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WRITING

MATHEMATICAL PAPERS IN ENGLISH

JERZY TRZECIAK Copy Editor

Institute of Mathematics Po l ish Academy of Sciences

Gdansk Teachers' p~ ~':" ~ :-:'

Acknowledg7lLents. The author is gra teful to Prof\:~sor Zo ria DCllkow­ska. Professor ZdzisJaw Skupicll a nd Dalliel Davie:) fur th eir llclpful

cr iti cism . Tlul!1ks are also due to Adam ~rysior and ~Ll!"cill .\daw:,ki for suggesting several il1l I' rm'clllcnt s, and to Hcmyb Walas for her

painstaking job of typesetting the continllously varying manuscript.

Published by Gdallskie Wydawn ictwo O§wiatowe

(Gcl a llsk Teachers' Press )

P.O. Box 59, 80-876 Gdallsk 52, Poland

Cover design by Agnieszka Polak

Typeset by Henryka vValas

P;illt.ccl ill Poland by Zaklady Graficzne w Gdallsku

© Copyright by GdaI1skie Wydawnictwo Oswiatowe, 1993 All rights reserved . No part of this publication may be reproduced in any form \vithout the prior permission of the publisher.

ISBN 83-85G94-02-1

CONTENTS

Part A: Phrases Used in Mathematica l Texts

Ahstract and introciuction ... .. . . . .... . . . . ...... . . . . . .. ... .... . .. ... .. . 4 Definition ........... . ....... .. ..... . ... . . . ...... . .. . .. . . . .. . .. . .. .. . . , G

Notatioll . . . . . . .. ... .. ... . . .... ..... ... . ... ...... .... .. .. ...... . . . ..... 7 P ro perty ... . . . . ................. ..... . . . ..... . . .. . ........ . . . ..... .. . . 8 Assumptio n , cond ition, convention . ... . . . ... ... ..... ... . . ..... . ..... . 10 Thcorem: general remarks .. ... . . ... . .. . . .. .... . ............ .. .. . .. ... 1 ~

Theo rcm: illtrociuctofy plfrase . . -:-: . : .. ~-:-...... -.. ... .. ..... . .. ....... 1:1 Theorem: formulation . .. . . ..... . ... . ..... .. . ..... . . ... . ......... ..... 1:1

Proof: beginning . . .. . . .. .... .. . . .. ...... . ... .. . ... . . ... . . . ..... . . .. . . 1·1

Proof: argu me nts ..... . ... . .. . ....... . .. . . . ... . ... . ... . ... . .. . .. . . . . . 15 Proof: consecutive steps ... . . . . ..... . . ... . .. . ...... ... . ... . . .. . . .. ... . J (j Proo f: "it is sufTic ient to .. ... " .......... ... .. ... . .. ............. . . . .. . . 17 Proof: "it is eas ily seen t ha t ..... " .. . . . . ... ........ .. . . . ..... . .. ... .... 18

Proof: conc lusion and remarks .... .. .. ... ..................... . ... . . . . 18

n.efcrenccs to the literatu're . ... .. ... . . . .. . . . . ... ... . . . . .. . .... .. . .. . . . 19

Acknowledgments ................. .... . . . ... ... . . .. . ... . ... ..... .. ... 20

How to shorten the paper . . ....... . .... ..... . . ..... .. . . . ... . . .. . ... .. 20

Editorial correspondence .... . . . ..... ....... .. . . ................... . .. 21

R eferee's report .. ... ... .. ... .. . . . .. . . . . ..... ..... .. .. ... . . .. .... . ..... 21

Part B: Selected Problems of English Grammar

Indefini te a rt ic le (a, an, -) .. .. . .... . ... . . ... . . . . .. . ....... ......... :':1 Dcfinite artic le (t he) .. . . . .. . . ... . . . .. . . . . . . . .. . .... . .. ......... . .. :' I

Article ornissioll .. . . . .. .. . . . .. . . _ ..... .. . . . .. ..... . .... ........ . ' ..... :>;) Infinitive .... . . . .. .... .. ...... . ... . . . .... . . . . . . . .... . ... .. .. . . ...... . . 27 l ng-form .. .......... . . ... ..... . .. .... . . . ........ . . .. ... . ... . . . ... ... . 29

Passive vo ice ...... .. .. .. ......... . ... .. .... . ..... . .. .... . ....... .. .. . 31

Quantifiers . ..... . . .. . .. . ... .... ... . .. ....... . ... . .. .. . . .... . . .... .. .. 32

Numher , quantity, size .... .. . .. . .. ...... .. ... . . . .. . .... ... . . . .... . ... 34

How to ;cvoid repetition . . . .... ....... .. ... .... . .. .. . .. . . . . ... . . . ... . . 38

\ Vord order .. .... . . ........ ..... .. ... . .. ...... . . ... .. .. . ......... . . . . 40

\Vhere to insert a comma ... .. . . .. ....... .. . . . .. . . . .. ... . . ... .. . . . . . . 44

Some typic;).] e rrors ........ .. .. ....... . .... . .. ... . .. . . . . . ... .. ..... . . 46 Index . .. . .......... . . . ...... .. .... ...... .. . ...... . . .... . ...... . .. ... . 48

r? ",,,:.~,

PART A: PHRASES USED IN MATHEM ATICAL TEXTS

ABSTRACT AND INTRODUCTION

\\"' IHove that in so me fami li es of compacta thcre are no universa l e lelllen ts. It is a lso s hown that ..... SO lll e r levant coun terexamples are indicatcd.

