Notes on English Composition

Introduction

This is the first in a planned series of notes on English composition. These notes are intended as a reference for the writing of academic works to be used by ``non-native" physicists and mathematicians. They are specifically designed to help Japanese writers, but hopefully they will be found useful by a wider audience.

In writing these notes, it is not my purpose to create a comprehensive guide to scientific writing. Something of this nature would be both very difficult to write and of little practical use. It is also not my intention to explain standard points of English grammar or elementary rules of English composition, as there are already many textbooks on these subjects. My goal in writing these notes is to create a practical tutorial that addresses specific common mistakes made by non-native (and particularly Japanese) writers. Using examples taken from academic papers, I will consider the most common problems I have encountered and offer my advice on how to correct them. In this way, I hope to help these writers to improve their writing by eliminating mistakes one by one.

I hope that those who read these notes will think of them as not just a guide to scientific writing but as one piece in a more general English education. According to my personal experience, real improvement in one's understanding of a foreign language can only be gained by studying many concrete examples. Abstract speculation on language structure and the formulation of general grammatical rules may be interesting from a linguistics point of view, but they are of little use if the goal is to improve one's communication skills. ``Theoretical" discussion usually becomes only an obstacle to language instruction, and I will therefore try to keep such discussion at a minimum. The ultimate goal for students of a foreign language should be the development of an intuitive understanding, without which writing remains a mechanical operation that can never do justice to the ideas it is used to convey. I believe that the example-based instructional format I will use here is the most effective for the purpose of developing such an understanding. I hope that by contemplating these examples, readers will eventually develop an insight that transcends the set of ``microscopic" lessons they provide.

Over the last four years, I have proofread on the order of 1,000 papers, most of these written by Japanese physicists and mathematicians. During this time, I have found that a large percentage of the mistakes that I see are repeated again and again. Some of them seem almost universal. In many cases, these problems result from the tendency of authors to directly translate Japanese into English, while in other cases they arise from what seems to be some pervasive misinformation existing in Japan with regard to English usage. Most of these problems are fairly simple and are easily corrected. It may be wise to restrict the focus of these notes to such simple problems, and eventually this may be what I decide to do. However, I would also like to make an attempt at addressing some more difficult problems that arise from the complexities of the English language itself.

I consider these notes to be an open-ended project, as I am sure that there is no end to the useful examples that could be added to them. Partly for this reason and partly due to my own lack of organization, the monthly installments to appear on the Progress webpage will consist of collections of very incomplete notes on a variety of perhaps unrelated topics. In addition, their content will undoubtedly reflect my personal writing style, my preferences, and my prejudices. I hope the readers will overlook the shortcomings of my very imperfect presentation. I welcome and will do my best to honor any requests for material to be covered, and I invite comments of all kinds.

I. On the contrary

This is the first in a series of notes in which I discuss expressions that are commonly misused by Japanese writers.

The expression ``on the contrary" is used very often in the papers I proofread, and it is almost always used incorrectly. This expression can only be used to introduce a statement that expresses a meaning that is opposite to what has been previously stated or to what is expected. Its use gives emphasis to the statement in which it appears and to the falsity of that which this statement denies. The following sentences demonstrate its proper use:

1. Today is not cold. On the contrary, this is the warmest day thus far this spring.

2. This interaction does not lower the energy of the system. On the contrary, it increases the energy by such a large amount that the approximation used above becomes invalid.

In most of the cases that ``on the contrary" is used in the papers I read, the intended meaning is that expressed by ``contrastingly", ``in contrast", ``by contrast", ``while", ``but", ``however", etc. Consider the following (incorrect) examples:

1. Anderson investigated the full system in the weak-coupling limit. On the contrary, we consider the simplified system described by (1) and study it in the strong-coupling limit.

2. This equation can be easily solved. On the contrary, that derived in Sect. 1 can only be treated numerically.

3. Bose-Einstein statistics describe integer spin particles. On the contrary, the particles in which we are interested are always of half-integer spin.

4. In the case discussed above, t₀ > t₁. On the contrary, in the present case, t₀ < t₁.

5. The primitive form of analysis used in the previous section does not allow us to draw any definite conclusions with regard to properties of this solution. On the contrary, with the form of analysis developed in this section we are able to determine upper bounds on the values it assumes in each of the intervals in question.

Note that in none of these cases does the second sentence contradict the first. For this reason, ``on the contrary" is inappropriate. Rather than contradiction, in each case above, the second sentence presents a situation which is in contrast with or in some sense inconsistent with that presented in the first sentence. In 1, the situation in which Anderson investigates the full system and that in which we investigate a simplified system are not mutually exclusive. In 2, since we are considering different equations, there is nothing contradictory about claiming that one of them can be solved easily while the other cannot. The following are some possibilities for corrected versions of the above examples.

1. Anderson investigated the full system in the weak-coupling limit, while we consider the simplified system described by (1) and study it in the strong-coupling limit.

2. This equation can be easily solved. Contrastingly, that derived in Sect. 1 can only be treated numerically.

3. Bose-Einstein statistics describe integer spin particles, while the particles in which we are interested are always of half-integer spin.

4. In the case discussed above, t₀ > t₁. However, in the present case, t₀ < t₁.

5. The primitive form of analysis used in the previous section does not allow us to draw any definite conclusions with regard to properties of this solution. Contrastingly, with the form of analysis developed in this section we are able to determine upper bounds on the values it assumes in each of the intervals in question.

In this note I discuss the use and misuse of the three related words ``keep,'' ``maintain'' and ``preserve.''

The word ``keep'' is often mistakenly used by Japanese authors in situations that ``maintain'' or ``preserve'' would be appropriate. In fact, in the papers I read, this word is more often used improperly than properly.

In written English, and in particular scientific writing, it is best to avoid words with very broad meanings and to use in their place more precise expressions: ``perform,'' ``carry out,'' ``undertake,'' or ``execute'' instead of ``do,'' ``take on,'' ``obtain,'' or ``become'' instead of ``get,'' etc. The situation is the same in the case considered presently. Because the word ``keep'' has many meanings, its use can be ambiguous, and it is therefore better to use more narrowly defined (and therefore more precise) words such as ``retain,'' ``maintain,'' ``preserve,'' ``conserve'' and ``prevent.''

Misuse of ``Keep''

The following are typical ways in which the word ``keep'' is misused:

1.^* The system keeps its symmetry under the transformation ${\cal{T}}$.

2.^* The qualitative behavior of the system is kept even when the temperature is raised by as much as 10 Δ T.

3.^* The delicate balance of these two opposing terms is kept until the critical temperature, because until this point, the operator P keeps the only relevant quantity, z.

4.^* In this way, the system is kept from reaching a global equilibrium.

