,2006, Ed Packel (Lake Forest College)
Join us this summer for a Frisco, Colorado,
and Stan Wagon (Macalester College), both accredited Wolfram Research instructors with much experience, teach an introductory course (for those with little or no intermediate course on the use of
Mathemalica.
Mathematica background) and an
Anyone who uses mathematics in teaching, research,
or industry can benefit from these courses.
The introductory course will cover topics that include getting comfortable with the front end, 2- and 3-dimensional graphics, symbolic and numerical computing, and the basics of
Mathematica
programming. Pedagogical issues relating to calculus and other undergraduate courses are discussed. The intermediate course covers programming, graphics, symbolics, the front end, and the solution of numerical problems related to integration, equation-solving, differential equations, and linear algebra. Comments from participant Paul Grant, Univ. of California at Davis: The instructors had boundless enthusiasm for and an encyclopedic knowledge of Mathematica. I was
able to complete an animation project that I will be showing to my students. I highly recommend these courses to anyone wanting to learn Mathematica from the ground up or enhance their knowledge of the program.
For more info, contact packel©lakcforcst.edu or
[email protected], or visit rmm.lfc.edu.
16
THE MATHEMATICAL INTELLIGENCER
M a the m a tic a l l y B e n t
C o l i n Ad a m s , Editor
The Theorem Blaster Colin Adams T h e proof is in t h e pudding.
Opening a copy of The Mathematical Intelligencer you may ask yourself
uneasily, "What is this anyway-a mathematical journal, or what?" Or you may ask, "Where am !?" Or even "Who am !?" This sense of disorienta tion is at its most acute when you open to Colin Adams's column. Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.
Column editor's address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01 267 USA e-mail:
[email protected] I
s your theorem overweight? Need to trim the pages of your paper? Want to fit into that volume on semi-simple Lie Algebras? Then you're in luck! Be cause we are offering to the public, for the first time ever, the amazing Theo rem Blaster®. You've heard about it in the Math Lounge. You've seen it on Math TV. But now, due to a special pur chase agreement with the American Mathematical Society, we can offer it to you at drastically reduced prices. Overweight theorems can be dan gerous in the long term. Sluggish and impenetrable, they cause colleagues to lose interest in your work, leading to publication rate slowdown and severe career trajectory tailspin. But now you can have the theorem you always dreamed of. Yes, if you want hot-looking theorems, you want the Theorem Blaster. Here are just a few of the many testimonials from our satisfied customers: "My proofs were ponderous and un wieldy. Many mathematicians re fused to read them. But now I can't keep my preprints on the shelf. Col leagues can't keep their hands off them."-John Conrad, Princeton "Little children teased me. 'You're hiding your lack of mathematical in sight inside layers of impenetra ble notation,' they chanted in their barely audible sing-song voices as they played. I hated it, and I was in timidated, avoiding all contact with children. But now I walk past the playground with pride, my three page preprint held high for all the toddlers to see."-Robert Lazarus, Yale
"For many years, I could only sub mit to journals with no upper page limit, like the Transactions. Now I'm publishing two-page notes in the Proceedings of the AMS. Thank you, Theorem Blaster."-Karen Ham mond, Duke University How does it work? By isolating the log ical components of the core theorem and honing the fundamental connec tive tissue, the patented triple-action technology of the Theorem Blaster fo cuses its energies exactly where you need help the most. It burns off unde sirable symbols and equations. It re moves unsightly constants. It squeezes out the excess verbiage. As a special bonus, the Theorem Blaster comes with our special video "Five Days to Beautiful Proofs and The orems." Watch as our Ph.D.-trained personnel guide you in the use of your Theorem Blaster. In five days, you can say with pride, "Hey, there, random person walking down the street! Take a look at this proof!" Always lacked the confidence to show your proofs off at the Mathemat ical Sciences Research Institute in Berkeley? Afraid to take that long bus ride up the hill? In the pit of your stom ach was the fear that seminar partici pants might point and laugh at the cir cuitous route you took to prove your theorem. Or even worse, they might get up and leave ·in the middle of your· talk. Well, live in fear no longer. Go ahead, get on the bus! Because they will be begging you for more theorems by the time you are done! Well-known math trainer Steven Krantz has spent years teaching mathe maticians to write clearly and concisely. "I was amazed by the Theorem Blaster," he said. "It's like liposuction for math. My book A Primer of Math ematical Writing is superfluous in the face of this amazing invention." Do you have a closet filled with cumbersome machines, each bought on the promise of sudden size reduc-
© 2006 Springer Science+ Business Media, Inc . . Volume 28. Number 1 . 2006
17
tion? The Corollary Cutter, the Lemma Loser, the Conjecture Cruncher? Well, throw them all away. You won't need them anymore! And what about Mathersize@? Or n- ladies®? All those reduction fads are out the window! That's right. You don't need them anymore! A top university released the results of a study demonstrating that the The orem Blaster® is the fastest and safest method to reduce papers to a reason able size. They compared it to other re duction methods and found that the theorem survival rate with the Theo rem Blaster® approaches 95%, way
outperforming the competition. And, believe it or not, as a special deal for those ordering today, in addi tion to the Theorem Blaster, and the video "Five Days to Beautiful Proofs and Theorems," you will also receive instructions for the world-famous Analysis Diet®. That's right. The diet where you just drop analysis from your mathematical diet and watch the bloat come off. Epsilons and deltas, gone! cr-alge bms, gone! Radon-Nikodym deriv a tives, gone!
