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by Richard C. Heyser
The existence of bimodality of response is a direct clue that the perception of quality is a nonlinear process. It should also be a clue that any objective measurements which are based upon linear theory will not be worth a hill of beans when we try to correlate those measurements with subjective value judgments.
After bimodality, the next most prevalent response characteristic which we note in human behavior is tri-modality. Patterns of stable compromise may emerge between equally stable, but opposing, strengths of opinion. In some cases this may show up as an evolutionary transition between what Thomas Kuhn has called Paradigms.' A middle ground may be taken between the tradition of old ideas and the promise of new ideas.
When this happens, the press of time or of evolving evidence will tend to re solve the compromise position to one paradigm or the other.
Three regions of stable response may occur under a number of circum stances. When a formerly clear-cut situation splits into discernible camps, those who occupy the outermost camps, which represent the extremes of opinion, will tend to assign rubrics to themselves and to their most extreme counterpart. They are generally self flattering and of a nature which denigrates the opposing view. These are the heraldic flags of strong opinion and the field of audio bears many such flags.
A compromise opinion or reaction to control factors will seldom be identified with strong labels. This is because the trimodal position is less frequently occupied then either of the two extreme positions which surround it. This does not detract from the fact that such a compromise position can be legitimately taken and possess high stability.
It might normally be thought that since trimodality is the next degree of behavior complexity from bimodality, it would arise in the next higher level of control dimensionality. But this is not so. Two factors and one response yields the cusp catastrophe. Three factors and one response yields the swallowtail catastrophe; however, the swallowtail catastrophe does not exhibit much in the way of stable behavior, let atone trimodality.
Trimodality shows up when there are four factors and one response.
And, sure enough, when we attempt to analyze those situations leading to a compromise behavior we find that four, not three, factors are involved.
These factors have been given the names: Splitting factor, normal factor, bias factor, and butterfly factor.
The manifold of stable response is a hyper surface in the five-dimensional behavior space. The bifurcation set-places where a change in behavior occurs --lies in the four-dimensional control space. Obviously we cannot sketch a hypersurface in five dimensions, but the math game can be played without such limitations. The catastrophe map thus formed has been given the name Butterfly Catastrophe, in recognition of a certain abstract shape which appears when lower-dimensional "slices" of the bifurcation set are sketched.
In order to provide an audio example of this higher dimensional catastrophe, let us again consider the case of a music listener who has an audio reproducing system. Let me assume that the control space is characterized as follows: Music enjoyment is a normal factor, time spent listening is a splitting factor, cost of new equipment is a bias factor, and product awareness is a butterfly factor.
These are the four factors which will control the following behavior, desire to purchase a new audio reproducing system.
The first two control factors, normal and splitting, are identical to those of the two-dimensional example which we discussed earlier. The response is also the same if we presume that a sufficient dislike of one's existing audio system will relate to the desire to re place that system with one which sounds better.
The terms which are new in this example are the bias factor and the butterfly factor. Bias is that factor which, if all else stays put, tends to multiply the response. I have presumed that one's bias to purchase something is inversely related to the price of purchase. Hence cost is a bias factor. The butterfly factor has the effect of an almost inexorable buildup in pressure which tries to upset the status quo. In this case I have assumed that the listener also reads magazine articles on sound reproduction and cannot help but see advertisements and product claims.
Thus, product awareness becomes a butterfly factor.
It is a feature of Thom's theory that all of the lower dimensional forms of abrupt behavior can be found in the higher dimensional catastrophes. Progressing upward in dimensionality adds new types of catastrophe to the inventory. The fold catastrophe (which we did not discuss, but is the either-or hysteresis jump available in a one-dimensional control space), the cusp catastrophe (control dimension two), and the swallowtail catastrophe (control dimension three) can be found under certain conditions when there are four control factors. Obviously I cannot go into any sort of detail in this brief discussion, so I will present what I believe may be the more important patterns for audio.
We cannot show four dimensions, so let us concentrate on the bifurcation set (places of abrupt decision) as it appears for the two-dimensional slices with the coordinates of music enjoyment and time spent listening.
We will follow the pattern for various cost and product awareness situations.