5.^* The integral of this function keeps its value for all time.

6.^* This measurement keeps the total angular momentum of the system.

7.^* Of course, the system keeps its total energy.

8.^* The center-of-mass distances are all kept during the evolution.

9.^* The propagator keeps the values of α and β.

10.^* The material keeps its structural integrity until the amplitude of the driving force is increased to f₀.

These sentences are better written as follows:

1. The system maintains its symmetry under the transformation ${\cal{T}}$.

2. The qualitative behavior of the system is maintained even when the temperature is raised by as much as 10 Δ T.

3. The delicate balance of these two opposing terms is maintained until the critical temperature, because until this point, the operator P preserves the only relevant quantity, z.

4. In this way, the system is prevented from reaching a global equilibrium.

5. The integral of this function is independent of time.

6. This measurement preserves the total angular momentum of the system.

7. Of course, the system maintains its total energy.

8. The center-of-mass distances are all preserved during the evolution.

9. The propagator preserves the values of α and β.

10. The material maintains its structural integrity until the amplitude of the driving force is increased to f₀.

Maintain vs. Preserve

These words are closely related, and for this reason they are commonly confused. In some cases this does not lead to any problem, as they can often be used interchangeably, but there are many situations in which only one is appropriate. In this section I attempt to make their distinction clear.

``Maintain''

In general, it is better to use ``maintain'' when describing the action of some physical or mathematical object on itself. For example, as in 1 above, it sounds more natural to say ``The system maintains its symmetry'' than ``The system preserves its symmetry.'' While it cannot be said that the latter is wrong, it seems to suggest that ``it'' here is not referring to the ``system'' but to something else.

It is also better to use ``maintain'' when in reference to a general state or condition. This can be seen by considering the following examples.

1. A near-equilibrium state is maintained throughout this process.

1.^*A near-equilibrium state is preserved throughout this process.

2. Laminar flow is maintained below R = R₀.

2.^*Laminar flow is preserved below R = R₀.

In these examples, ``preserve'' is very inappropriate.

Finally, it is better to use ``maintained'' to convey the meaning that there is some influence being somehow defended against. For example, while ``The structure was maintained against the constant eroding force of the ocean'' is very natural, ``The structure was preserved against the constant eroding force of the ocean'' is quite unnatural.

``Preserve''

While use of ``maintain'' seems to imply a defending (oneself or something else) against some influence, ``preserve'' implies the somewhat different idea of keeping/leaving unchanged.

There are two cases in which ``maintain'' cannot be used instead of ``preserve.'' First, ``preserve'' should be used when that which is unchanged is some specific quantity or quality. This is demonstrated by the following:

1. The value of the parameter α is preserved in time.

1.^* The value of the parameter α is maintained in time.

2. The velocity of the flow in the region between points a and b is preserved when the electric field is applied.

2.^*The velocity of the flow in the region between points a and b is maintained when the electric field is applied.

The second situation in which ``preserve'' must be used is that in which there is some action applied to a system that produces some kind of change, but there is some property of the system which is not changed and which is being singled out for discussion. This is a very common situation in mathematics and physics:

1. The operator ${\cal{Z}}$ preserves the rotational symmetry of this state.

1.^*The operator ${\cal{Z}}$ maintains the rotational symmetry of this state.

[While such a statement is possible, the meaning is entirely different from that of 1. Here, it would seem that there is some other influence (perhaps some kind of perturbation, etc.) against which ${\cal{Z}}$ is acting in order to keep the rotational symmetry unchanged.]

2. The number of vertex points is preserved under the mapping in question.

2.^*The number of vertex points is maintained under the mapping in question.

3. This change of coordinates preserves the total volume of all the regions.

3^*This change of coordinates maintains the total volume of all the regions.

4. Only c₁ and c₂ are preserved under this transformation.

4.^*Only c₁ and c₂ are maintained under this transformation.

5. This perturbation preserves the total energy.

5.^*This perturbation maintains the total energy.

(Again, note that 5^*is possible, but the implication is that if the perturbation were not applied, the total energy would change. The implication of 5 is much different.)

Proactive vs. Reactive

There is another difference in nuance between ``maintain'' and ``preserve'' that could perhaps be surmised from the above discussion but has not been made explicit. While these nuances are often not present when these words are used in mathematical discussion, I think it is helpful to gain a more intuitive understanding to make this difference clear.

When describing some object or property that is being either ``preserved'' or ``maintained'' in the presence of some perturbing influence, use of ``maintain'' implies stabilizing action in response to this influence, while use of ``preserve'' implies action in anticipation of it.

The word ``maintain'' is more appropriate than ``preserve'' in the following situations:

1. The dog stood on top of a fence and maintained her balance in the windstorm.

2. The woman managed to maintain her sanity during a very difficult time.

In the situations here, the dog and the woman are acting in response to some destabilizing forces.

Now consider the following:

4. During the processing of the food, certain chemicals are added to it to preserve its flavor.

5. The quality of the book has been preserved against aging by placing it in an ultra-clean environment.

In both cases here, some action is taken in anticipation of the effect of aging.

Now, let us see how the meanings of the above sentences change if we switch ``maintain'' and ``preserve'':

1.^*The dog stood on top of a fence and preserved her balance in the windstorm.

2.^*The woman managed to preserve her sanity during a very difficult time.

These both sound quite unnatural. It seems that perhaps the ``windstorm'' and the ``difficult time'' are not the perturbing forces whose effect is to be prevented by the ``preservation'' in question.

4.^*During the processing of the food, certain chemicals are added to it to maintain its flavor.

This does not sound unnatural. In fact, the meaning changes little when replacing ``preserve'' by ``maintain''. However, in the former, the adding of chemicals seems to be part of the processing, while in the latter case, it seems to be something separate from the processing, perhaps something done only after it was discovered that the flavor of the food had changed.

5.^*The quality of the book has been maintained against aging by placing it in an ultra-clean environment.

This sounds quite strange, because placing the book in the ultra-clean environment is clearly something done in anticipation of the influence of aging, while use of ``maintain'' seems to imply that this is done in response to this influence.

In this short note, I address the very common problems involving use of the word ``order'' when this word is used to indicate the approximate size of some quantity. Fortunately, the rules governing this use are quite simple, and this problem is thus easily remedied.

``on the order of'' vs. ``of order''

The basic rule governing the usage of these expressions is the following: When reference is being made to a pure number (i.e. a dimensionless quantity), ``of order'' should be used, and when reference is being made to a dimensional quantity, ``on the order of'' (This is the appropriate American English phrase. In British English this becomes ``of the order of.'') should be used. When proofreading papers written by Japanese authors, I encounter many variants of these expressions. For the most part, these are all rather awkward. In general, I strongly suggest adherence to the above basic rule when using this word.

