JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vo]. 27, No. 1, JANUARY 1979

On Applied Mathematics R, [SAACS

1

Communicated by P. L. Yu

Abstract. This paper covers some aspects, problems, and episodes of applied mathematics intended to be enjoyable, instructive, and advisory to the young. Key Words, Advice to young applied mathematician, pure versus applied mathematics, careers, optima, models, frustration loop, aerodynamic drag, extreme cases, simplicity, secrecy.

1. Introduction My m a n y years as an applied mathematician have engendered these lines. I hope that they will be enjoyed by some and will be instructive to more, but they are aimed primarily at the young, from students to those still young in their careers.

2. Pure Versus Applied Mathematics My title at Johns Hopkins was Professor of Applied Mathematics. Although this domain certainly earned my bread, it has not d i m m e d a devotion to pure mathematics. I admire practitioners in both fields. What I d e s p i s e - - a n d I have tried so to persuade my s t u d e n t s - - a r e the bigots. A t one extreme is the pure mathematician with such pride in his purity that to him others are peons who do numerical routines on computers. A t the other, is the mathematician so applied as to gauge all in terms of economic utility. Both extremes exist, but, as I have been glad to discover, rarely. T o me, the ideal, sometimes elusive, is balance. It is owning two hats and knowing the right times and places for the wearing of each. At my first jobs, I was a lone mathematician, first a m o n g engineers, then a m o n g physicists. W h e n I joined the Rand Mathematics D e p a r t m e n t , I was still a bit of a loner, for the backgrounds of m y colleagues were largely academic. 1Professor Emeritus, The Johns Hopkins University, Baltimore, Maryland. 31 0022-3239/79/0100-0031503.00/0 © 1979 PIenum Publishing Corporation

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Many were o u t s t a n d i n g - - s o m e have b e c o m e famous s i n c e - - b u t often balance was lacking; not so much that the theoretical scale pan was heavier as that the two pans did not coordinate well. T o clarify, if I can, this dichotomy of attitude, let me take for an example the existence theorem. It has great impact in its proper settings. O n e such is the t h e o r e m ' s holding for a large class of cases. For example, without the result that, for every finite matrix, a saddle point of mixed strategies exists, game theory would not have been born. For a particular problem, we, wearing our applied hat, can find things very different. W h a t good is knowing the existence of an answer that we can't find? Of course, we have the assurance that our p r o b l e m was correctly set, but are we not then really concerned with a class? On the other hand, if we have found a solution, there is no need for a separate existence proof. These italicized words seem so obvious that they appear only because I have encountered mathematicians who seem unaware of them. I think our standard academic training invests the existence question with an overdone sanctity. I have heard ecstatic cries f r o m my colleagues: " T h e solution does not exist!" Often, the important but ignored question is " W h y n o t ? " ; its answer may be very simple. An actual example: the strategy choice of a game player is to act at any one of infinitely m a n y opportunities. It was ascertained with glee that his optimal strategy did not exist. It did in the finite version where there were but n opportunities; it d e m a n d e d probability 1In for each. Thus, the reason for nonexistence is simply that we cannot m a k e an equiprobable choice over an infinite set. But there are easy ways to do so approximately, and each yields a close practical approximation to the original game, "practical" because in real life we would not face an infinitude of choices and would have to play a finite version. My early work on differential games was spurned by m a n y R a n d colleagues, because it lacked an existence theorem. As the subject grew, it seemed to e m b o d y an unending series of novel p h e n o m e n a ; how could an existence t h e o r e m e m b r a c e them all? Such attempts as I have seen since are either so narrow as to exclude these p h e n o m e n a or so broad as to be fatuous. My agonies were dispelled by Samuel K a r l i n - - o n e of the best living applied m a t h e m a t i c i a n s - - w h o suggested what I soon called K-strategies. T h e y quantized games elegantly in theory and realistically in practice. With them, I evolved the verification theorem, which enables us to prove that a formal solution, once attained, is actually a solution in Karlin's precise sense. This approach I thought far better suited to differential games than the conventional existence theorem. T o m y a m a z e m e n t , except for my own students, it seems to have been totally ignored. Is this because the existence t h e o r e m is rooted too deeply in our tradition? O r was my explication unclear? See the third printing of my b o o k Differential Games (Ref. 1) for an improved version.