Under these conditions the desire to purchase a new audio component will depend upon the relative amount of experience that we have with natural sound as compared to the amount of time we spend listening to the sound reproduced from our present audio system. If, at the point of relative experience shown here as 0, we substantially increase our exposure to reproduced sound, without changing the amount of time spent attending live concerts, then we move out on the trajectory L. Travelling on I. tends to reduce the desire to purchase a new system. Increasing the exposure to natural sound will move us in the direction M.
Of particular interest is the situation that arises if we travel the trajectory M1. Figure 1(d) is a sketch of what happens when we move along M1. As we begin to attend more and more live concerts, our general desire to purchase a new audio system will increase. This increase will take on catastrophic jumps at low cost and high cost situations, but not at medium cost. The low cost catastrophe corresponds to impulse buying, while the high cost situations may be a prestige reaction.
In Fig. 1, I have plotted the bifurcation set as a function of decreasing cost of a new system and with almost no product awareness. The behavior is cusp-like with the cusp swung farther toward the direction of live music enjoyment as the cost of a new audio system increases. If we trace the journey marked M (for music enjoyment), our desire to purchase a new system will increase as we increase our enjoyment of that music. If we trace the journey L (for listening to our existing audio system), the desire to purchase will diminish with increased listening.
This is the same sort of situation discussed in the previous example of the cusp catastrophe.
As an additional set of curves, I also plot the desire to purchase (height of the manifold) as a function of cost.
Now we can see a situation emerge which we might not have anticipated.
At low levels of listening to reproduced sound, the curve of increased live music experience (MI) cuts the bifurcation set at low cost and high cost, but misses it for medium cost. For a given small amount of listening to reproduced sound, there are response catastrophes at low cost and at high cost situations, but there is a region of cost in which purchase desire is continuous with no jumps. In all cases the desire to purchase goes up as the cost of a new system comes down, but there is a certain range of time spent listening where our opinions take a jump.
This jump at low cost is the sort of behavior which can lead to impulse buying. We all recognize the situation; one day we run across a bargain too good to pass up. We did not really in tend to buy a new cartridge or "whatever," but the cost was "right" and the impulse hit us.
The tendency toward impulse buying, according to this butterfly catastrophe situation, will fade as the price rises. Impulse will give way to a smooth curve of deliberation of worth versus cost. But as the price continues to rise we will again enter a region in which our increasing enjoyment of music will cause a sudden jump in purchase desire. This does not mean we will buy the higher priced product, but our desire will "gain ground" faster than we might anticipate as our enjoyment of good sound increases.
The effect which a greater amount of time spent listening to our present system has on the desire to purchase is to offset cost. The trade-off between enjoyment of music, cost, and more listening, is shown in Fig. 2. The interesting fact which emerges from this situation is that while the general de sire to purchase is diminished by more listening to our present audio system, the potential for impulse purchase is greater. Not only do the low-cost impulse catastrophe and the higher cost catastrophe merge to eliminate a smooth change in desire, but the magnitude of behavior jump is much greater due to the increased listening experience.
If we believe the mathematics, the person most likely to whip out his checkbook and make a surprise purchase (even to himself) is the one who does a lot of listening to reproduced sound at home and who has recently become more interested in the enjoyment of live music. That does not seem so surprising, but the magnitude of the desire catastrophe is a surprise.
The Effect of Advertising
Now let us look at the emotional effect of product awareness. What role does advertising and product chest thumping have on the desire to purchase? In order to visualize how product awareness (the butterfly factor) can precipitate trimodal behavior, Figs. 3 and 4 sketch the way in which the cusp catastrophe becomes modified with the introduction of the butterfly factor. As product awareness begins to increase, Fig. 3, the simple fold of the cusp catastrophe begins to warp and convolute. Continued increase, Fig. 4, puts a third sheet in the manifold and converts the bifurcation set (the projection of the "edges" of the fold where opinions must jump) into a complicated multi-cusped pattern.
This pattern has been likened to an abstract sketch of a butterfly, and is the basis for the name given to this particular response catastrophe.
If product awareness were to increase, the middle sheet would continue to produce a pattern like Fig. 3, but with the line segment break in the bifurcation on the other part of the major cusp line.
This figure illustrates the influence which advertising has on desire to purchase an audio component, under conditions of constant cost but for the situation in which we suddenly begin to go to more live concerts. If we are passing along the trajectory N and are at the locations shown by the x, then advertising can be used to bump us up to a higher desire for purchase.