It is o f inte rest to know whethcr ... .. We wis u to invest igate ..... We are interested in finding .. ... Our purpose is to .. . .. It is natural to try to relate .. .. . to ... . .

T his work was intended as an attcmp t to motivate (at moti vat ing) The aim of lIlis paper is to bring together two areas in wh ich .. .. .

I I !i "1 l il ill :1 II III" III i I d ' '' 'I I ill II

INI I/ ,' 11.1 1 111', 1 ' '1i11 / ",, 1111 11 1

review some of the st.andard facts on .....

have compiled some basic facts .. summarize without proofs the

rele vant material OIl . . .. .

g ive a bri ef exposition of ... . . bri efl y sketch ..... se t up nota tion and terminolo;;y. d isc uss (study/trea t /e ' amillc)

til e C; I S (~ .....

il ll.rocl\lcc the notion of .... . II " dl'\,r lop tlt e theo ry of ... . .

wi ll look 1Il0re closely at .. .. . lI'i li 1)(' c() Ilce m ed with .. .. . JlI (lcel'li wit h the s tudy of .... . illlli (":1.lC' Ir w tilC'se techniq ues

III ;\Y b · llsed to ... . ex te lId til e res ults of ..... to ... .. de rive a lI int eres ting formub for

it is s how n that .. .. . some of the recent resu lts arc

rev iewed in a more genera l sett ing. . some applications arc indica ted . our m ain resu lts arc s tated and proved.

con tai ns a brief summary (a discuss ion ) of .... . deals witlL (discusscs) _the casc~~ . .. . ___ _ is intended to m oti\·ate our investigation of .....

Srct ion 4 is dc\·otcd to the study of .... . provides a dctailed exposit ion of .. .. .

I es tablis hes the relation between ... . . presen ts sOllie preliminari es .

\\ ' . ·11 1 touch only a few aspects of the t heory. e WI restrict our atten tion (the discussion/ourseh·es) to .....

It is Hot o llr pll rpose to s tudy ..... No atte mpt h:\.S be 'n made h ere to develop .. .. . It is poss ible tha t ..... but we will not develop this point here. A more comple te theory lllay be obtained by .....

H I this top ic exceeds the scope of this paper. owever. .. .

, we wtll not lise tlus fact III any esse ntIal way.

T l b . ( .) I idea is to apply ... .. Ie aS lc malIl . . d. . geometnc mgre !Cut IS .. .. .

The crucial fact is tbat the norm sa tisfies .... . Our proof involves looking at .....

I b:\.Sed on the concept of .. ...

The proof is s imilar in spirit to ..... adapted from .....

This idea goes back at least a5 far as [7)

We em phas ize that .. ... Il is worth point ing out that . ... . The important point to note here is the forlll of .. .. . The advantage of using .. .. . lies in tlw fact that .... . The est imate we obtain in the course of proof sce IHS to be of indepenrient

inte rest. Our theorem provides a natural and intrinsic cil;u acter iza t.ion of. Our proof m akes no appeal to .. ... Our viewpoint sheds some new ligbt on ..... Our example demonstrates rather strikingly that . .. .. The choice of ... .. seems to be the bes t adapt ed to our theory.

The problem is t hat . ... . The main difficulty in carrying out this construction is th a t .....

' In this case the method of ..... breaks down . This class is not well adapted to .... . Pointw ise converge nce presen ts a more delica te p robleI1l .

Tbe results of this paper were announced witbout proofs in [8). The detai led proofs will appear ill [8) (elsewhere/in a forthcoming

publication) . For the proofs we refer the reader to [G] .

It is to be expected that .. .. . __ One may conj ec ture that . ... .

One m ay ask whetber t his is st ill true if One ques tion still unanswered is whether ... . . The affirmat ive· solution would allow one to .... . It would be d es irable to . ... . but we haw not bee n able 1. 0 do

thi s. These results are far from being conclusive . This question is at present fa r from bcill~olved.

,.( .:;Y ~"~i~/

5

Om method h<ls the disadv<l:ltagc of not I)f'ill;!; i:!trillsir . The solutio!l bll.; short of pr,.'\·idil:::; :1:; l~xph· !t fdrlllilh . \ rhat is still he!.;i :1::; i.-; :ell ex plici 1 ,j, ' ~nirl i,ll! of .

:\ s for pr(,I"I'fju isiks, the rC':lIler is l'Xr<'c'kd 1\1 11t ~ fal~lili:\i' '-':it;; ..... The first t\\"o ch<lpters or" ... .. CCll1stiwtp. ~ll:;:cil"lit pr,.p;'i·<ltio!l . :\0 prelilllill:ery kllo',\"led~,~ of ..... is r<' <i'.lirl'l;.

To facilitate ;tcccss to tile' ilJd i':idlliti t')pic~. the ch;lprcl·., iUI.: rClldel"t!d as 5elf-(1)Ilt:liiled as jl<ls., i\'[(".

For the (tl;:\'e lli ellce of tLe r,'ad,' r we l"l'pe<lt the relc\':l!lt ll1aterial from [7] witl 'O \lt proofs , th\ls lll:tking our cxpo:,iJiull self-colltil.incd .

DEFINITION

.-\ ~e t 5 is dcn.,, > if ..... A set 5 is c;1JJ,~d (said to be) ric71se if .. ... \\'e c;t ll it set dense if ..... We call In tire product mcasU1·C. [Note: The term defined appears last.]