The following examples demonstrate the appropriate use of ``order'' in the present context.

Dimensional quantities

1. The temperature of the system is on the order of 10 K.

2. This state has an energy on the order of 50 eV.

3. The characteristic timescale of the phase separation process is on the order of hours.

4. The average velocity of the particles is on the order of .1 c.

5. This energy scale is on the order of the Planck mass.

6. The overestimate is on the order of the size of the system.

7. There are on the order of 100 particles in the reaction chamber.

Dimensionless quantities

8. The first term here is of order 1.

9. This quantity is of order ε, the small parameter in terms of which we are expanding.

10. The size of the system is of order N^α, where N is the total number of particles.

11. The dimensionless velocity of the front is of order z^3/2, which in the system described by Fig. 1 is approximately 1.5. This corresponds to a velocity on the order of .2 mm/sec. in the `typical' physical system considered in the previous section.

12. This quantity is of order ρ_v/ρ_h, where ρ_v and ρ_h, are the vertical and horizontal dimensions of the apparatus.

13. All quantities appearing on the right-hand side of this equation are of order unity.

14. The number of particles in the reaction chamber is of order 100.

Several of these examples deserve some discussion. First, in 6, since we may be thinking of the size of the system as a dimensionless quantity, it may seem that this could also be written as follows:

6.^* The overestimate is of order the size of the system.

However, in fact this sentence is very problematic. The reason for this is that even in the case that we are thinking of the size of the system as a pure number, ``the size of the system'' is not. This points out that often the distinction between the two cases considered here is more a matter of English semantics than mathematical meaning. To make this point more explicit, if we change this sentence to read something like

6^'. The overestimate is of order N^1/2, the size of the system.

where N is understood to be a pure number, then there is no problem.

Next, let us consider examples 7 and 14. These seem to be making precisely the same statement, and it may thus seem strange that we use ``on the order of'' in one case and ``of order'' in the other. Again, this is a problem of linguistics rather than mathematics or physics. In 7, ``order'' is used in reference to ``100 particles,'' while in 14 it is used in reference to ``100.'' This difference is also reflected in the fact that in 7 the verb is plural (``are''), while in 14 it is singular (``is'').

Additional examples

For reference I give here some examples demonstrating the proper use of ``order'' in some situations related to those considered above.

1. Here a is of higher order in δ than b.

2. The values σ and θ are of the same order.

3. The temperatures T₁ and T₂ are on the same order.

4. A and B are of vastly different orders.

While ``of'' and ``on'' are interchangeable in 2 and 3, the former is perhaps better with dimensionless quantities and the latter with dimensional quantities.

I. Some discussion of verb tense

This note contains discussion regarding some of the most common mistakes involving verb tense. The topics covered here are in no way exhaustive, and the order in which they are presented has no particular significance. In future months further discussion on this topic will appear.

Verb tense in an abstract

In an abstract, the present tense is most appropriate when discussing the work you present in the paper. This is because abstracts are usually written with an implied ``in this paper" in front of all statements regarding the work you present. (For this reason, phrases like ``in this paper" should not appear explicitly in abstracts.) Thus, we have statements of the following kind:
``A general model is presented and its behavior is studied for the case of a harmonic potential."
The implied meaning is, ``In this paper, a general..." Since the contents of the paper exist in the past, present and future, the appropriate tense is the present.

When referring to previous work in an abstract, the same rules apply as when referring to previous work in the main text.

Verb tense in referring to previous work

There are several cases which must be discussed concerning reference to previous work. The first thing which should be pointed out, however, is that reference to one's own previous work and that of others should be done in a consistent manner. It is best to treat all previous work on an equal footing in this regard.

(1) Reference to a ``dead" field, or to research which is no longer current

Although such reference is probably fairly rare, I discuss it here. In this case, it is appropriate only to use the past tense:

1. Galileo studied the motion of the planets.

2. This topic was the studied extensively in the mid-1800's.

3. Many authors made the mistake of ignoring elastic effects in the early years of this field.

(2) Reference to general aspects of work on a current field of research

Here, it is best to use the present perfect tense. This carries with it the meaning that the activity has some extended duration in time and that it is continuing today. The implication of 2 above is that this ``topic" is no longer the subject of research. The implication of 3 is that people no longer make this mistake. These situations contrast with those described by the following examples:

1. Smith has studied this system with much success.

2. This topic has been the focus of a great deal of study in recent years.

3. Many people have made the mistake of ignoring the effect of non-linear terms.

The implication of 1 is that this is study of a current topic, and allows the possibility that Smith is still studying this system (or perhaps something related). However, it is clear that she has obtained some results and carried some aspect of the work to completion. The implication of 2 is that this topic is still the focus of a great deal of study. The implication of 3 is that at least some people are (probably) still making this mistake.

(3) Reference to specific actions or specific activities associated with research performed in the past

In contrast to the situations considered in the previous section in which reference is made to general aspects of past research, we now consider situations in which reference is made to specific acts that people have carried out in their research. In such a case, the past tense is appropriate because the act in question was carried out in the past. Consider the following:

1. Thompson computed these coefficients systematically.

2. These measurements were made at 1.2 K.

3. We proved this result with the help of Theorem 1 above.

In 1, the present perfect tense is also possible: ``Thompson has computed these coefficients systematically." The difference between this and the sentence above is a matter of time. If Thompson did this, for example, several years ago, and the result is well known, the past tense is best. If, however, Thompson just published results on this, and these are perhaps not yet well known, the present perfect tense is better. Now, suppose the tense of 2 is changed to present perfect: ``These measurements have been made at 1.2 K." In this case, the meaning is that these measurements have been made a number of times (perhaps by several different people), while the meaning of the original sentence is that the measurements were probably done just once and perhaps by only one person (or group). In contrast to 1 and 2, the meaning of 3 is almost completely unchanged when the past tense is replaced by the present perfect: ``We have proven this result with the help of Theorem 1 above." However, in the case that this proof was done some time ago (and perhaps is well known), the past tense is better.