Fig. 5(a)--If there is minimal advertising, we are at the position shown as (a). We need to attend more live concerts before experiencing a substantial increase in desire to pay the cost of a new audio system for listening at home.
Fig. 5(b)--Advertising has the effect of rolling the manifold tack toward lower levels of natural sound experience and of introducing am intermediate sheet. Even with no further in crease in live music experience, advertising has bumped us up to position (b). We are in a pocket of compromise and have had the incipient catastrophe moved closer to our position. It now takes less of an exposure to pop us up all the way to a higher desire-to-purchase.
Fig. 5(c)--If the pressure of advertising is increased, it may be possible that as the pocket of compromise moves out, the wrinkle in the third sheet pops us up to position (c), a position we could never have achieved without the influence of advertising.
Beyond this point, further pressures in advertising will continue to force the bifurcation set farther towards higher levels of live music listening. This means that there is an optimum level of advertising which can trigger us to a locally highest desire-to-purchase, but beyond which further sales pressures will tend to diminish desire. In extreme situations the additional warping caused by high levels of advertising may lose the sale by popping the point (c) back down to some place like (d). Supersaturation by advertising not only turns us off, but actually raises the amount of listening we must do in order to want to purchase that component.
The bifurcation set which this produces has the shape indicated in this sketch. The term "butterfly" is taken from the shape of this bifurcation set, which has been likened to that of an abstract butterfly.
Figure 5 shows sketches of the bifurcation set and purchase desire plotted for increasing amount of product awareness, but at fixed cost for a new system. A trajectory of increasing mu sic enjoyment for a critical range of listening is shown by the line N. At a certain level of awareness and at a certain level of cost, the desire to purchase will experience a trimodal behavior. This third mode represents a compromise struck between a strong desire to purchase and a weak desire to purchase. The feeling one might have is "I sure would like to have that system, but it is just more than I can afford." Trimodal behavior will disappear if any of the four control factors change moderately, but it will disappear the quickest with a change in the butterfly factor--product awareness. There is, in other words, a critical threshold of advertising which is required to cause the greatest increase in desire to purchase. If the listener can be bumped up to a compromise reaction, it is easier to push him upward to a purchase with a modest increase in advertising than with a proportionately larger drop in price.
As we pointed out, the shape of the bifurcation set at the place where tri modal response sets in is the basis for the name Butterfly Catastrophe. The central region of this set --the body of the butterfly --is often called the pocket of compromise. The effect of the butterfly factor is to drive the bifurcation set from a single cusp shape toward and through this butterfly shape. The effect of the bias factor is to magnify the response at any given set of conditions.
By opening up an intermediate level of stability between an otherwise large jump, the butterfly factor is a trigger mechanism. If we were at place (a) in the response shown in Fig. 5, the size of the jump we would take if we ever got to the fold in the behavior manifold would be quite large if the bias factor (cost) were sufficiently strong.
But we do not have to change our listening habits if the butterfly factor is now increased. Symbolically, we are standing on a floor with the ceiling well above us, and the butterfly factor now ripples our floor and bumps us up to an intermediate shelf level be tween floor and ceiling. Position (a) now changes to position (b) as in creasing product awareness opens a pocket. It is now much easier for a modest increase in normal factor to take us to a jump point where we are not at the ceiling--a ceiling we might never have reached without that assistance.
I have called product awareness the butterfly factor because that is the drive which advertising, equipment reviews, and sales claims provide. We are constantly bombarded with advertising and product claims. Most of the time this has no effect on us, and we tend to wonder why advertisers spend so much money and time. The power of advertising, according to this butter fly factor behavior model, comes into play when we become interested in possible purchase of a new product.
Months can go by with the same ad appearing month after month and we barely notice it; then, through music enjoyment and time spent listening, we start to pick up an interest in possible purchase. We start to notice the ads (product awareness begins to increase), and we begin to compare prices (cost factor entering) even though a few months prior we paid no attention to prices or what was new.
This is quite consistent with human behavior, and there would be little reason for mentioning something so obvious to us all. But the butterfly model indicates a condition which we should be aware of. The butterfly factor plays such a dominant role in precipitating a large catastrophe that it "comes on strong" after we have achieved a certain threshold in desire to purchase.