The function 1 is given (defined) by 1 = ..... Let 1 be given (defined) by 1 = .. .. . We define T to be .4B + CD.

requiring 1 to be const;1llt on .. .. .

"[hi;; lIlap is defined by the requirement that 1 be constant on .. ... [Note the infinitive.]

imposing the following condition: .....

The k7!t;lh uf <l sequence is, by definition, the 11l1ll1ber of .. ... The length of T, de110ted by I(T), is defined to be .... . D\' tl~e lcngth of T we mean .....

1 I

I is ... .. , W Jere I, ,t f -

Defi11e (Let/Set) E = L1 we la\e se - ... .. , f being the sol u lion of ... .. with f satisfying .....

\\'(' wi il 'ol,ider I the behaviour of the family 9 defined as follows . . ..,.~ the height of!J (to be defined later) and .....

To 1l1(~ ;1~llr " the growth of!J we make the following definition.

I we shall call I

In Ihis \\';\:; we ol)tain what :vill be referred to as the P-s?Jstcm. 15 known as

S· I r l:t~ norm of f is well defined. Illce ..... , the definitiOll of the llorm is unambiguous (makes sense).

G

It is immateri,,-I which J\J we choose to denne F as 101lg as 1\1 contains x. This product is indepelldent of which Ill ell lber of !J we choose to define it. It is Proposit ion 8 t h<lt makes this denllit.ion ,,- lIowable.

Our definition agrees I w~th the olle givell in [7] if II is ... .. wllh the cla.sslcal one for .. ...

I this coincides with our previously introduced

Note that terminology if J{ is convex. this is in agreement with [7] for .....

NOTATION

_ We \,'ill denote by Z I Let us denote by Z - tJleset ..... Write (Let/Set) 1 = ..... Let Z denote [Not: "Denote 1 = ..... "]

Here

The closure of A will be denoted by clA. We wi ll use the symbol (Jetter) k to dellote ..... We write H for t.he value of .. ... We will write the negation of]J as -'p. The notation aRb means that ..... Such cycles are called hOlllologous (written e ~ e').

Here and subsefjuelJ tly, Throughout the proof, In the sefjuel,

, I denotes I I h stands for t Ie map .....

From now on,

'vVe [ollow the not;ttioll of [8] (used in [8]). Our notation differs (is slightly different) from th at of [8]. Let uS introduce the temporary notation F J for 919 .

Wit h tIl(! 11ot;1tio11 J = ..... , I With this notation, we have ..... In the notation of [8, Ch. 7]

If f is real, it is cus!ol1l:try to write ..... rather thil.1l .....

For simplicity of llot:ttion, To (simplify/shorten) notation, I3y ,,-buse of no t.ation, For abbreviation,

We abbreviate Fallb to b'.

we

write f instead of ..... use the same letter f for .... . continue to write f for .. .. . let 1 stand for .....

We denote it briefly by F . [Not: "shortly"] We write it F [or short (for brevity). The Radon- Nikodym property (RNP for short) implies that ..... \Ve will write it simply x when llO confusion can arise.

/':;r t-..("

7

II \~ ill (,, \II Se no confusion if we use the same lett er to designate " "" '"Ille r o f A and its rest.riction to [\' ,

\ \ '1' ', 11.111 WI i I I' the above exp ression as I II", "I I/IVII I'x press io ll m<l.)i be written as t = ..... \\ '1' 1 ,1 11 IV I iii' (,1) ill tbe form

' I II" (:II 'I'k illl li l'l'S !<tilel components of sections of E ,

1' lllIt il'IIIIIIIOloIl,Y'

'1'1 ", 1' ,X PII ':,:, io ll ill it a li cs (ill italic type), in large type, in bo ld print; III 11 ,'" ' 111.11" '01':1 ( ) ( rOlillcl brackets) , . iii 1/1.\111 ' 1:1 \ I ( hCjU I\ r e brackets), 111111 ,11 '1':, ( ( r lilly I>I 'al' kets ), in angular brackets (); WII hili L111' IllIllIl ~i; ', II :;

( ', qlil.iI 11'1.11'1 '1 11I1pI ' I r ; 'iI' Idters i= smail let. tcrs = lower G\SI! It: t.ters ; (;oLlli r (;1'"11;111) lei LI' I ~ ; sc ripl (ca ll i~ra rhic) le t.ters F; s l' cri :ti I{ Ulil l lIl l<'t l le l S If'!

Dot " prime " as t. erisk = s t. ;ll' ., tilde - , bar - [over a symbol], ha t - , ve rti cal s troke (vertica l bar) I, s l;Ls h (diagona l s troke/slalll)/, da.5h - , sharp #

Dotted line . , , , .. , das hed line _____ , wavy line __ ~

8

PROPERTY

such that (w it h the proper ty that) [Not : "such an element that"]

with the following properties: ..... sat isfy ing LJ = .... , wit h N J = 1 (with coordinates J:, 1j , ;::)

of norm 1 (of the form .... ,) whose norm is ..... all of whose subsets are ..... by means of which 9 call Iw COIIlPIlt.I~d for whi ch this is true

tbe (an) element at wh ich 9 bas a local ma.ximum descr iber! by th e equations .. ... g;ivclI by LJ = ... .. depe nding olIly on ..... (independent of .. ... ) 1I0 t in A so small that (sm a ll eno ugh that) ..... as above (as in the prev io us theorelll) occurring in the cone condition

[Note the do uble "r" .] guaranteed by the assumption .... , so obtained

we have jus t defined we wish to study (we used in Chapter 7)

tbe (an ) e lemen l to b e defined later [= which wi ll be defined] in questio n under study (consideration)

..... , tbe constant C being iudependent of .... . [= where C is .. .. ,]

..... , the supremum being taken over a ll cubes .....