(4) Reference to work in progress

Here, there are two situations. The first is that in which the work has been going on for a long time but there are yet no results (or at least no published results), or there are some results but the work is not completed. In this case, we use the present perfect progressive tense:

1. Simpson has been studying this system numerically.

2. We have been calculating these terms analytically.

The implication of 1 is that Simpson is not yet done studying this system. The implication of 2 is that we are not yet done calculating these terms, and therefore that there are probably many of them and that calculating them takes a great deal of time. The second situation is that in which the work has not yet been taking place over a long time, or that work on it has just begun. In this case, we use the present progressive tense:

3. We are presently studying this system in the deep quench regime.

4. Experiments on such systems are now being performed.

additional note

When referring to previous work, the following is best kept in mind. When referring explicitly to a particular work, use the past tense, as I have done here. Thus, for example: ``This problem was studied by Einstein [1]." However, when you refer to a whole body of work without making explicit reference to any particular work or works, it is best to use the present perfect tense. For example: ``Study of this problem has been carried out in many settings." The reason for this distinction is that in referring to a specific work done by a specific author(s), you are referring to something done at one specific time. Thus you need the past tense. When you refer to a body of work in general, however, you are referring to something which has taken place over an extended period of time. Thus the present perfect tense is more appropriate.

Verb tense in referring to one's own results

The results of a particular work (whether right or wrong) are timeless. Because they do not change with time, when referring to any such results, it is proper to use the present tense:

1. Our results are in agreement with those of Stevens.

2. The results of our experiment show that the electronic structure of the system is not as simple as is generally believed.

3. The proof we have given in Sec. 1 implies that the set S is dense in D.

However, it should be kept in mind that when referring to the actual act of obtaining these results (calculating, measuring, proving...), rather than their meaning or implication, one must use the past tense or the present perfect tense:

4. According to the results obtained in Section 3, we can conclude that the equation in question is structurally unstable.

Here, in referring to the act of deriving the results, ``obtained" is used, since this was done in the past, but in referring to the implications, which are timeless, we use present tense.

5. We calculated the value of the exponent β using the method described above.

6. In this paper, we have shown that the instability of the system cannot be described using the simple linear analysis introduced in Ref. [2].

7. In Sect. III, we present the results of numerical simulations of this gel system carried out using this model.

Verb tense in referring to behavior of an equation or experimental system

This case is similar to that just discussed. The main point to keep in mind is that the behavior of a given mathematical model or experimental system does not change in time. [Although the results of experiments on a given system may change, the behavior of the system itself (when considered in the abstract sense) does not.] Thus it is necessary to use the present tense:

1. As shown by Sato, these solutions diverge in the t → ∞ limit.

2. The behavior of the system we presented in the previous section is common to a large class of reaction-diffusion equations.

3. In the experiments we performed on these systems, we found that they are quite insensitive to temperature changes in the range in question.

Contrast the above examples with the following:

4. In Sato's numerical study of these equations, he discovered that the solutions in question diverge in the long time limit.

5. The behavior of the system presented in the previous section was found to be common to a large class of reaction-diffusion equations.

6. In our experiment, the system was quite insensitive to temperature changes in the range in question.

The difference between 1 and 4 is that in 1, the main clause is discussing the behavior of the solutions -- the fact that they diverge for large t. This is true at any time, past, present or future. In 4, however, we are discussing the discovery made by Sato. This discovery was made in the past. Similarly, 2 and 3 are relating the nature of the behavior of the respective systems, while 5 and 6 are describing what was found and what was measured, respectively.

The skillful use of verbs is perhaps the most distinguishing property of good writing. For this reason this topic will be the focus of a large number of the notes in this series. This is the second of these.

In this note, I discuss several verbs that are to various degrees synonymous with ``do''. These verbs are, of course, very common, and because they are often misused by Japanese authors, I consider their treatment to be quite important.

The verb ``do'' itself is quite nondescript. Its meaning is very broad, and therefore imprecise. For this reason, it is best avoided when there are more descriptive words that can be used in its place. In particular, I suggest making more use of verbs like ``perform'', ``conduct'', ``carry out'', ``execute'', ``undertake'', etc., when these are appropriate. In this note, I discuss the appropriate and inappropriate use of these verbs.

I should point out that the verbs considered here certainly do not constitute a complete set of alternatives to ``do'' and that the example sentences appearing below in no way exhaust the possible ways of using these verbs.

perform

The following are all acceptable uses of ``perform'':

1. The experiment was performed at room temperature.

2. We performed the calculation in two regimes.

3. Numerical computations were performed for a variety of systems.

4. We performed a set of manipulations on the equations in question and obtained the following result.

The following are all inappropriate ways in which to use ``perform'':

5. We perform an approximation of the equation in question.

6. The measurement was performed at a scattering angle of 12.5°.

7. We perform a proof of this theorem in the next section.

8. Discussion is performed in Sec. II.

9. We perform research on systems of this type.

The sentences 5 and 6 are not good because in each case, the action being ``performed'' does not consist of a set of steps (at least in most natural situations).

The situation with 7, 8 and 9 is perhaps somewhat more subtle, as we now discuss. To ``perform'' usually implies the following of some prescription or previously devised plan. An experiment is performed according to some plan. A calculation is performed following some prescription or set of predefined rules. (Note that ``perform'' also is used with regard to actors, who, in fact, follow some predetermined script.) A proof, discussion, and research on the other hand, do not follow some predesigned set of rules (in most meaningful cases, anyway). Thus ``perform'' is not appropriate.

The above sentences are better rewritten as follows:

5. We make an approximation of the equation in question.

5'. We approximate the equation in question.

6. The measurement was taken at a scattering angle of 12.5°.

6'. The measurement was made at a scattering angle of 12.5°.

7. We construct a proof of this in the next section.

7'. We give a proof of this in the next section.

7''. We prove this in the next section.

7'''.This is proven in the next section.

8. Discussion is given in Section II.

8'. Discussion appears in Section II.

9. We conduct research on systems of this type.

conduct

The following are appropriate ways to use ``conduct'':

1. We conduct research on spin systems.

2. The experiment was conducted at temperatures just below the phase transition.

3. We conducted a survey of all the presently available experimental results.

4. We conducted a series of numerical simulations.

The following are inappropriate ways to use ``conduct'':

5'. The calculation was conducted.

6'. A simulation of the systems was conducted.

7'. We conduct a proof of this property in the following section.

8'. Measurements were conducted for a range of field strengths.

The verb ``conduct'' is used almost exclusively in reference to experiments and research, or, more generally, activities that require the coordination of many sub-activities. (Note that the conductor of an orchestra is responsible for coordinating the activities of all of the members of the orchestra.) Thus, for example, 4 above is feasible because each of the series of numerical simulations can be thought of as such a sub-activity. The problem with 5'-8' is that, although each of the activities in question can be thought of as consisting of sub-activities, these sub-activities are more naturally thought of as (perhaps somewhat mechanical) steps that do not necessarily require some kind of overall coordination. The following represent some appropriate ways to rewrite the above:

5. The calculation was carried out.

5'. The calculation was performed.

6. A simulation of the system was carried out.

6'. A simulation of the system was performed.