Once we have been pushed high enough on the behavior manifold, there is little time left for rational analysis leading toward a purchase. At that critical stage we are likely to be triggered into purchase by almost any product claim, either in print or verbally by a salesperson. Geometrically, the gradient of the response manifold has its steepest value just before a catastrophe, and the steepest gradient of all will generally be due to product awareness in the audio situation we are considering. All it might take to precipitate a catastrophe is some small increase in desire to purchase, such as perceived cosmetic improvement over competition or an exaggerated product claim. At that crucial stage we, the audio purchaser, are likely to accept product reviews or hearsay comments or advertising claims which we might normally reject as pap or worthless.
Wild claims and come-ons are not in tended for the person who has no de sire for purchase, but are geared for the last stage of purchase intent when' we are most vulnerable.
A good salesman is much like a shepherd who, through artful means, keeps the prospective purchaser on a path which will lead to a commitment to purchase. Mathematically, it is the salesperson's task to keep the customer heading in the upward slope of the response manifold. Skillful employment of product awareness can steepen the slope to the place where a customer can be triggered by another wise minor increment in product claim. "Would you prefer this model in walnut or mahogany?" is one such technique which directs the customer upward on the manifold by concentrating his attention on alternative positive features of the product and diverting consideration away from the customer's natural contemplative act, namely, whether he wanted to acquire this product at all. Any situation that offers the opportunity for the customer to consider "not buying" as an alter native, is heading the customer the wrong way on the response manifold (from the salesperson's view) and is to be avoided. This is particularly import ant prior to the point where the customer can be triggered to a desire catastrophe, but, as we shall see, is still important after the catastrophe occurs.
It is not what it is, but what it appears to be, that precipitates a response catastrophe. Once we are triggered, we have jumped to a higher sheet on the response manifold, and once there we are at a place of much higher desire and of lower gradient in response. Product awareness and cost have done their deed and may now be modestly altered without triggering a disillusionment catastrophe. The sales slip can add up to a bit more than we thought it would ("... of course, there are shipping charges and . . ."), and product awareness can show modest negative factors ("... the color scheme you have chosen is not in immediate stock and will take a few weeks for delivery"), but we are hooked at the moment the catastrophe occurs and will tend to drive toward a purchase anyway.
Generally, product awareness still has the highest gradient among the control factors, even after we have been triggered. That is why a "cooling off" period, a time lag, is so important before we sign a commitment to purchase. What has happened to us is something familiar in human affairs. A desire catastrophe is a response similar to "falling in love." Once we have "fallen" for something, it is often said that we are blinded against negative factors. That is not completely true, ac cording to the catastrophe model.
What has happened is that the gradient of response is significantly reduced when we jump. If we jumped from a place with a large enough splitting factor and normal factor, the increment in desire is so large that it could be likened to a transition to blinded love--the change in gradient is that large.
But if the continued press of product awareness is now directed in a negative manner, the gradient will tend to increase downward. Depending on the interaction of the bias and butter fly factors, the response may stabilize or it may be triggered to a disillusionment. A "cooling off" period may then be said to allow us to "come to our senses" in the rational consideration of our desire (or need) for the audio component.
The role which advertising (and product reviews) plays in this process can be helpful if we inject a delay be tween initial response and eventual commitment. (Within the framework of Catastrophe Theory, there is a pro cess called delay which relates to the conditions under which a transitional response change must occur and refers to a "smoothing" effect on such changes --it is not this smoothing de lay to which I refer.) While it is apparent that the initial impact of product awareness can be that of precipitating a response catastrophe, blatant claims and product puffery will have less of an effect during a "cooling-off" period and, in fact, can turn a sales away if it must compete with accurate product claims when the purchaser is allowed to exercise rational judgment.
A good salesperson instinctively knows that a customer who wants to "think about it" will probably not come back once he walks out the door. Thus, a salesperson who knows what he is doing will not present us, the customer, with an opportunity for a cooling-off period. We must take this step ourselves.
During a cooling-off period between a desire catastrophe and commitment to purchase, increased product awareness can drive us back toward a disillusionment catastrophe if the added knowledge reveals things which we ourselves would consider undesirable. That is most important.