There is no guarantee that the bloat will stay off. Persons with heart conditions or difficulty digesting de Rham coho molgy should consult their Ph.D. advi sor before using the Theorem Blaster.
Your proofs will shrink at a dra-
ScientificWorkPiace· Mathematical Word Processing •
matic rate. Watch as lemmas and corol laries are exposed to view for the first time. It will bring out definition, in fact, lots of definitions. This opportunity is not available in any math department. Don't delay! Call our toll-free number today. Why not order one for a friend as well? He will appreciate the help. We all will!
Animate, Rof4te, Zoom, and Ry New in Version 5.5
• Compute and plot using the MuPAJY' 3
LAlEX Typesetting • Computer Algebra
computer algebra engine
• Animate 2D and 3D plots using MuPAD s VCAM
• Rotate, move, zoom in and out, and fly through 3D plots with new OpenGLe 3D graphics
• Label 2D and 3D plots
so that the label moves when you rotate or zoom a plot
• Create 3D implicit plots
• Import !HEX files produced by other programs • Use many new l.i\TE)C packages
• Get help from the new, extensive troubleshooting section of 'ljlpesetting Documents with Scientific WorkPlace and Scientific Word, Third Edition The Gold Standard for Mathematical
Animated Tube Plot • Totnlkt ananlmattd t�o�btplot 1
2
3
tht umt varleblts
The n.lll animlllion shows a knot being dr�WR. �
of a button allows you to typeset your
Typt en expreuion in one Ot two varitbles
With tht lnnrtion point in the t)ll),.n .ssio , Open 1M Plot ProperUe$ dialog end, on
Plot 3D Animated + Tube
r
-IOn,82). The geodesic segment [I,A - 1!2 BA - 112 ) is parametrised by y0(t) (A - 112BA - 112)1, by what we said about the commuting case. So, the geodesic (A,B) !fA'"{J),fAv.{A- 112BA- 112)) is parametrised by
=
lllog I - log (A - 112BA - 112)112 lllog(A - 112BA - 112)llz. The ma trices A - 112BA - l/2 and A - 1B have the same eigenvalues. So, this can be expressed as
= A 112 (A- 112 BA - 112)1 A 112 , 0 :::; t :::; 1.