7. We give a proof of this property in the following section.

8. Measurements were taken for a range of field strengths.

undertake

The following are acceptable uses of ``undertake'':

1. Study of this system was undertaken with the hope of settling the remaining unanswered questions.

2. We undertook this research knowing little about the difficulty inherent in its experimental observation.

The following are inappropriate ways to use ``undertake'':

3*. We undertook an experiment on this system.

4*. Calculation of these exponents was undertaken numerically.

Use of the verb ``undertake'' implies that the thing which is being undertaken is in some sense of grand scope. It usually is reserved for very large projects requiring years of effort. It is also best used in reference to more abstract activities (e.g., ``studies'', ``research''), as opposed to more concrete activities (e.g., ``calculations'', ``experiments'').

3* and 4* can be better rewritten as:

3. We performed an experiment on the system.

3'. We conducted an experiment on the system.

3''. We undertook an experimental study of the system. (This is most appropriate for a set of a large number of experiments.)

4. Calculation of the these exponents was performed numerically.

4'. These exponents were calculated numerically.

carry out

The following are appropriate uses of ``carry out'':

1. The authors carried out a calculation and obtained the following results.

2. We carried out a series of experiments that confirm the theoretical predictions made in Ref. [1]

3. Analysis of this system is carried out below.

This term is usually used in reference to some activity consisting of a set of (at least to some extent) clearly defined steps. It is similar to ``perform''. (Note that ``perform'' could be used in all three of the examples above.) There is a slight difference in nuance, however, as ``carry out'' seems to imply a slightly more mechanical following of well-defined steps. Also note that for 2, ``carried out'' could be replaced by ``conducted'' with little change in meaning. Again, the difference here is that use of ``carried out'' seems to imply that there perhaps has not been an effort to coordinate the various experiments or their results. For example, if the results of this series of experiments must be used together to obtain some result, ``conduct'' is better. However, if the ``theoretical predictions'' referred to here can be ``confirmed'' independently from the results of a series of independent experiments, ``carried out'' is better. The following are inappropriate uses of ``carry out'':

4*. We carry out discussion of this point in the following section.

5*. Research has been carried out on this topic for the last ten years.

6*. We have carried out a study of hadron systems ignoring the effect described above.

7*. In this paper a confirmation of the results obtained previously is carried out.

8*. Measurements were carried out near the critical point.

9*. Let us carry out an approximation of this interaction.

10*. In this paper, predictions are carried out concerning the nature of crystal growth under such extreme conditions.

For 4*-7*, the activities in question do not follow a sequence of clearly defined steps. The situation for 7*-10* is somewhat more subtle. The ``measurements'' referred to in 8* could be considered as resulting from a sequence of well-defined steps, and thus it may seem that ``carry out'' is appropriate. The problem, however, is that while a measurement may involve a set of well-defined steps, it is not equivalent to such a set. Rather, it is most natural to consider a measurement as the culmination of a set of steps. Thus, while the sentence

The steps necessary to make the measurements near the critical point were carried out.

sounds quite natural, 8* is somewhat unnatural. The problems with 9* and 10* are similar: In both cases, while it is natural to think of ``carrying out'' the steps through which the approximation or prediction is obtained (if indeed it consists of a set of well-defined steps), the approximation or prediction itself is not equivalent to these steps. (Note, by contrast, that a calculation is most naturally thought of as being equivalent to the steps it involves.)

The following are appropriate ways to rewrite the above sentences:

4. We give discussion of this point in the following section.

4'. We discuss this point in the following section.

5. Research has been conducted on this topic for the last ten years.

6. We have undertaken a study of hadron systems ignoring the effect described above.

6'. We have studied hadron systems ignoring the effect described above.

7. In this paper a confirmation of the results obtained previously is given.

8. Measurements were made near the critical point.

9. Let us make an approximation of this interaction.

9'. Let us approximate this interaction.

10. In this paper, predictions are made concerning the nature of crystal growth under such extreme conditions.

execute

The verb ``execute'' is most naturally used in reference to specific actions. While it is similar to ``carry out'', ``execute'' is of somewhat narrower applicability. The main difference between these two verbs is that, with regard to a process such as a calculation or experiment consisting of a number of steps, while ``carry out'' can be used in reference to the calculation or experiment itself as well as to the steps composing it, ``execute'' can only be used in reference to these steps. Thus, while both of the sentences

1. We carried out an experiment on this system.

2. We carried out the steps of this experiment.

are quite natural, only 4 below represents an appropriate use of ``execute'':

3*. We executed an experiment on this system.

4. We executed the steps of this experiment.

The following are appropriate uses of ``execute'':

5. The steps in the experiment were executed in the following order.

6. The computer program executed commands as shown in the diagram given in Fig. 1.

7. Though there seems to be some ambiguity in this regard, the steps in the calculation must in fact be executed in a particular order.

The following are inappropriate uses of ``execute'':

8*. This integration can be easily executed after applying the transformation discussed above.

9*. This proof is executed in the following section.

10*. Analysis of this spin system has been executed by a number of people.

11*. We execute a similar argument in the present paper and reach a similar conclusion.

12*. A transformation into a more convenient coordinate system is executed below.

These sentences are better rewritten as follows:

8. This integration can be easily performed after applying the transformation discussed above.

9. This proof is given in the following section.

10. Analysis of this spin system has been carried out by a number of people.

11. We make a similar argument in the present paper and reach a similar conclusion.

12. A transformation into a more convenient coordinate system is performed below.

I. Abbreviations

This short note addresses a very common but very easily remedied problem, the excessive use of abbreviations. I have found this to be a very common problem in papers written by Japanese authors, and while it may not seem to be terribly serious, in many cases this overuse is so extreme that it presents a real obstacle to readers. My advice is simple: Use abbreviations sparingly.

The limited use of abbreviations for well-known and commonly used terms is obviously desirable for the purpose of conciseness, but their overuse only leads to confusion. While the employment of a large number of abbreviated terms can be timesaving and convenient for the author, it is anything but convenient for readers. In addition, this practice is stylistically very poor.

I believe that in many cases, people are simply unaware of the great number of abbreviations they introduce into their papers and the problems that this often presents to readers. In general, I think it is best for authors to limit themselves to ``standard'' abbreviations (PDE, RG, UV, QED, WKB, etc.) and avoid creating their own. Of course, there is no problem in introducing one or two non-standard abbreviations, but one should be alerted when sentences like ``The solution of WKE found by UIJA loses meaning in the IAH limit.'' begin to appear. There is a second type of problem involving the use of abbreviated expressions that I would like to briefly mention, although this is perhaps somewhat outside the expressed scope of these notes. When proofreading papers, I often come across such abbreviated mathematical expressions as the following:

1. ``f<sub>σ, μ, τ</sub>'' in place of ``f_σ, f_μ and f _τ''

2. ``t<sub>i,k</sub> > t<sub>j,l</sub>'' in place of ``t_i > t_j and t_k > t_l''

3. ``We assume that γ<sub>a,b</sub> becomes negligible as t approaches t<sub>0,1</sub>.'' in place of ``We assume that γ_a and γ_b become negligible as t approaches t₀ and t₁, respectively.''