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I have always tried to keep my hand in pure mathematics too and turned out an occasional paper. There are some I never got around to, which is a reason why I welcome retirement. What might be called pure-applied mathematics appeals to me and a few others. It consists of problems rooted in the world but appealing, not because of their utility, but for the same reasons that all mathematics is appealing. For me, they have been largely physical: the theory of a flag flapping in the wind, the form and ripples of sand dunes (I have not solved either). But there are other aspects for others; the traveling salesman problem could have begun as pure-applied mathematics, although it has had broad applied consequences. This case shows the distinction to be not always clear or enduring. I have recently revived an old concept of mine, the three-vortex problem. (For vortices, see Section 7 on aerodynamic drag.) It is like the three-body problem, in that each vortex moves in response to the two others, and unlike it in that the induced motion is lateral rather ensuing from attraction. The problem interests me in the sense that general mathematics does; I see no utility in its resolution. It is "applied" in that I can at least envision the relevant experiments. Would the three-body problem with the inverse cube instead of square law so interest me? No, because it has no basis in reality. I would prefer a fully general law of attraction and maybe a more general number of bodies. Then, matters would be so general that I could not hope to find specific orbits, but would seek instead much broader (and necessarily shallower) conclusions. These, if I found them, would be theorems, and we are now in (or close to) pure mathematics. Such, I think, comes near the elusive distinction between pure and applied: precise, verifiable--at least thinkably so--results in one; broader and less detailed conclusions in the other. There is no reason not to like both.

3. Careers The career choice between pure and applied besets every mathematics student. Obviously, pure mathematics means teaching and journal research only, while the applied choice includes these and more. Let me risk some blunt counsel. T h e r e are so many advanced fields today that pursuit of either career, particularly the pure choice, can require intense specialization. Avoid becoming too narrow. Too much originality and innovation is not encouraged today. Grants have funds for nought but the beaten track. Accept them only if you need the

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financial help, but becoming dependent on them can limit your mind to the stereotype and your career to the inconsequential. Journals too eschew deviation. For their intolerance of innovation, editors are fond of the phrase "high standards." Rigid hours beset the nonacademic mathematician. Part of the financial tally of an institution or firm is in m a n hours. This unit is defined as an employee's being on the premises for an hour. You, as I, will probably find it impossible to perform mathematics daily from nine to five, but you do not have to. My own recourse at times was to devise light pastimes that looked like mathematics (I lacked the gall to read novels openly). Sometimes they actually were mathematics, and one, which I did long ago, is due for publication, 2 but mostly they were intellectual persiflage. Write a report, internal memo, or whatever the institution provides, for every problem you solve, even if it occurred over lunch and was settled in a moment. Your written output is a basic factor in advancement and salary raises. Similarly, publish all you can. A long list of publications is one of the most impressive items on a job application; I suspect, even more so outside the university. D o not expect too much from your achievements. The output of think tanks--there are many organizations that are partially such--is reports. Just what their destinies are has always mystified me. Each has a distribution list. My vision of a distributee has his secretary neatly and systematically tucking each report into a file where it remains forever. He, having prestige enough to make distribution lists, certainly has not the time to read the deluge of consequent documents. T h e r e have been times when I, a colleague, or a group put exhausting time and effort into a report of whose content we were proud and in whose utility we had confidence. It was discouraging to have it simply disappear, but we were soon engrossed in something else. When your first journal article appears, you await its impact with satisfaction and hope. Note, however, that journals today are legion; each specialty seems to have several. You will almost surely find that your contribution has but added a drop to the ocean. All the preceding is of course superficial. The choice is within. If you have that wonderful appetite for abstract mathematics and the talent to create it, a career therein, despite worldly drawbacks, is optimal for you. If you have the craving but inadequate talent, the career assumes some aspects of a luxury: enjoyment without production. I grant that assessing your own talent may not be easy. But the decision is yours. Really good advice is rare and perhaps nonexistent. 2 The Distribution of Primes in a Special Ring of Integers

(Mathematical Magazine).