-
This shows that the geometric mean A#B defined by the fonnula (2) is nothing but the midpoint of the geodesic join ing A and B in the Riemannian manifold !f1>n· Thus while (2), (3), and (4) might have ap,Reared as over-imaginative non commutative variants of �. very natural geometric con siderations lead to the same notion of mean as is given by (2). Note that for each t, y(t) defmed by ( 13) is a mean of A and B corresponding to the functionf(x) :xf in the for mula (6). Those means are not symmetric, however: (I) fails unless t 1/2. This discussion also gives an explicit formula for the metric 8 2 . We have o2 (A,B) 8 2 (I,A - 112BA - 112)
=
=
=
The inequality (9) captures an essential feature of lfl>n : it is a manifold of nonpositive curvature. To understand this, consider a triangle with three vertices 0, H, and K in §n· Under the exponential map, this is mapped to a "triangle" with vertices I, exp H and exp K in !f1>n· The lengths of the two sides [O,H) and [O,.K] measured by the norm ll· llz are equal to the lengths of their images [I, exp H) and [I, exp K] measured by the metric 82 . By the EMI (9), the length of the third side [ exp H, exp K] of the triangle in !f1>n is larger than (or equal to) IIH Kllz. The general case of a geodesic triangle with vertices exp A, exp B, exp C in !f1> n may be re duced to the special case by applying the congruence fexp(-A/2) to all points and thus changing one of the ver tices to I. This is often described by saying that two geo desics emanating from a point in !f1>n spread out faster than their pre-images (under the exponential map) in §n· It is instructive here to compare the situation with that of IUn, a compact manifold of non-negative curvature (Fig ure 1 ). In this case the real vector space i§n consisting of skew-Hermitian matrices is mapped by the exponential onto IUn. The map is not injective; it is a local diffeomor phism. Using the formula ( 1 1) with H and K in i§n, we reduce
=
exp(iA)
2
2. 5
2
1. 5
0. 5
0
0.5
1 .5
2
2.5
Figure 1. Three curvatures, showing a comparison of a Euclidean (curvature zero) triangle in §2 with its images under exp(-) in P2 (nonposi tive curvature) and exp(i·) in Q.J2 (non-negative curvature). The colours indicate matching vertices. Note that the geodesics emanating from exp(A) spread out faster than Euclidean ones (compare the straight lines at A), whereas those emanating from exp(iA) spread more slowly.
© 2006 Springer Science+Business Media, Inc., Volume 28. Number 1 , 2006
35
Returning to IPn and the geometric mean, it is not diffi cult to derive from the information at our disposal the fact that given any three points A, B, and C in 1Pn we have (15)
A#C
B
c
B#C
Figure 2. Geodesic distance from A#B to A#C is no more than half that from B to C. Joining the midpoints of the sides of a geodesic triangle in IP'n results in a triangle with sides no more than half as long. Iterating this procedure leads to the construction of Ando, Li, and Mathias, described in the text.
H to diag(iA 1 o now
•
•
•
, iAn) with A1 real. Instead of (12) we have
sin(A i - AD/2 i· (Ai - AJ)/2 k i
Since [sin x[ :S 1, the inequality ( 10) is reversed in this case, X as is its consequence (9), provided elf and eK are close to each other.
o 2 (A#B,A#C)
1
:S 2 o2 (B, C).
This inequality says that in every geodesic triangle in IPn with vertices A, B, and C, the length of the geodesic join ing the midpoints of two sides is at most half the length of the third side. (If the geometry were Euclidean, the two sides of (15) would have been equal.) Figure 2 illustrates ( 15). We saw that the geometric mean A#B is the midpoint of the geodesic [A,B]. This suggests that we may possibly de fine the geometric mean of three positive definite matrices A, B, and C as the "centroid" of the geodesic triangle Ll(A,B,C) in 1Pn. In a Euclidean space �. the centroid x of a triangle with vertices x1 , x2 , X3 is the point x �(x1 + Xz + x3). This is the arithmetic mean of the vectors x1 , x2 , and x3. This point may be characterised by several other properties. Three of them are:
=
(M 1) x is the unique point of intersection of the three medians of the triangle ll(x1 ,x2,x3), as in Figure 3; (M2) x is the unique point in � at which the function attains its minimum; (M3) x is the unique point of intersection of the nested sequence of triangles [lln} in which ll 1 Ll and ll1+ 1 is the triangle obtained by joining the mid=
expA
�--__,.
A
expC
� c
Figure 3. In the hyperbolic geometry medians may not meet. While the medians of a Euclidean triangle intersect at the centroid, the corre sponding median geodesics of a triangle in IP'n may not intersect at all. A 3-D wire model would make it clear that, generically, the medians do not even intersect in pairs.