The shape of the behavior manifold, and where we are on that manifold, is different for each of us. Two persons can react differently to the same product if they bought it during the passion of sales and then took it home for listening. At the moment of desire catastrophe, both persons might be equally convinced that this is the product for them. But the person with the higher gradient of response will become more quickly disillusioned with that product if additional negative features are revealed. Product awareness on audio components includes knowledge about sonic imperfections. If those imperfections are of a type which will detract from the enjoyment of sound, in the frame of reference of a particular purchaser, then it is helpful in the long range satisfaction of that purchaser that he know about such imperfections before buying the product, not after. The person who might become the most unhappy about the performance of a product is the one (with the higher gradient) who can best benefit from a cooling-off period during which he is allowed to compare product claims and do comparative listening.
If there is a moral which catastrophe theory teaches us, it is that the time to pay attention to product claims is be fore we are in the store and exposed to the heat of salesmanship; then we should allow ourselves a cooling-off period after we fall in love with a product.
If, during the cooling-off period, the customer becomes aware of enough negative factors, he may find himself dropped into a pocket of compromise.
His trimodal condition lies between outright rejection of the product and wild acceptance. Because the drive of product awareness has been stalled by opposing trends, cost may now play a more dominant role in popping the desire to purchase upward to a higher sheet.
Since the bias factor (cost) tends to magnify response, a reduction in price (shopping around for a better deal) may set up a condition where an in crease in splitting factor (time spent listening) triggers a jump upward in desire to purchase.
There is no surprise here; shopping around for a lower cost is, or should be, a requisite for any rational purchase. However, the combined effect of bias (cost) and splitting/normal (music listening) factors can now set up another trigger condition. A good salesperson can pull the butterfly trigger by letting a customer know that demand for the product has so thoroughly outpaced deliveries that this demonstrator model is the last one in stock. Desire catastrophe! Pull out the checkbook! What salesperson could be cruel enough to turn down the tearful request of a customer who wants to take home the prize of his desire. This particular ploy also side tracks any further opportunity for a cooling-off period, so beware.
Approximate C. T.
The foregoing analysis applies when there are four distinct, identifiable factors and one response. One's own personal emotions, when considering purchase of an audio component, may involve more than four factors, and there may, indeed, be several responses which those factors elicit. In setting up an example which illustrates the application of Catastrophe Theory to audio, I have chosen what I consider to be the most significant factors and have tried to relate them to the mathematical terms to which they most nearly correspond.
It is my opinion that, at this time, the most important use we can make of Catastrophe Theory is to uncover trends in response under the influence of conflicting factors. When there are many factors, but four of them are dominant and relate to each other as does the normal, splitting, bias, and butterfly factors, and when there is one dominant response, we can use Catastrophe Theory to determine the most likely behavior. The influence of other, less dominant, factors will not change major aspects of the probable behavior, but will color the details of that response.
While not explicitly stated in most technical discussions of Catastrophe Theory, one can often simplify the analysis of a problem by reducing the dimensionality to that of the most dominant behavior space. For example, if there are 10 apparent factors, but only four of them stand out as dominant, then the situation can be reasonably approximated by the butterfly catastrophe. If, in turn, two of these four factors are considerably more important under a given set of circumstances, then we can resort to the cusp catastrophe.
The reason I suggest we use this simplification wherever possible (which I personally call "approximate catastrophe theory") is because of the enormous increase in detail complexity which occurs with rise in dimensionality. It simply gets out of hand and we may tend to lose the forest (general trends) for the trees (fine details).
When there are two responses, the type of catastrophe maps which are involved produce hypersurfaces which Thom calls "umbilics." Within Elementary Catastrophe Theory there are six umbilic catastrophes, ranked ac cording to dimensionality of the behavior space. Time does not permit us to discuss the application of umbilic catastrophes to audio, although there are several of these and, perhaps, at a much later date we can discuss them.
Mathematics of Emotion
In this discussion we have essentially been considering a mathematics of human emotion. Improperly used, such a theory could cause consider able mischief; however, one of the quickest ways to defuse this potential weapon is to be aware of its existence.
That, in part, is why I have not hesitated to discuss what might otherwise be considered a touchy subject.
Computer-programmed advertising campaigns, or, for that matter, political campaigns, may not be very far off. As a consequence we must be aware that any assault on our pocketbook must begin with a play on our emotions.
That is the game, and we all play it.
And as long as we all understand the rules, the match is balanced and the game is fair.