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4. Optima So many of the applications of mathematics today seek maxima or minima. T o some extent, this was always true; in fact, almost any problem whose answer is a n u m b e r - o r set of them, such as a sought function--bears a formulation equivalent to a maximizing task. But the extent has burgeoned in more recent times when the goal is the direct hunting of an optimal procedure. There are several facets that have always left me uneasy. The first and greatest was that, whenever a cogent real decision was cast into a mathematical model, what ensued was the question of maximizing some one quantity. What quantity? The real world has a way of confronting us with large sets of critical factors. The design of an o p t i m a l - o r , let us say, just good--fighter aircraft depends on its speed, range, rate of climb, maneuverability, amount and kind of firepower, accuracy, sensing abilities, and . . . . Basic and baffling as this quandary is, it was for far too long neglected. I rejoice now that my former student and cowriter in this issue, Professor P. L. Yu, has taken it into his able hands and created his theory of multiple criteria decisions. Further exposition is better left to him. The second facet is what I have elsewhere (Ref. 1, page 350) called the flat laxity principle. It is simply the old idea that an interior maximum of a smooth function occurs a t a point where the derivative(s) is (are) zero. For this very reason, small changes in the independent variable(s) induce a much smaller change in the function; in other words, for practical purposes, the maximum is not critical. This is the reason, I believe, that in the world around us we see such an astonishing paucity of useful maxima or minima arising because of zero derivatives. The classical calculus of variations is included in this observation. Other disciplines often are too. Thus, in linear programming, we are given many linear inequalities, which means we are restricted to a polyhedral region in some n-dimensional Euclidean space. We seek to maximize over it some linear function, which, by rotating coordinates, we may speak of as the height. Now, if our polyhedral surface has a large number of facets, it approximates a smooth one; if we seek its maximal height, we are not too far from the old "derivative = 0" arena, and hence the implication that maxima may not be very critical. T h e r e are still other cases. I can account for them by experience, rather than logic; often I have unfolded with great effort a theoretic maximum, not engendered by a null derivative, only to find that its advantage over the common prototypical cases was miniscule. However there is more to the theoretical maximum than just the biggest possible useful value. In recent years, I have come to think that here are both

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Fig. 1. Frugal light ray.

x

Path length

Fractional error

-0.3 -0.2 -0.1

4.2686 4.2538 4.2454 4.2426 4.2452 4.2528 4.2650

0.0061 0.0026 0.0006 0.0000 0.0006 0.0024 0.0053

0.0

0.1 0.2 0.3

significant and puzzling questions. As a very simple illustration, let us consider two points A and B lying a b o v e a line 2, (Fig. 1); they have coordinates ( - 1 , 1) and (2, 2). We seek the path of minimal length from A to B which meets 2, between. That minimum replaces maximum should not trouble us; it is but a question of sign. T h e solution follows simply from F e r m a t ' s principle of least time: taking 2 ' as a mirror, our answer will be the classic reflected light ray f r o m A to B which meets 2, at 0, where the angles of incidence and reflection each equal 45 ° . Some lengths of other paths, which m e e t 2, at distance x from 0, have been calculated for comparison and also their fractional errors. For example, if x = 0.2, approximately the path sketched, the error is less than one-fourth of one percent. Thus, if we had some practical reason for seeking a short path, it would hardly be worthwhile to insist on x = 0 too stringently. Yet, the preceding optical law is the crux of the reflecting telescope. The construction of their astronomical versions is a m o n g the most precise endeavors of man. H o w do we reconcile these two extremes of laxity and precision? T h e third aspect of maximization concerns uncertain or probabilistic outcomes and is a quandary which statisticians know very well. In these cases, what almost everyone does is maximize the expectation. We are t e m p t e d to do so, I think, because of the obliging way in which this quantity

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behaves; it can often be found without delving into the entire probability distribution. Let me give a minute but typical instance of the latter. What is the expected number of coin tossings until the first head appears? We toss once; the probability is ½that the outcome is 1 (heads); but, if this toss is tails, we face a new start with one toss already consumed. Thus, E =

+ ½ ( E + 1),

and we merely solve for E. Be assured I have used this species of reasoning on much more intricate problems. H o w worthy a criterion is expectation? Would you gamble your annual salary--double or nothing--if your win probability were 0.55? You decline? But you have a positive expectation.

5. Models Whenever an applied mathematician tackles a real-world situation, he makes simplifying assumptions. This simplified version of reality is called a model It is a most convenient word, for it is his only recourse.