I strongly recommend that such abbreviated expressions be strictly avoided. They are stylistically very poor and can be quite misleading.

In this note I address a stylistic problem that I often encounter.

Consider the following sentences:

1. We therefore derive the relation

v = ∑ _ia_i σ_i,

    where the a_i's are positive integers.

2. We thus obtain

v = σ₁ γ + σ₂ β ,

    where the σ's are real numbers.

This use of the expressions ``a<sub>i</sub>'s'' and ``σ's'' is stylistically poor. Although it may be too strong to say that such expressions are ``wrong,'' there is always a better way to write sentences of this type. (Sometimes such sentences are written ``...where a_i's are...'' and ``...where σ's are...'' These are simply wrong.)

The first problem here is that in English, use of ``'s'' implies possession, not plurality. Of course in the present case, the reason for the appearance of the apostrophe is simply that if we change ``a_i's'' to ``a<sub>i</sub>s,'' there is the possibility that the ``s'' will be misinterpreted as being a part of the mathematical expression. In any case, this type of usage is syntactically awkward at best. (Perhaps the conclusion regarding this point is simply that we should avoid mixing mathematical and English symbols.)

The second (and more important) reason the above usage should be avoided is the following. Let us suppose we wish to think of the a_i as ``coefficients.'' We could make this explicit by rewriting example 1 as perhaps something like, ``...where the coefficients a_i are positive integers.'' Here it is obvious that ``a<sub>i</sub>'' is a collective (plural) noun. Note that this fact does not depend on the explicit presence of the word ``coefficients'' before ``a<sub>i</sub>,'' as is easily seen if we simply change the position of this word, for example as follows: ``...where the a_i are positive integer coefficients.'' Here again, it is clear that ``a<sub>i</sub>'' represents a collective noun. In fact, it is not necessary for ``coefficients'' to appear at all, as the meaning of the following is clear: ``...where the a<sub>i</sub> are positive integers.'' In this case, it is natural to think of there being an implied ``coefficients'' appearing in front of ``a<sub>i</sub>.'' Of course, the same can be said for the original sentence in example 1 above. But if we make this explicit, we obtain, ``...where the coefficients a_i's are...,'' or perhaps, ``...where the coefficient a_i's...'' However, it is obvious that neither of these makes sense, and thus the problem with this example becomes clear.

For example 1 some of the alternatives are the following:

1. ...where the coefficients (parameters, constants, functions, etc.) a_i are... (This is probably the clearest way of expressing this.)

2. ...where the a_i are... (This more concise. Note that there is no danger of misinterpreting ``the a_i'' as being singular.)

3. ...where the a_i (i = 1$\cdots$N) are...

4. ...where the a_i (i = 1$\cdots$∞) are...

For example 2, the best expression is clearly, ``...where σ<sub>1</sub> and σ<sub>2</sub> are...'' (Note that it is best to avoid something like ``...where σ_1,2 are...'' This is very misleading.)

I. Discussion on problems with pronoun usage

Among the papers I proofread, probably the most common type of problem I encounter involves pronoun usage. There are a number of problems that seem to arise repeatedly in this regard, and I will attempt to address them individually, but all of this discussion can be summarized with one piece of advice: When using a pronoun, make sure it is clear to what this pronoun is referring.

Pronouns are used in place of nouns. They are useful for the purpose of conciseness, but their ambiguous use can create great problems for the reader. In this note, I discuss several points concerning pronoun usage. Further notes will appear in the future.

The topics to be covered in my discussion of pronouns include problems with the general use of pronouns, problems with relative pronouns, and problems with the pronoun one. I briefly consider all of these in this note.

Problems with the general use of pronouns

ambiguous reference

The most familiar pronouns, it, this, that and their plural forms, are, of course, very common and indispensable in both written and spoken English. In general, however, they are overused by Japanese authors. The source of the problem appears to me to be over ambitious and sometimes misdirected attempts at conciseness. I suggest that authors be careful when striving for conciseness. While it is obviously a desirable quality in a composition, conciseness should never be substituted for clarity. Clarity should always be of primary concern. It often seems to me that authors use pronouns simply because they feel they should, perhaps in an attempt to make their writing sound more like native English. This is a mistake. Writers should never force a style on their writing. Rather, the goal of clearly explaining one's ideas should always be the focus. Style follows.

The most common problems involving the use of pronouns is exemplified by the following:

When we transform this function by applying the operator ${\cal{T}}$, we obtain a solution to a certain class of equations that possesses the important property we discuss in the following section. It was studied by James and Chu [1].

This example is actually very typical. I believe one reason that sentences like this appear is that their authors are too familiar with the material they are discussing, and for this reason the ambiguity of their statements is not apparent to them.

There are two problems with these sentences. The most obvious problem involves the use of ``It'' in the second sentence. We discuss this sentence in the present section. The second problem is addressed in the next section.

The possible interpretations of the second sentence in the above example are the following:

(i) James and Chu studied ``this function''.

(ii) James and Chu studied the operator ${\cal{T}}$.

(iii) James and Chu studied this ``solution''.

(iv) James and Chu studied this class of equations.

(v) James and Chu studied this ``important property''.

(vi) James and Chu studied the transformation of the function in question under ${\cal{T}}$.

In fact, a good case could be made for any of these interpretations, although the most natural are (i), (iii) and (v). The resolution of this ambiguity is simple: ``It'' should simply be replaced by ``This operator'', ``This function'', ``This solution'', ``This class of equations'', ``This property'', or ``This transformation'', as the case may be.

I believe that the single example considered here is sufficient to demonstrate the most common problem that occurs with the general use of pronouns. The important point to keep in mind is simply whether there is any ambiguity with regard to the noun to which a pronoun refers.

reference to nothing

There is a somewhat different type of problem that also occurs with the general use of pronouns. I often encounter situations in which pronouns are actually used in reference to nothing. While this problem is less common than that discussed above, it deserves some consideration.

Consider the following sentence:

The ${\cal{O}}$(ε²) calculation is not straightforward. In this case, the standard procedure yields useless results: Although application of this procedure produces a valid description of the system, this description is no simpler than that provided by the original equation. It necessitates a reevaluation of the standard procedure and the construction of a more general method.