... .. , the limit being taken in L,

is so chosen tha t .. .. , is to be chosen later. is a su itable constant.

.... " where C is a conveniently chosen element of .....

The operators Ai

invoh'es tbe de rivatives of , .. .. ranges over all subsets of .... , may be made arbitrari ly small by .... ,

have (share) many of the properties of ..... have still better smoothness prop€:-ties. lack (fail to have) the smoothness properties of ..... still have norm 1.

Inot m erely symmetric but actually se lf-adjoint. not necessarily monotone. both symmetric and positive-definite. not cont inuons, nor do they satisfy (2).

[Note the inverse word order after "nor" ,] are neither symmetric nor positive-d efinite.

only nonnegative rather than strictly positive, a.s one may have expec ted,

allY self-adjoint operators, possibly even nnbounded .

still (no longer) self-adjoint.. not too far from being se lf-adjoint .

preceding theorem indicated se t [B ut adjectival clauses with

prepositions come after a nou n, The above-mentioned group resulting region e ,g. "llle group defin ed in Sect ion 1" .] req uired (des ired) element

13otli--X-allCi Yarc finite :- - - -- - --­

Neit her X nor Y is fi n ite. X and Yare countable, but neither is finite, Neither of them is finite. [No te: "Neither" refers to two a lternat ives,] None of the fUllctions Fi is finite,

X is not finit e ; nor (nei ther) is Y.

9

x is not finite, nor is Y countable. [Note the inversioll .]

X is empty I ; ,() i~ . ~" : , iJut 1 IS !lot.

X belongs to )' I : ~() cl(~l~S. Z . I . uut Z aoes not.

ASSUMPTlmJ, CONDITION, CONVENTION

\Ve will make (Ileed) the follo\\'ing assumptions: ..... From IlOW on we make the assum p tion : .. ... The following assumptioll will be needed throughout the paper.

_ Our baSiC assllm[Jtioll i.'i..S1L~Jul lowillg-,-Unless otherwise stZltecl iUntil further ;;-otice) we ;L'is\l~le that ... -.. --~ -~~ In the [,(;Illainder of tbis section we assume (require) g to be .... . In order to get asymptotic results , it is necess;try to put some restrictiolls

on f. \Ve sh;tll m;tke two st;tDding assumptions on the maps under consider;ttioll.

It is required (;tssumed) th;tt .. .. . The requirement on g is that .... .

I is subject to the condition Lg = O.

.. ... , where g satisfies the cond ition Lg = O. is merely required to be positive.

the requirement that g be positive. [Note the infinitive.]

requiring g to be .. ... Let us orient M by

imposing the condition : .. ...

for (provided/whenever/only in case) P i= 1. unless p = l.

(4) bolds the condition (hypothesis) that .... . under the more general ;:;ss:1nlptioll that .... .

some further restnctlOns on .. ... additional (weaker) assumptions.

satisfies (fails to satisfy) the assumptions of ..... ha.'i the desired (asserted) properties. ()i·o·:ides the desired diffeomorphism .

F still satisfies (need not satisfy) the requirement that ..... meets this condition. do~s not necess;trily have this property. satIsfies all the other conditions for membersh ip of X.

There is 110 loss of generality in assuming ..... vVithout loss (restriction) of generality we can assume .... .

10

This involves no loss of generality.

, siuce otherwise .....

\Ve can certainly assume thilt .. ... , for .. ... [= because] , for if not, we replace .. ... \ Indeed, .... . .

i\'eit her the hypothesis nor the conclusion is affected if we replace .. ...

By cllOosing b = a we may actually assume tuat .... .

If f = 1, which we may assume, tben .. ... For simplicity (convenience) we ignore the ciepeudence of F on g.

[Eg. ill IlotZltion] It is convenient to choose .....

~ - - \ \ 'e C;tLl assunre; by decreasing k if necessary, that .....

F meets 5 transversally, say ilt F(O) . There exists a minimal element , S;t)' 71, of F. G acts on H as a multiple (say n) of V .

For definiteness (To be spec ific ) , cOll sid er .....

This coLld ition

is not particularly rest r ictive. is surprisingly mild . admits (rules out/excludes) elements of .. .. . is essential to the proof. cannot be weZlkened (relaxed/improved/omittedj

d ropped).

Tbe tbeorem is true if "open" is deleted from tbe hypotheses. The assumption .... . is superfluous (redunclant/unnecessar ily restrictive).

\Vc will no\v show how to dispense with the assumption on ..... Our lemma does not involve allY assumptions about CUrva llir ' .

We bilve been working under the assumption that .. ... Now suppose that this is no longer so. To study the general case , take .. .. .

For the general case, set .... .

The map f w ill be viewed (reg;trded/thought of) as I a fUlll clO r .... . rca lZll1g .... .

I

think of L as being constant. From now on we reg~rd f as a map from .. ...

tZlC\t/y assume that .....

It is understood that r i= l. We adopt (adhere to) the convention that 0/0=0.

C;;r f' 11

THEOREM: GENERAL REMARKS

. an extension (" birly straightforward gf'neralizatioll/a sharpened \'Crsion/ a rcfillement) of .....

IS a reforlllulation (restatement) of ..... in terms of .... .

all aualogue of .... . analogous to .... .