The human mind is incapable of thinking other than about models. This is such an obvious truism that I should think e v e r y o n e - - n o t merely applied mathematicians--must be aware of it. Are they? I have always been perplexed by this question; perhaps, some day I shall take a poll. Whenever I have felt confident of the obvious '°yes" answer, I seem to get a shock. The last one found me on the examining committee of a doctoral dissertation. The candidate had undertaken an economic analysis of his country, Iran. All the salient features he could handle were in it; those he could not were qualitatively estimated. I thought it an incredibly good job and was amazed to hear my colleagues--senior university faculty members all--carp because of this or that omitted detail. The student had selected his model. Could he have done the real Iran with its millions of citizens each with his own economic goals, abilities, wealth, mores, and psyche, to say nothing of the country's resources as well as world economy, its future technology and events, a n d - - f o r really authentic reality---each of its myriad citizens too? Such is clearly beyond hope. Yet, here was a young man who had worked alone two years on a project fit for a large sophisticated organization. And he was criticized for being short of complete verisimilitude! Even with much simpler subjects, there can be more than one model. We are seldom aware of this, for our education usually settles for the best

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Fig. 2. Heat exchangers; two models.

one alone. My first experience with an alternative model was naive, but, being quite young, so was I. Mr. Carrier had conceived his contact mixture theory. An application is the heat exchanger: air blown through a channel might be heated during its course. We seek its rising temperature as a function of the distance traversed and envisage the air stream receiving heat through the bottom plates as in Fig. 2. The usual model postulates a boundary layer: a stagnant layer of air adjacent to the plate, as in the upper sketch, which acts as a heat resistor. This means that the rate of heat passing through it is proportional to the temperature difference between the plate and the air at x. It is not hard to embody this image in a simple differential equation which yields the air temperature as a function of x, the distance from entrance of the channel. The contact mixture theory quite differently supposes the air to be comprised of little packets (molecules is too presumptuous a term; let us eschew any spurious particle physics). They contact the hot plate at a rate constant throughout passage. Each is either hot or cold--at plate or initial air t e m p e r a t u r e w b l a c k or white on the lower sketch. At its first contact, a cold packet becomes hot; at later contacts, it simply remains so. The air temperature at x results from the ratio of hot to cold packets there. I set up and solved the differential equation for this model and was immaturely astonished to find the same solution as before, provided, of course, a simple relation held between the thickness of the boundary layer and rate of packet contacts. The relation set my young mind abuzz: there

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needn't be a boundary layer at all! It could be condemned as a myth that arose because we didn't understand the details of thermal physics. Years later, I learned of the role of the boundary layer in aerodynamics. Here, it is vital to wakes and turbulence, unconcerned with heat. But, naive as was my youthful reaction, it had a glimmer of something deeper. As distinctive models may lead to the same conclusion, we naturally ask ~'which model is the truth?" This question has no meaning; there is no such thing as a true model, for a model is but a limited version of an unattainable reality. The criterion of a model is not its truth but how well it works. Let us take a famous example and tinge it with a bit of instructive fantasy. In the seventeenth century, a prominent quandary was the cause of the planetary paths round the Sun. Newton, in what I deem the greatest intellectual achievement ever, through his laws of gravity and motion proved that the orbits obeyed Kepler's carefully observed taws. In the same era, D~scartes postulated his vortex theory: the planets were carried by an invisible fluid swirling about the Sun. The desuetude that befell this theory is due not to its falsity, but to D6scartes failure to employ it quantitatively and mathematically toward conclusions like Newton's. Let us imagine he had done so: definite postulates about the fluid and some astute analysis had led him to the Keplerian ellipses. Then, the modern science of dynamics might contain not a word about gravity, but students enrolled in Physics 1 would do their homework anent the invisible Cartesian fluid. It would become as familiar and unquestionable an entity to them as gravity is to us.

6. The Frustration Loop Whoever coined vicious cycle gave the language a phrase badly needed by the human psyche. Would that I could make the phrase of this subtitle as well known, for it can depict a horrid pitfall facing the applied mathematician. When I was young I would undertake any assigned problem with naive overflowing confidence. There were the times o f neat, brief answers, but often a week would pass leaving but palimpsests of scrawled scratch paper and a turbid brain. An ardent renewal of effort brought but another such week; then, unbelievably, another. I would add my three weeks of salary and stare with appalled guilt at the sum. My employer had received absolutely nothing for this money! I would fire, one after the other, all my mathematical weapons, and then strive for brilliant originality. Driven by conscience to new frenetic efforts, I put in more weeks; it was astonishing how they could accumulate. B u t . . . b u t . . , but I could not give