36
THE MATHEMATICAL INTELLIGENCER
points of the three sides of t:.J (Figure 2 mimics this construction in the non-Euclidean setting of IP'n). To define a geometric mean of A, B, and C in IP' n we may try to imitate one of these definitions, now modified to suit the geometry of IP'n· Here fundamental differences between Euclidean and hyperbolic geometry come to the fore, and (Ml), (M2), and (M3) lead to three different results. The first definition using (Ml) fails. The triangle t:.(A,B,C) may be defined as the "convex set" generated by A, B, and C. (It is clear what that should mean: replace line segments in the definition of convexity by geodesic seg ments.) It turns out that this is not a 2-dimensional object as in ordinary Euclidean geometry (see Figure 4). So, the medians of a triangle may not intersect at all in some cases (again, see Figure 3). With (M2) as our motivation, we may ask whether there exists a point X0 in IP'n at which the function
J(X)
= 8�(A,X)
+
8 �(B,X) + 8 � (C,X)
attains a minimum. It was shown by E lie Cartan (see, for example, section 6. 1.5 of [Be]) that given A, B, and C in IP'n• there is a unique point Xo at which f has a minimum. Let G2(A,B,C) X0, and think of it as a geometric mean of A, B, and C. This mean has been studied in two recent papers by Bhatia and Holbrook [BH] and Moakher [M]. In another recent paper [ALM], Ando, Li, and Mathias define a geometric mean G3(A,B,C) by an iterative proce dure. This iterative procedure has a nice geometric inter pretation: it amounts to reaching the centroid of the geo desic triangle 11(A,B,C) in IP'n by a process akin to (M3).
=
Starting with /1 1 as the triangle 11(A,B,C) one defines /12 to be 11(A#B,A#C,B#C), and then iterates this process. Figure 2 shows the beginning of this process. The inequality (15) guarantees that the diameters of these nested triangles de n scend to zero as 112 . It can then be seen that there is a unique point in the intersection of this decreasing sequence of triangles. This point, represented by G3(A,B,C), is the geometric mean proposed by Ando, Li, and Mathias. It turns out that the two objects G2(A,B,C) and G3(A,B,C) are not always equal (Figure 5 illustrates this phenome non). Thus we have (at least) two competing notions of the centroid of 11(A,B,C). How do they do as geometric means? The mean G3(A,B,C) has all of the four desirable properties (a)-(8) that we listed for a mean G(A,B,C). Properties (a) , (/3), and (8) are almost obvious from the construction. Prop erty ( y)-monotonicity-is a consequence of the fact that the geometric mean A#B is monotone in A and B. So mo notonicity is preserved at each iteration step. The mean G2(A,B,C) does have the desirable properties (a) , (/3), and (8). Property (/3) follows from the fact that rX is an isome try of (IP' n,82) for every X in GLn. However, we have not been able to prove that G2(A,B,C) is monotone in A, B, and C. We have an unresolved question: Given positive definite matri ces A, B, C, and A' with A 2: A', is G2(A,B,C) 2: G2(A ' ,B,C)? An answer to this question may lead to better under standing of the geometry of IP'n, the best-known example of a manifold of nonpositive curvature. Certainly this is of interest in matrix analysis. Computer experiments suggest an affirmative answer to the question. Finally, we make a brief mention of two related matters. The Frobenius norm is one of a large class of norms called
0.25 0.2 0.15 0.1 0.05
0.2 0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 4. Conv (A,B,C) is not two-dimensional. In the hyperbolic (nonpositive curvature) geometry of l?m the convex hull of a triangle (formed by successively adjoining the geodesics between points that are already in the object) is not a surface but rather a "fatter" object.
© 2006 Springer Science+Business Media, Inc., Volume 28, Number 1 , 2006
37
unitarily invariant norms or Schatten-von Neumann norms. These norms 11 · 11"' have the invariance property IIU A VII"' I AII"' for all unitary U and V. Each of these norms corresponds to a symmetric norm
ence- Bus>ness Media, Inc
Figure 2. R. M. Schindler: The McAimon House, 1 935, 271 7-2721 Waverly Drive, Los Angeles; The Mackey Duplex, 1 939, 1 1 37 South Cochran Avenue, Los Angeles.
linear space-forms are highly inter
in 48-inch units, but after ± 12", ±24" re
understood in the mind as a continu
locked. Their union creates a rhythm
fmements, results always lie within a 12-
ous quantity in three dimensions. Aris
of complex combinations. Thus it is of
inch grid. A variety of L-shapes and other
totle bisects "quantity" into two parts:
ten hard to distinguish where the build
arrangements
magnitude and multitude. For Aristo
ings start or end; frequently they lack
dated. Such a unit is useful to measure
tle, magnitude is that which can be
a clear main fa