In this case, ``It'' appearing at the beginning of the third sentence refers to nothing. There is no word or words appearing in the previous sentences for which ``It'' is acting as a substitute. Of course, one could argue that the meaning of the last sentence is not completely unclear, as we can guess that ``It'' is being used in reference to the situation described in these sentences. However, it is not a good practice to force the reader to guess in this way. In addition, this is an example of very poor style. In general, pronouns should only be used in place of nouns that appear explicitly in the preceding sentence or sentences. The best way to remedy this problem is to replace ``It'' with something like ``The complication presented by the ${\cal{O}}$(ε²) calculation''.

Problems with the use of relative pronouns

example 1

Let us now return to the first example considered in the previous section. The second problem with this example involves the use of the relative pronoun ``that'' in the first sentence. It is not clear whether the important property referred to here is one of the ``solution'' or of the ``class of equations''. In many cases like this, if the reader is sufficiently familiar with the material, she can guess the intended meaning even when the sentence is grammatically ambiguous. However, it is best to avoid situations like this whenever possible. Below appear some possible ways of rewriting these sentences. After each, the words to which the pronouns ``that'' and ``it'' correspond are indicated.

1. When we transform this function by applying the operator ${\cal{T}}$, we obtain a solution to a certain class of equations. This solution/class possesses the important property we discuss in the following section, which was studied by James and Chu [1].
(solution/class, property)

2. When we transform this function by applying the operator ${\cal{T}}$, we obtain a solution to a certain class of equations. This solution/class possesses the important property we discuss in the following section and was studied by James and Chu [1].
(solution/class, solution/class)

3. This function was studied by James and Chu. When we transform it by applying the operator ${\cal{T}}$, we obtain a solution to a certain class of equations. This solution/class possesses the important property we discuss in the following section.
(solution/class, function)

4. When we transform this function by applying the operator ${\cal{T}}$ (studied by James and Chu [1]), we obtain a solution to a certain class of equations. This solution/class possesses the important property we discuss in the following section.
(solution/class, operator)

5. When we transform this function by applying the operator ${\cal{T}}$, we obtain a solution to a certain class of equations studied by James and Chu [1]. This solution/class possesses the important property we discuss in the following section.
(solution/class, class)

6. When we transform this function by applying the operator ${\cal{T}}$, we obtain a solution to a certain class of equations. This transformation was studied by James and Chu [1]. The solution it produces possesses the important property we discuss in the following section.
(solution, transformation)

7. When we transform this function by applying the operator ${\cal{T}}$, we obtain a solution to a certain class of equations. This transformation was studied by James and Chu [1]. This class of equations possesses the important property we discuss in the following section.
(class, transformation)

example 2

Let us now study a slightly different type of problem that often arises in the use of relative pronouns. Consider the following sentence:

We treat the first term in this equation as a perturbation, which can be interpreted in the following manner.

The problem here involves the relative pronoun ``which''. It is not clear whether this refers to ``perturbation'' or to our treatment of the term in question as a perturbation. Of course one may argue that the interpretations of these two would be essentially the same thing, and therefore that there is really no ambiguity here. This may be true, but even in this case, this sentence is an example of sloppy writing that should be avoided. It should be rewritten as one of the following:

1. We treat the first term in this equation as a perturbation. This term can be interpreted in the following manner.

2. We treat the first term in this equation as a perturbation. This treatment can be interpreted in the following manner.

example 3

A similar but more serious problem exists in the following sentence:

In the nonconserved case, the result of the computation is in good agreement with experimental results, but in the conserved case, there is strong disagreement, which supports the conclusions stated in Ref. [2].

In this case, it is not clear whether (i) the fact that there is agreement in the nonconserved case but not in the conserved case, (ii) the fact that there is disagreement in the conserved case, or (iii) the type of disagreement in the conserved case supports these conclusions. Both (i) and (ii) are quite natural interpretations. The third interpretation is somewhat unnatural, because in order for this sentence to possess this meaning, the comma appearing after ``disagreement'' would have to be removed (and, strictly speaking, ``which'' should be changed to ``that''). However, a reader confronted by such a sentence would probably doubt the author's knowledge of such fine points of grammar. In any case, the two most natural interpretations are more clearly expressed by the following:

1. In the nonconserved case, the result of the computation is in good agreement with experimental results, but in the conserved case, there is strong disagreement. These results support the conclusions stated in Ref. [2].

2. In the nonconserved case, the result of the computation is in good agreement with experimental results, but in the conserved case, there is strong disagreement. This disagreement supports the conclusions stated in Ref. [2].

Misuse of the pronoun one

According to my experience, the word ``one'' is overused as a pronoun by Japanese authors. In this section I give some discussion of the proper and improper use of this term.

The following are all proper uses of this pronoun:

1. Consider two Riemannian manifolds, one that is conformally flat and one that is only locally conformally flat.

2. Each strictly positive, monotonic steady state solution corresponds to physically realizable behavior for the θ > 0 system, while each strictly negative one corresponds to physically realizable behavior for the θ < 0 system.

3. It is important to note that the type of local solution in question is one in which slow motion and fast motion are formally distinguished.

4. An energy relation for Ψ₁ together with one for Ψ₂ can be derived quite easily.

In each of these cases, the use of ``one'' in place of that to which it refers leads to a more concise sentence. Also note that there is no ambiguity with regard to this use.

The following represent inappropriate use of ``one'':

5*. We consider an equation that contains a locally acting operator and a globally acting one.

6*. In the simplest case these methods give a consistent result, but in more realistic cases they give an inconsistent one.

7*. It is known that the a posteriori estimate is, in general, much better than the a priori one.

8*. We discuss both the hydrogen atom and the helium one.

9*. In the graphs displayed in Fig. 2, y is the vertical axis and K is the horizontal one.

10*. The functions φ₊ and φ_- correspond to the non-stationary states localized near the minima of the double well potential on the positive side and the negative one of x, respectively.
11*. Noting that ψ₀⁰ and ψ⁰₁ are an even function and an odd one of x, respectively, we can carry out the proof in two parts.

12*. In the analysis of this system, the extra differential one-form χ in Z₂ is introduced in addition to the usual one-form dx^μ in M₄, and therefore our formulation is very similar to that using ordinary differential geometry. Contrastingly, in Martin's original one the Dirac matrices γ_μ and γ₅ are used to describe the generalized gauge field.

In these examples, the main reason that ``one'' should not be used is simply that there is no reason to do so. The purpose of pronouns is to provide brevity and simplicity. However, in all the cases above, ``one'' is used in place of a single word. In these examples, its use does not lead to more concise statements, and for this reason there is nothing to be gained by this use. In situations like this, use of this word sounds forced and quite artificial. In addition, its use often creates writing that is difficult to follow. For example, in 11*, upon encountering the phrase ``Martin's original one'', the reader will likely be forced to return to the previous sentence to determine that ``one'' here is used to mean ``formulation''.