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up n o w - - m y debt of nonearnings was far too g r e a t - - I must keep on trying . . . . This torturing miasma is what I mean by a frustration loop. Can I help the novitiate to avoid it? A warning--a pointing to its existence--is the best I can do with certainty, for some occurrences are inevitable. The reason is not obscure. Our technical education cannot help but try to convey the superb output of many man-years of research which total more than a human student can absorb. What is omitted are the failures: the methods that do not work, the legion of problems that the marvelous tools do not solve. Could it be otherwise? Hardly. The numbers of successes to be taught more than consume what time there is; none is left for the even larger numbers of failures. In the Carrier Engineering Department, just after my engineering BS, there was a problem to be solved by a differential equation. Now, I had taken a course in differential equations; it told us how to set one up germane to a real-world problem and how to solve it; I was a master. Therefore, I wrote my equation including in it every detail of the problem. In average handwriting, its length was about two feet. And, hard as I find it to believe now, I was chagrined to find no solution rattling out as they had done for the homework problems in the textbook. Please restrain the polite snicker. The course had given no hint that not all differential equations can be solved in elementary closed form. With the famous special functions--Bessel's, Legendre's, hypergeometric--that had been coined just to fill such gaps, my acquaintance was still to come. In my own courses on applied mathematics, I once resolved that students should hear of some of these frustrating experiences. It was hard to do. First, I had naturally kept no written records of my unsuccesses. Second, like all humans, I found it much easier to remember my past glories than my ignominies. I gave my students not much above generalities. Writers of textbooks would win my admiration if they put more stress on the limitations of their topics. But this is awkward in a competitive economy like ours, where salesmanship often marks the winner. But there are really meretricious cases; one well-known text on elasticity covers a class of problems with no hint of solution methods. Rather than admit such, the author gives instead a string of solutions and finds problems to fit them, a spurious and useless subterfuge. The only key to avoiding the frustration loop is experience. Education, as I have tried to show, is rarely of much help. As I grew older, I learned to make better assessments of what I could and could not do. There were times when I would eschew attempting a problem that in earlier days I would have plunged into. However, it is not always simple to assess a problem which--as

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Fig. 3. Ideal flow around a body. do all to some extent--leads into virgin terrain; even quite recently, I have erred. The two important judgments are whether to start and when to stop. A short effort abandoned is better than a futile long one. Often, when ensnared in the frustration loop, I would ask myself: what would the great minds--Euler, Gauss, Riemann, Poincar6, yon K a r m a n - have done in my position? Years later, I got a glimmer of the answer in regard to the last, as will be told in the next section.

7. Aerodynamic Drag This story exemplifies much of applied mathematics: its scope, beauty, importance, insight, limitations, and computer adaptability. More than a century ago, d'Alembert analyzed the flow of a fluid around a body such as is sketched in Fig. 3. H e calculated the ensuing pressure at the boundary and integrated it to get the total force exerted on the body. His result was zero. This is known as d'Alembert's paradox, and it is one indeed. It implies that sailing ships would not sail, windmills would not turn, and hurricanes would be undetected, save for the slight chill on the skin from the undue evaporation of moisture. The mystery is readily explained, but it leads to others. First, we must know our model. The fluid is to be incompressible and nonviscous; the planar flow is to be irrotationat, which means that the circulation 3 is zero around any closed curve lying with its interior in the fluid. This is a pretty model theoretically, for it is virtually identical to the theory of analytic functions; it is practical as it served aviation well until sonic speeds were approached (sound requires compressibility). The classical analytic existence and uniqueness results obtain, and d'Alembert's solution is the only one possible. 3 Circulation= integral over curve of tangential component of flow velocity.

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The resultant force component along the undisturbed flow velocity is called the drag. W e consider it alone here. Why did d'Alembert find zero drag? Elementary physics gives the answer: a nonzero drag would do work (it may help to think of a stationary fluid with the body trolled through it, resisting the pull with its drag). The work must convert to energy. W h e r e is it? It cannot turn into heat, for we have excluded viscosity, the fluid version of friction; it cannot become kinetic energy in the fluid, for far downstream the flow approaches its undisturbed constant velocity state. Thus, no energy; hence, no drag. However, the latter form of energy can occur if we give our model a remarkable twist; a turbulent wake will then trail the body. T o understand it, we must first glance at vortices. A vortex at a point consists of the fluid's flowing about it in concentric circles at speeds decreasing inversely as the radius. This speed law is the only one ensuring zero circulation around curves not containing the vortex; for all curves that do, the circulation will be a constant called the strength of the vortex. Thus, a vortex is a singularity of the flow; in the analytic format, it is a simple pole with imaginary residue. We need two laws (Helmholtz), nicely provable. Law 1. Conservation. If there is a change of circulation around a body, it must shed vortices of negatively equal total strength. Law 2. Free Vortices. A vortex unattached to a body will move in the flow as if it were a fluid particle. Thus, if there is a set of free vortices in an otherwise steady-state flow, each will travel as a particle subject to this flow and to those arising from the other vortices. Such is turbulence. T o simplify our vista of drag, we shall consider only the flow about a vertical flat plate. Figure 4-1eft shows the d'Alembert flow about its upper end (or edge). The ends of the plate are singularities; the flow speed becomes infinitely large as they are approached. W e sense a drawback in our model. W e posited zero viscosity. But, be it ever so small, the viscous forces will become unboundedly large at the unbounded velocity gradients near the plate ends. This deviation from reality will be repaired, as Kutta showed, if at the ends the flow is tangential as shown in Fig. 4-right. We can expect real flows to act thus, for the enormous viscous forces they would otherwise encounter will so compel the fluid particles. W e have encountered the unexpected: a discontinuity in nature. Z e r o viscosity implies the d'Alembert flow (left), but the limit of a viscous flow as the viscosity approaches zero leads to the Kutta flow (right).