Up to now each of us has had to learn the rules by experience, either our own or what we observe from others. Now, a weird, far-out, abstract mathematics has come into existence which, while not originally developed for that purpose, seems to model some of the primal rules of human emotion.
As a service to the readers of Audio, I believe it is better to point out the existence of this new game plan and to let you know what might be coming, than to suppress the knowledge in hope that overzealous advertisers would never find out about it.
On a more pleasant note (that is, less sinister) we can now begin to appreciate the part that human emotion can play in our judgment of the subjective listening quality of audio components. All of us, I am sure, have experienced the situation where we "liked" the performance of a particular component one time, then something happened and we "didn't like" the same component when we heard it again. Perhaps we learned something which caused us to change our minds.
It is only human; a new factor was introduced. But the component did not change --the measured technical performance did not change--we changed. Or, more properly, our response changed.
In light of what we have been discussing, this does not seem so mysterious. Yet think how capricious this might seem to a technically oriented "flatlander" who found that nothing had changed in the technical performance of that component.
The investigation of these delights and other properties of perception are yet to come. They can be the subject of a future discussion.
The final point I would like to discuss in this three-part series involves the reason I am personally interested in Thom's geometric theory. The intangibles of audio include perception, cognition, and valuation. Perhaps with Catastrophe Theory we can, for the first time, begin to understand how one's personal subjective impressions of quality can be linked to conflicting circumstance.
We can begin to realize that it is not inconsistent for us to "like" something one time, then "not like" it another time, even under seemingly identical conditions. It is not where we are, that determines the intensity of our emotions, but how we got there.
The old complaint "why can't some one measure what I hear?" now takes on a different tone. We can "measure" under stable conditions, but that may not be sufficient for the determination of subjective valuation. Education, knowledge, experience, and training are factors involved in subjective valuation. It is often said that we can hear with different ears. It now appears that this is quite true.
We can be fooled into perceiving one thing while actually being subjected to something else. The stage artistry of a master illusionist can make us "see" things contrary to reality, such as flowers from thin air or a per son sawed in half and then reassembled in front of us. Similar artistry can allow a ventriloquist to "project" a voice so that we "hear" it coming from an impossible location. And even as early as the 1920s a live vs. reproduced experiment was conducted in which an audience was substantially unable to distinguish whether the sound they heard coming from a stage was that of a live performer, whom they could see, or the playback of that performer from an Edison acoustic phonograph. The perceived sense of reality in all such cases is an illusion supported by factors other than those of the principal sensory organ which is involved.
So, too, it is in this present business of audio reproduction. It is not reality, but the illusion of reality which the present audio industry depends upon.
We must perceive an acceptable illusion in order to have any success in so called high fidelity reproduction.
The type of analysis embodied in Catastrophe Theory might begin to ad dress this very important audio problem from a new direction. We can be gin to look at the human process of perception, cognition, and evaluation as responses under sets of factors, some of which may be conflicting. We might also begin inquiry into a very important problem in present audio, namely, what are those factors necessary to support an acceptable illusion? It is not a retreat either from reality or from technology to accept the existence of human emotion. Nor should technology ignore the role played by emotional response in establishing the quality of the listening experience.
Returning to the theme we discussed in the first part of this three part series, those of us who pursue endless quests of measuring vanishingly smaller amounts of system distortion for sine-wave and square-wave signals are scrambling around under the wrong lamppost. The wrong lamp post, that is, if we want to correlate such measurements with subjective quality. It is the wrong place to look for two reasons: First, those distortion measurements are derived from linear theory and are, at best, on shaky ground; second, such measurements take no cognizance whatsoever of the intangibles of the listening experience.
The light is not so bright and cheery when we walk into the bushes and search for audio truth with nonlinear tools.
Catastrophe Theory is one of those newly evolving mathematical tools which show promise of being able to provide us with a framework for handling those types of distortions, either real or imagined, which lead to instability of perceived form. It can score the most heavily in those situations poorest handled by linear methods. I really do not know whether Catastrophe Theory can be of any lasting value in our understanding of the dilemma of perception. But we will never find out until we give it a try.
1. T. S. Kuhn, The Structure of Scientific Revolutions, The University of Chicago Press, Chicago, 1970.
(Source: Audio magazine, May 1979; Richard C. Heyser)
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