' I 'hi:: tll l'on' lIl a partial converse of .... . a ll answcr to a question raised by .....

dea ls with ..... (,IISllr s th e existence of ..... ('X PI' sscs th e equivalence of ..... p l'ov' icl cs a criterion for .. .. y i(: ld: ill[ I'malion about .. ". Illakcs it leg it imate to apply .....

Th e tit 'o rCIII s ta tes ( ;L"iS ' r l ,/s hows ) th a t ..... l to ug ldy (Loosely ) spe:tkillg, th e fUl'lIllila says t hat .....

\\,1 f . (3 7)' ttl to saying that ... .. . ' len IS opell, . .IllS ,\lllOUIl S to th e fact that ... ..

Here is another way of stating (c): .... . Ano th er way of stating (c) is to say: .... . II 1\ ('q uiva lent formu\atiou of (c) is: .... . T I\( :o rc IllS 2 and 3 may be sUlllmarized by saying that ..... }\ ssc r l.io n (ii) is nothing but the statement that .... . (;('() lli elrically speaking, the hypothesis is that ..... : part. of t.he conriusion

is tha t .....

T he inte res t 1 I' . T I .. I " fi f I I . 111 the assertIOn .....

Ie pnUClpa Slgru canee 0 t. Ie emma IS tl t't II t 'Ih . t la I a ows olle 0 e POIll

The theorelll gaius in interest if we realize that .....

TI tl I

is s till true I'rl we drop the assu;nptioll ... Ie leo rem '1111 1 I .. I It I t s t l 10 (S It IS onlY assumec la .... .

I [ wc t;l!<C f = ..... I I th e standard lemma ..... . we recover _ c 1(l:p lar lll ;; f by - J, [I, Theorem Cl].

'l'l li :; s pec ializes to the result of [7] if f = g.

I

be Ileed ed in 1 ' , 'Ii i:: I (, ~; lI l l will prove extremely useflll in Section 8.

not be needed until

12

TH EOREM: INTR ODUCTORY PHRASE

\ Ve have thus p rove d ... .. Summarizing , we have ... ..

1

rephrase Theorem 8 as follows. \Ve can llOW state the analogue of .....

formulate our maill results.

We are thus led to the following strengtheuillg of Theorem G: .... . The remainder of this section will be (b'oted to the proof of .... .

The coutinuity of A is established by our next theorem. The following result Illay be proved in much the same way as Theo rem G. Here arc some elementary properties of these concepts. Let us mention two important consequences of the theorem. \Ve begin with a general result on such operators.

[Note : Sentences of tIle type "We now have the following lemnw.", carrying no inforlllation , can ill gelleral be cancelled.]

THEOREM: FORMULATION

If ..... and if .. ... , then .....

1 Suppose that .. "' 1

Let M be ..... Assume that ..... Then ..... , Write .. ...

Furthermore (Moreover), . .... In fact, ..... [= To be more precise] Accordingly, ..... [= Thus]

Gi\'en allY f i= 1 suppose that .. ... Theu .... .

1

t he hypotheses of .... . Let P satisfy the above assumptious.

N(P) = 1.

Let assumptions 1-5 hold. Then .... . Under the above assumptions, .... . Under the sallle hypotheses, .... . Under the conditions stated above, .....

provided III i= l. unless m = l. with g a constant

satisfying .... .

1 Then ... ..

Under the assumptions of Theorem 2 with "convergellt" . replaced by "weakly conv ergent" :~~ .. . - ---

Under the hypotheses of Theorem 5, if moreover .....

Equality holds iu (8) if and only if ..... The following conditions are equivalent: .....

[Note: Expressions like "the following inequality holds" can in general be dropped.]

rpi .'. ~,~ ,

13

PROOF: BEG INNING

We I fi t rs · Let us

pro\'c (sh()\,,/rrcall/obscr\'e) tklt .... . pro\'(~ " rC l !ll(,l~ci form of the thcore:lI. outli!lC C;i \'t: t!:r : mai n ideas of) U!\, p roof. examill e BI .

II To ~cc (pro':e) this. Id f = .....

Out. A = n. \:I\!l:':)\'e th is i\s,f()!:~\:',~~ _ 1,1,., b proved b~ ,\. )l ldo g - ... ..

I To t his end. consider .. ...

\V f t t If [= For tbis Dllrposc; not: \! Irs COlllpl1 e '1 To do this. tah; .....

"To this aill l"]

- fl)r this Jllirpose,we se t .... .

To deduce (3) froD! (2) , LIke .... . \ \ 'e claim that .. ... Indeed , .. . .. We begin by proving ..... (by recalling the notion of .. ... )

Our proof starts with the observation that . .. .. 1')1e procedure is to find .. ... The proof consists in the construction of .....

straightforward (quite involved). The proof is by induction on n.

left to the reader. based on the followillg observatioll.

The main (bas ic) idea of the proof is to take .....

The [lroof I falls l1at~r ~lly i:ltO three parts . wdl be clJvlded mto 3 steps.

We have diviued the proof into a sequence of lemmas.

S I the assertion of the lemma is false.

uppr)se l' h t , contrary to our c aJm, t a .....

Conversely (To obtain a contradiction), I tl t () I

suppose la .... . !l t Je contrary,

Sllppose the IClllIlIa were false . Then we could fiIlt! .. ...

If t · n we wou ave .....

I t here existed an x ..... , I ld h x were no Ill, tI Id b it were true that ..... , .lere wou e .....

Assume the formula holds for the degree k; we will prove it for k + 1. Assullling (5) to hold for k, we will prove it fo r k + l. We only give tbe main ideas of the proof. We give the proof only for the case n = 3; th e other cases are left to the

reader.