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7 ! I w

Fig. 4. Flowspassing the upper edge of a plate.

How will the Kutta flow arise? Only with circulation around the plate. As such will soon lead to vortex trails, my account thereof will carry as much of yon Karman's lecture as its distant m e m o r y permits. "People seem to think," he began, "that I was named after the von Karman vortex trail." He took us back to G6ttingen, early in the century, when aviation was new and a fluid flow theory acutely needed. The laboratories there could render flows visible. One apparatus was irksome; the flow would not remain steady, but oscillated despite all efforts at adjustment. H e r e came yon Karman's first insight: the flow should change with time! Let us return to our plate. 4 It can be shown that, for the Kutta condition to hold (at least stably so), there must be a suitable time-varying circulation around the plate. By Law 1, the plate must always shed opposing vortices; by Law 2, they must be carried downstream. H e r e then, is the turbulent wake. As the kinetic energy it bears must ensue from the drag, here then is the resolution of d'Alembert's paradox. Smooth bodies, lacking the sharp ends of the plate, also suffer drag. In this quick tour, let us dodge the boundary layer theory needed. The above resolution is a rather strange one. Qualitatively, all is clear and even more. If we knew the full flow pattern at any instant--all velocities and the plate circulation growth r a t e - i t will be determined at an instant closely following, a case of the usual genesis of differential equations. Let us pause for a brief update. Were the smooth flow replaced by its velocities on a large, finely spaced but bounded lattice of points in the plane and were time also quantized into small interludes, the whole problem becomes a natural for the computer. Each state--finitely many (very) numbers--is calculated from the one preceding it in time. But, in von Karman's day, solving the differential equations was Herculean. His masterstroke was to bypass the unattaiaable (should I say: 4 Is this example my choice or von Karman's? I am not sure, but t think it is mine.

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I

Fig. 5.

I

Von Karman vortex trail.

cut the frustration loop) and perceive what was fertile. Instead of grappling with the intricacies near the plate, he looked far downstream and discovered his vortex trail. Here, far from the configurative details of the plate or other body, the wake ought settle into a pattern of stable equilibrium. H e found that there was just the one possibility: the two rows of vortices as sketched in Fig. 5. In it, the ratio of width to vortex spacing is a universal constant; the width, the (constant) vortex strength, and the flow speed are related in a simple formula. It is not hard to compute the drag therefrom and, as far as I know, this is the sole such rational success in planar ideal flows. But the mystery still lingers. Despite the grand achievement, I believe there is no known analytic route from the shape of the body to the parameters of the vortex trail. Measurements of drag coefficients, certainly by wind tunnel, possibly by computer, are not yet obsolete.

8. "When in Doubt, My Dears, Take an Extreme Case" There are times when these words, from Lewis Carroll's A Tangled

Tale, can be a boon. At the outbreak of World War II, aircraft propeller blades were made of solid aluminum. Hamilton Standard proposed a hollow steel construction: a central narrow core was to be brazed along its length to an outer steel shell of the ultimate aerodynamic shape. When an eight-foot blade of such construction rotates at umpty thousand RPM, the centrifugal force load is born over the length of the braze as a shearing stress. H o w is it distributed? This urgent question was handed to me for a quick answer.