In 6*. and 8*-11*, it is best to simply replace ``one'' with the noun to which it refers. The remaining sentences are best rewritten as follows:

4. We consider an equation that contains both a locally acting and a globally acting operator.

5. In the simplest case these methods give a consistent result, but in more realistic cases this is not true.

7. We discuss both the hydrogen and helium atoms.

I. Presently defined nouns

In this note I give some discussion regarding the use of articles. This is the first in a series of notes on this topic.

For non-native speakers, the proper use of articles is quite difficult to understand. For native speakers, it is difficult to explain. There are certain general rules governing the use of articles, but it is often unclear how these apply in specific cases. For this reason, I think it is best to explain their use through examples.

One of the general guiding principles involved in the use of articles is that the definite article, ``the," should be used when the noun in question is unique or in some sense singled out, and its nature as such is known by the reader. Thus, the sentence ``I ate the apple" is appropriate only if the reader has some preexisting knowledge of this particular apple. A situation that gives non-native speakers a great deal of trouble, however, is that in which the information that defines or singles out a noun is given within the sentence in which this noun first appears. In this note, I consider such sentences.

When defining information about a noun appears within the sentence in which the noun appears (a situation I refer to as ``presently defined"), the definite article should be used. In general, this information consists of some definition, identification or description that uniquely identifies the noun in question. It can appear in several forms that modify this noun (in particular, as an adjective, a noun, a prepositional phrase, or a relative clause).

Below I give a number of examples demonstrating the use of articles in such situations. The first sentence (demonstrating incorrect use) for each example was taken (in modified form) from a paper I have proofread.

1. Noun modified by an adjective

The following sentences provide examples in which uniquely identifying information is given in the form of an adjective.

1^*. Our model is not applicable to k $\sim$ 0 case.

1. Our model is not applicable to the k $\sim$ 0 case.

2^*. This occurs in a following manner.

2. This occurs in the following manner.

3^*. This behavior is described by a following equation:

3. This behavior is described by the following equation:

In the first example, ``k $\sim$ 0" is an adjective defining the case in question. Thus this example would seem to be clear. The second example is somewhat more interesting. Here, one might think that because the information actually defining this ``manner" appears after this sentence, ``the" should not be used. However, this is not the case. In fact, mistakes of this kind are quite common -- probably because of this type of reasoning. The important point here is not whether the information given actually allows us to obtain a full understanding of this ``manner," but whether it uniquely identifies it. In the present case, the adjective ``following" certainly does this. The situation for the third example is similar. This sentence was used to introduce a single equation that appeared directly below it. Thus uniquely identifying information is provided by the ``following," the definite article must be used.

2. Noun modified by a prepositional phrase

Nouns can also be uniquely identified by modifying prepositional phrases.

1^*. We thereby obtain a configuration in Fig. 3.

1. We thereby obtain the configuration in Fig. 3.

1.^' We thereby obtain the configuration depicted in Fig. 3.

Clearly the prepositional phrase identifies the configuration in question. Note that if there were multiple configurations displayed in Fig. 3, grammatically ``a" would be possible. However, in this case the meaning of the sentence as it stands would be quite strange, and something like ``We thereby obtain one of the configurations in Fig. 3" would be more natural.

3. Noun modified by a defining relative clause

There are two types of relative clauses, defining and non-defining. A defining relative clause provides information about the noun that to some extent identifies it. In some cases, this information is uniquely identifying, and in such cases the definite article should be used. The information provided by a non-defining clause is never identifying to any degree.

The following are examples of defining relative clauses that do provide uniquely defining information.

3a. Finite relative clauses

1^*. This is in contradiction with a fact that τ > 0.

1. This is in contradiction with the fact that τ > 0.

2^*. This is known to be true by a fact that γ vanishes beyond the bifurcation point.

2. This is known to be true from the fact that γ vanishes beyond the bifurcation point.

3^*. In a sense that we can choose these values freely, this does not present a problem.

3. In the sense that we can choose these values freely, this does not present a problem.

4^*. In a sense that the action of each of these operators is largely confined to a particular, isolated region, we can think of the system as being separable.

4. In the sense that the action of each of these operators is largely confined to a particular unique and isolated region, we can think of the system as being separable.

5^*. With a value of V = 1.21 that we derived.

5. With the value of V = 1.21 that we derived.

6^*. A solution that we derive in the next section exhibits similar behavior.

6. The solution that we derive in the next section exhibits similar behavior.

7^*. A person who first introduced this idea is not known.

7. The person who first introduced this idea is not known.

In examples 1 - 4, it is quite clear that the information provided uniquely defines the noun in question. The situation is perhaps not as clear for example 5, but here, the situation in which ``a" would be correct is very strange. For example 6, in the case that there are multiple solutions obtained in the section referred to, ``a" would be grammatically correct, but in this case, the sentence would be better written as ``One of the solutions we derive in the next section exhibits similar behavior." In the paper in which this sentence appeared, however, there was a unique solution derived in the section in question. In such a situation, ``the" must be used. Here, despite the fact that the reader has not yet seen this solution, the information provided here uniquely identifies it. In 7, though we do not know the identity of this person, it is clear that there can be just one, and the information provided here uniquely identifies him or her.

3b. Participle relative clauses

1^*. There are two solutions to this equation. A solution moving to the right has velocity v_r, and a solution moving to the left has velocity v_l = v_r / 2.

1. There are two solutions to this equation. The solution moving to the right has velocity v_r, and the solution moving to the left has velocity v_l = v_r / 2.

2^*. A form of this equation resulting from the transformation T is much simpler than the original.

2. The form of this equation resulting from the transformation T is much simpler than the original.

3^*. A function defined by these constraints appears in Fig. 1.

3. The function defined by these constraints appears in Fig. 1.

In example 1, despite the fact that we do not know the nature of these solutions, each is uniquely identified by indicating the direction in which it is moving. The original in example 2 would only be possible in the case that there are multiple forms of the equation produced by the transformation T. Such a situation may be possible, but it would indeed be quite unusual. In the paper in which this sentence appeared, this form was unique. For example 3, the indefinite article is not possible. Grammatically, it could only be used in the case that there are multiple solutions defined by the constraints. However, if this were the case, it would be incorrect to say that such a solution is ``defined."

3c. Infinitive relative clauses

1^*. A solution to be chosen is that which first assumes a negative value at x = x₀.

1. The solution to be chosen is that which first assumes a negative value at x = x₀.

Here it is clear that there is only one solution of the kind in question, and though we do not yet know what this solution is, the information provided here uniquely identifies it.