14

PROOF: ARGUMENTS

I definition, ... .. the defini tion of .... . assumptioll, ..... t he compactness of .... .

By Taylor's formula, ... ..

But I =g , which follows from ... .. as was described

(shown / ment ioned/ noted) in ... ..

a similar ::l.rgument, ... .. the above, .. ... the lemma below, ..... conti nuity, .... .

--- Since .hs compact~

Theorem 4 now

shows that .. ... yields (gi ves /

implies) I = ..... leads to 1 = ... ..

L1 = o. [Not: "Since ..... , then .... .-'] w~ ha.'ieLI_= O. it follows"that L1 = 0 we see (co llclude) that LI = o.

But L1 = 0 since I is compact. 'I'Ve bave If = 0, beca use ..... [+ a longer explanatio n] \Ve must have L1 = 0, fo r o therw ise we can rep lace ..... As 1 is compact we Iw.ve L1 = O. Therefore LI = 0 by Theorem 6. That L1 = 0 follows from Theorem G.

From

(5) this what has already

been proved,

we conclude (deduce/see ) th at .. ... we have (obtain) AI = N.

[Note: without "that"] it follows that ..... it may be concluded that .....

According to (On account of) the above remark, we have AI = N .

It-follows that I M = N Hence (TllUs/Consequently,/Th erefore) .

[hence = from th is ; thus = in this way; therefore = for t h i.- I (';(!,{) I I ;

it follows that = from the above it fo llows th a t]

and so AI = N. This gives M = N . We thus get AI=N. The result is M =N.

and conseq uently M = N . and, ill consequence , Ai = N.

(3) now becomes M=N. This clearly forces M = N.

F is compact, and hence bounded. which gives (im plies/

yields) 111 = N. [Not: "what gives"]

F = G = H I the last equality being a consequence of Theorem 7. , which is due to the fact that ..... ~

Si nce .. ... , (2) shows that ... .. , by (4). We conclude from (5) that ... .. , hence that .. ... , and finally that .. ...

E;;r t;:, ,:~~ -

15

Th!! 'quality f = g. which is part of the conclusioll of Tlteorem 7, implies thal ..... .

As in the proof of Theorem 8, equation (-I) gives ..... Analysis similar to that in the proof of Theorem 5 sI10\\'5

that ..... [Not: "similar as in"] A passage to the limit similar to the abo\'e illlplil's that ..... Similarly (Likewise), .....

Similar arguments apply It tl } 'I . I' 0 Ie case ..... Ie same reasoulllg app les

Tlte same couclusion can be drawn for .... . This foll olV by the s;].me method as iu .... . T ill! telill T f can be haudled in much the same wav, the only difIereIlce

Iwill t'. ill lli e ;]'Dalys is of ..... -II I llll! saJllI! manner we can see that ..... T lt l' rc:; t of lhe proof runs as before. W nolV apply this argument again, with I replaced hy J, to obtain .....

PROOF: CONSECUTIVE STEPS

Consider ..... Define I Choose.... . Let f = .... . Fi x ..... Sel .

evaluate .... . compute .... .

Let. us apply the forl1lllia to ..... suppose for the III0nWIIt. regard s ;L~ fixed and .....

iN(}l e: The imperative mood is used wheu you 0l'del' the reader to do sO l1l ething, so you should not write e.g. "Give an exalllple of .... . " if yo u mean "~Ve give an example of ..... "]

Adding 9 to the left-hand side Subtracting (3) from (5) Writing (Taking) h = H f Substituting (4) into (6) Combining (3) with (6) Combining these

[E.g. these inequnli ties] Replacin" (2) by (3) Letting n co Applying (5) 11I 1,!' I,(," ;1111~ i l l g J ,wd 9

yields (gives) It = ..... we obtain (get/have) J =.rJ

[Note: without "that"] we conclude (deduce/sec) that .. ... we can assert that .... . we can rewrite (5) as ... ..

I N,, /,· '1'111' ill :~ fOrlll is eit.her the subject of a sentence ("Adding ..... gi­V(':I" ), (II' Il'ljll ires the subject "we" (HAdding ..... we obtain"): so do /11)/ IV I ill; e.!;;. "Adding ..... the proof is complete."]

1,\ ',. rO lll.illlle in this fashion obtaining (to obtain) J= ..... W,· 1I1 ;\Y now integrate J.: times to conclude that .... .

J(j

Repealed app li c: tI.ioll of Lemma 6 enables us to write ..... \Ve now J r r' 'd I Y illduction. \Ve can 1I0W Pl(l 'ced annlogonsly to the proof of .... .

\Vc nex t I claim (show/prove that) .. ... sharpen these results and prove that .....

claim is that .....

Our next goal is to determine the number of ..... objective is to evaluate the integral I. concern will be the behaviour of .....

\Ve now turn to the case f =1= l. \Ve are now in a position to show..... [= We are able to] \Vc proceed to show that .... . The task is llOW to find .... . Ha\'ing disposed of this preliminary step, we can now return to .....

\Ve wish to arrallge that J be as smooth as possible. [Note the illfinitive.]

We are thus looking for the family ..... We have to construct .....

In order to get this inequality, it I :vill be ll~cessary to .... . IS convelllent to .... .

To deal with I J, I To estimate the other terIll. we note that ..... For the general C<L5e,

PROOF: "IT IS SUFFICIENT TO

I ffi I I show (prove) that .....