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Let us replace this archaic problem by the more universal one of seeking the shear distribution over a continuous joint running the length of two bars when they are being tugged asunder (Fig. 6). I learned, after I had tackled this problem, that its solution had long been known, but it was new to me. The joint may be soldered, glued, riveted, welded, or brazed. How will the shear load vary over its length? Carroll's dictum might be extended to "take both extreme cases," and this advice I followed. The extreme sketched on the left has rigid-absolutely inelastic--bars and a very flexible joint; we can think of it as a thick layer of rubber. On the right is the other extreme: the bars are now quite elastic and the joint absolutely rigid. To have it so the two bars could be thought of as one piece and that is why the joint is drawn as a broken line. Let us look at two ensuing shear distributions. In the rigid-bar case, every cross section of the bar-joint system is like any other because of the bar's rigidity. Therefore, the deflection of the joint is uniform over its length, and accordingly so must be the stress. Thus, as sketched, here we have a constant shear distribution. In the rigid joint case, near the center, we have what amounts to a single bar under tensile stress. This stress, as is well known, will be (very closely) uniformly distributed over a cross section of the (double) bar, and hence there will be no shear at all exerted on the (now imaginary, if you like) joint. Thus, the shear will be concentrated at the ends. The logic of this conclusion can be completed by recalling that the two bars are now a single body, but with two reentrant corners. It is known in elasticity theory that such corners usually are singularities of the stress functions and infinite stresses can occur there. These Carrollean extremes pointed the way. The critical parameter should be a measure of the ratio of the elasticities of bars and joint. The analysis accordingly yielded a family of functions which merged toward the two extreme cases for the appropriate extreme values of this parameter. A typical instance is plotted as the heavy curve in the lower figure. T o conclude, we revert briefly to the propeller blade where the principle, if not the details, was the same. The braze was very thin, yielding a

~IG~ Jo~/4T

Fig. 6. Shear over a joint.

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parameter value close to that of the rigid joint extreme. In fact, I found that more than ninety-five percent of the centrigual load was carried by the ~ inch of the joint at the hub end! What do we make of this extraordinary maldistribution? First, we realize that it takes us outside the validity of our model, which was based on linear elasticity, which assumes H o o k e ' s law; certainly, the freakishly high concentration of stress would imply a local excess of the elastic limit; a more extended spread would follow. The practical conclusions? They might be: be especially careful in manufacture of the inboard end of the braze; taper the outer steel at this end to avoid the reentrant corner which can be considered as the source of the trouble. But all became inconsequential. New air speeds brought jets. Largescale propellers became obsolete then on military aircraft and later on all large planes.

9. Simplicity Of complex problems the world is too full. But sometimes, with insight, the toughest will bear a simple, transparent answer. If we have reached the right answer by a superfluously elaborate analysis, we face embarrassment,

Fig. 7. Sub seeking ship.

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probably even m o r e so than if our answer were wrong. This once happened to me at the Center for Naval Analyses. A submarine S and ship T each travel at constant (vectorial) velocity. The former wishes to find the state---location and velocity---of T by sightings; thus, four numbers are sought: two coordinates apiece. All S can learn from her sightings are bearings: the angles 0i in Fig. 7. To find four quantities, certainly at least four sightings are needed. I showed that it could not be done with any n u m b e r of sightings. H o w ? The conventional way: we have four or m o r e equations to solve for four unknowns. I painstakingly proved the rank of the relevant Jacobian to be less than four. The C N A Director of Research, Frank Bothwell, saw the impossibility at once. If we use a coordinate system moving with S, then we may take S at the origin and so stationary. This is a case of the reduced space I had invoked so often in differential games. Why had not I thought of it here? All is now simple as the second sketch shows. If two ships move in parallel paths with a suitable speed ratio, S will get the same sightings for both (see Fig. 8.) As S cannot distinguish them, she can get the sought data for neither. Of course, the preceding is but the overture to a richly orchestrated problem. As S cannot get her information from a constant velocity course she must deviate therefrom to a path that is curved, of varying speed, or both. Unless this deviation is appreciable, the four equations will be nearly singular; this means large errors in T ' s state m a y induce but small ones in S's

Fig. 8. Sub stationary.

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sightings. A remedy is that S take more than four sightings. The theoretical redundancy will reduce the practical error. H e r e is a nice problem in statistics. How many sightings and under what conditions will engender suitably small expected errors? But our topic is naval warfare. To frustrate such detection, T too can deviate from constant velocity; he can randomly zigzag, as is actual naval practice. We are confronted by a really formidable differential game with incomplete information whose optimal strategies--if we can find t h e m - - w e would expect to be mixed. The simplicity died early. Mr. Carrier would test his engineers with the following exercise in simplicity. With how little complexity can you answer it? A hoop, starting from rest on a hill at height h, rolls without slippage or friction until it rolls on level ground. How fast is it going?