It ~u 1 cflies. t to make the following observation. IS SUI! clen ( . . use 4) together WIth the observatIOn that .....

We need only consider 3 cases: ..... We only need to show that .....

It remains to prove that .... . (to exclude the case when ..... ) What is left is to show that .....

I ~ ~ We are reduceclto proving (4) for ..... ---

. We are left with the task of determilling .... . The only poiut remaining concerns the behaviour of ..... The proof is completed by showing that ..... We shnll have established the !emnw. if we prove the following: If we prove that ..... , the assertion follows. The statement O(g) = 1 will be proved once we prove the lemma below.

/:---x ~' -~::-~ 17

PROOF: "IT IS EASILY SEEN THAT ..... "

clear (e\'i<ieli! /ir;1llIu::lte/oi;vio'lS) that .... . It IS casiI-.· ~CC!l t!;.l~ .... .

easy"to cLx!': t ilat .... . a sinlp le !!1~:.:ter to .... .

\Ve SCI: (check) at O!lC" ::, : ,; . " .. ..... , wi:!r! 1 is cl,~ar from (3). . .... , itS i~ ot<J.~y to check. F is ca.;ilv seen (Cilcckc<, to be sm oo th.

It folluw,; cas i'" (inllll<~dl:l'c1y) t i!:lt , .... Uf cour~l' (CI l'a riy/Olwiol\siy) ...... The rroo[ is "traigLr:,,[\\'ard (stallJardi i!1Jl1l,~diate) .

All easy camp':' ~!: in" {A trivi:11 '{erii:C!l.tion) sho',\'s that .. ... - !:':2) ,Hakes -it ob '.<oUS 1hQt. ~ r= By (21 it is obviolls that]

Tile fac:.tO f Gj I'oses ilU prQblel:1 lkC,-lii.-;r. Gis .. ...

PROOF: CONCLUSION AND REMARKS

..... , wlliciJ

proves the theorem. completes the proof. I:stablishes the form ub . is the desired conclusion. : .Vot: .. '.\' h,l.T .. j is our claim (assertion). [Not : "thesis"] gives (-1) when substituted in (5) (combined with (5)).

the proof is complete. this is precisely the assertion of the lemma .

..... , and the lemma follows. (3) is proved. . f = 9 as claimed (required).

ThIS CO l!t[,',J ids our assumption (the fact that ..... ) . .. ' .. . conrra ry to (3).

, • 1 •. 'bl [N t ., It'''] . ..... '.VU1 C,1 15 IlllpOSSJ e. a : 'w la IS

...... '.vLie!! con t r:\dicts the maximality of .....

.... .. :l cOill.r'ldiction.

The proof for G is similar. C may be handled in much the same way. Silllilar considerations apply to C.

TI f I works (remains valid) for .... . Ie Silme proo still goes (fails) when we drop theassumptioll .....

The l1letbod of rroof carries over to domaills .... .

The rroof a/)o\'e gives more, namely f is ..... A s li ght change ill the proof actually shows that .....

18

Note tbat we have actually p roved that .... . [= We have proved more, namely that ... '

W I ll oll ly the fact that ... .. e 1;"\"(, user the existence of only the ri ght-hand derivati \·c .

for f = 11 it is no longer true that .. ... the argument breaks down .

The proof strongly depended on the assumption that

00tc that we did . not really ha\'e to usc .. .. . ; we could bave applied .....

For more details we refer the reader to [7]. The details are left to the reader. We !cal'e it to the reader to n:rify that... . [Nole: "it" necessary] This fillishes the proof, the d'.! wiled verification of (4) being left to tbe

reader.

REFERENCES TO THE LITERATURE

(see [or instance [7 , Th. 1]) (sec [7] and the references giVl'll tl ll 'I" )

more details) (sec [Kal] for the definition of ..... )

the complete bi bliograpby)

TIH~ bes t general reference here I .. wa~ proved hy L lX [t>]. TIlP. sUllldard work Oll .. ... IS .. ...

The cl:Lssical work Iwre This can be found ill

Lax [7 , eh. 2] .

is due to Strang [8].

goes back I to the work of .... . as far as [8].

was motivated by [7]. This construction generalizes that of [7] .

follows [7]. is adapted from [7] (appears ill [71) ' has previously been used by Lax 7] .

a rece nt account of the theory a treatment of a more genera l case a fu ll er (thorough) treatmell t

For a dee rer discussion of .... . we refer the reader to [7]. direct constructions along more

classical lines yet another method

We introduce the notion of .. ... , following Kato [7]. We follow [Ka] in assuming tilat .....

0 7 ~.: .. 19

' I'hl! lll aill results of this paper were announced in [7]. Silll il:tr results havebecn obtained independently by LiLX and arc to be

pub lished in [7] .

ACKNOWLEDGMENTS

T he author I :vishes to ~xpress his thauks (gratitude) to ..... IS greatly Illdebted to .. ...

for

his <1.ctive interest iu the publication of tllis paper. suggestiug the problem and for many stimulating conversations. sever<1.l helpful comments concerning ... .. drawing the author's attention to .. ... poi nting ou t it mist<1.ke in ..... hi s coll abor:tt ion in proving Lemma 4.

' I'lt l' :t IlL hor g ra tefull y ac knowledges the many helpful suggestions of .. .. . li m illg Lit pr para tion of tile paper.

This is p<1.r t of the a uthor's Ph .D. thesis, written uuder the supervision of .... . a t the University of .....

The author wishes to tlt ank the University of .. ... , where the paper WiLS

written, for financial support (for the invitation and. hospitality).