10. Secrecy Much of my career has been with institutions dealing with military matters and accordingly inundated by secrets vital to the nation's welfare. I pass to the beginner some handy information (which was at least valid ten years ago when I departed for academe). There are four categories: Top Secret, Secret, Confidential, and Unclassified. The procedure for assigning one to a particular document varies. Industries often do it on their own; at Rand, each paper, with ink still wet, was sent to Air Force Headquarters for a verdict. Whatever the means, an inviolable mystique pervades the pages from then on. T h e r e is ambiguity at the outset: what does the last of the four categories mean? You request a clandestine document. You too have a category which must be high enough for the granting. You find the paper to be unclassified. This can mean either (1) its level is still undecided, possibly because its content is so cosmic that more time is needed or (2) it has been classified as unclassified, which means it is open to the public. As such a basic ambiguity is unlikely an oversight, I like to think it due to some bureaucrat with a keen sense of irony. I wish I knew him. Each of my colleagues had his favorite clearance story. One was of a general who found a magazine article sensitive--a favored w o r d - - a n d ordered his secretary to stamp it T O P S E C R E T and lock it up. That other copies were on every newsstand was beyond notice. A person banned from reading his own works is not uncommon. The reason is the new clearance investigation going with a job change; the longer and more highly the government has trusted him, the longer the

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reinvestigation. The renovitiate is confined to an isolation ward sealed from each arcane paper even if it is his own. I was so sealed when I joined Hughes Aircraft. My first output was entitled The Optimal Deployments of Widgets over Enemy Defenses. What is a widget? It is a bomber decoy. As the assigning military officer had said that "decoy" was sensitive, I did an analysis of widget strategies. Of course I did not know what they were, despite the unofficial whisper in my ear. The output was naturally forbidden to me. At Rand, the desire to publish research sometimes led to titles of bizarre innocence. A US-USSR military study might emerge as The Battle of the Chambermaids and Chauffeurs. A consequence, sometimes, was the plaint of a dutiful congressman on the squandering of public monies on trivia. An early Rand report on differential games comprised, under some title about air-ground warfare, what I finally called the War of Attrition and Attack in my book. The Air Force deemed it SECRET. When I needed it in a subsequent report, it was as The Snowball Fight: aircraft factories became snowball makers; bombers became snowball throwers. The analysis-ascertaining how each side should best allocate its strength between these two purposes--was unchanged. But now it became: UNCLASSIFIED. A criterion of security, I strongly suspect, is often how well incompetence is hidden; I wonder in what proportion. Once, at Lockheed, I turned out a report which seemed of especial value. It points in a way to the converse of the above thesis: the report was switched from S E C R E T to C O N F I D E N T I A L to ease circulation as it had import enough to be in some demand. But the major criterion is the medium. I commend it as being useful to know.

Handwritten material, regardless of content, is immune to security. 1 learned this at Rand, when checking a freshly typed report against my handwritten draft. At closing time, something forbade a return to my o n c e , and I committed the heinous sin of leaving papers on my desk top. The next morning brought an urgent summons to the Security ONce. A long, stern lecture: what if a foreign spy had been on the prowl last night?; think of the calamity to national security (this was flattering indeed; the report contained only ideas of mine); I was barely escaping immediate dismissal and possible imprisonment. The report was cautiously unlocked and reluctantly handed to me from the safe. Chastened, I returned somberly to my desk. Upon it lay undisturbed my pencilled draft, a verbatim replica of the precious national secret alertly spotted by the guard. This was handy knowledge. At the Pentagon, safes (file cabinets) had to be locked each night with both a key and combination. The latter was changed monthly; we were allowed no written memo; I seldom could

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remember the digits. The upshot was my minimal use of files. Instead, I retained my (and in a few cases those of others) yellow paper manuscripts in my desk drawers. These were inspected nightly, but my pencilled packets were never touched.

11. Conclusions Any work that purports to be instructive should offer its readers at least one exercise. Publication has been mentioned twice herein: its benefit to the career and its orthodoxy and frown at innovation. How can the young, when teeming with originality, reconcile these two attributes? The pages of this journal have borne some splendid research. Now, research produces conclusions, and they appear in final sections as they should. This paper, although not research, must not breach the format. Therefore, as training for writing that is both creative and conformant, I conclude with this exercise: Write the conclusions for this article.

References 1. ISAACS, R. Differential Games, Third Printing, Robert E. Kreiger Publishing Company, Huntington, New York, 1975.