HOW TO SHORTEN THE PAPER

, lI era l rul es:

J . n.emember: you are writing for an expert. Cross out a ll that is trivial or routine. ~. Avoid repeti ti on: do not repeat the assumptions of a theorem at the beginning

of its proof, or a complicated conclusion at the end of the proof. Do not repeat the ~s uillptions of a previous theorem in the statement of;\ next one (instead,

writ e e. g. "Und er t.he hy potheses of Theorem 1 with f replaced hy g, ..... ") . Do not re peat the same formula-use a label instead .

3. C heck all formulas: is e<lch of them necess<1rY?

Phrases you can cross out:

We denote by IR the set of all re:tl numbers. lYe have the following lemma. T he following lemma will be useful. .. .. . the follow illg inequality is s<1.tisfied:

P h r <l~ ~ yo u ca n shorten (sec <1. lso p. 38):

20

I.e'/. Ill' <1.11 a rb iLrary but fi xed positive number""" Fix c; > 0 I,e'/. II:; fi. Htb ilra ril y x E X """ Fix x E X 1,I'i. 11:; firs t obs(·rvc· th :tt """ First observe th<1.t IV" wi ll lirs t compute """ \Ye fi rs t compute

II I' II{"(; we 11(1Xe :r = 1 """' Hence x = 1 II (, ll cc it [o llolVs tha t x=l""" Hence x=l

Tak illJ', in l.o I\ITO I III~ (tl) --> By (4) Oyv illIlP of( ,I) ,13y (4) Oy 11!lalio ll (d) , lIy (4)

III tit illll ' l vii i [0 , II ~. In [0,1] There (' X i S I. ~l :t f ili i ·tion f E C(X) --> There exists f E C(X) For every po ill t)J !II -v-+ For every p E AI F is defill (:d I>y tit , for mula F( x) = ..... -v-+ F is defined by F( x) = ..... Tbeorem 2 a ll cl Theorem 5 "" Theorems 2 <1.ud 5 This follows frolI l (1), (5), (6) and (7) --> This follows from (4)- (7) For deta ils!i ' [:31. [4J aud [5] ..... For details see [3]-[5] The derivaLi v wi Lit respect to l ..... The l-deriv<1.tive A fun ct ion [ cI <1.SS C 2

"" A C 2 function

For a rbitra ry 1; ~. For all x (For every x) In the case n = 5 ..... For n = 5 This leads to a contradiction with the maximality of f

........ .. , contrary to the maximality of f Applying Lemma 1 we conclude th<1.t "" Lemm:t 1 shows tll<1.t ... .. , which completes the proof --> .....•

EDITORIAL CORRESPONDENCE

I would like to submit I the enclosed manuscript " ..... " I am submitting for publication in Studia Mathematica.

I have also included a reprint of my article .. ... for the convenience of the referee.

I wish to withdraw my paper ..... <1.S I intend to make a m<1.jor - revision of it.

I regret any inconvenience this may have caused you.

I am very pleased that the paper will appear in Fundamenta. Thank you very much for accepting my paper for publication in .... .

Please find enclosed two copies of the revised version. As the referee suggested, I inserted a reference to the theorem

of .... . We have followed the referee's suggestions. I have complied with almost all suggestiolls of the referee.

REFEREE'S REPORT

The author proves the interesting result that ..... The proof is short and simple, and the article well written. The res ults presented are original.

21

The paprer is a good piece of work Oll a subject that attracts considerable attention.

I ;lm plca..c,cd to I j' r bl' " T. I'. ~ pI .. , ... ,rt'commcllC It. lor pc lcatlOll III 1l "" e'L'loLl:'OI S .. ~1 I' I strongly tuella. at..:ematlca.

The only remark I wish to make is that condition B should be formuhted more carefully.

:\ fe\\' minor typographi cal errors arc listed below. I !J,lxe indicated \'arious corrections 011 the manuscript.

The results obtained arc not particularly surprising and will be of limited interest.

'Tile re<-l'lts- ' I correct but ouly moder<1.tely intrrcstilig. __ _ , ., ~ I "re I, ,. c· f k -

i r<'.: :-, cr cas:; mOlll 11cations 0 :;own w.cts .

The cx:ullple is worthwlli:e but not of sufficient interest for a research article.

The Eng lish of the paper needs a tborough revision. T he paper does !lot meet the standards of your journal.

T 1• '). f I I as stated . ueorClll - IS it se. h' j' III t IS genera Ity.

Lel1lma 2 is known (see .. ... ) Accordingly, I recommend that the paper be rejected.

22

PART B: SELECTED PROBLEMS OF ENGLIS H GRAMMAR

INDEFINITE ARTICLE (a, an, -)

Note: You use "an or "an" depending on pronunciation and not spelling, e.g. a uuit, an x.

1. Instead of the number "one":

The four centres lie in a plane. A chapter will be devoted to the study of expanding map~ .

For this, we introduce an aux il iary variable z .

-~ 2:-Me3ning "member of- a class of objects" ,- "some" ,-"one of":

Then D becomes a locally convex space witll dual space D'. Tbe right-hand side of (4) is then a bounded functi on. This is easily seen to be an equivale nce relation. Theorem 7 bas been extended to a class of bound ary val\l e jll'Oh ll' III '; The transitivity is a consequence of the fac t that .... . Let us now state a corollary of Lebesgue's theo rem for ..... After a change of variable in the integral we get .... . \Ve thus obtain the estimate ..... with a constant C.

in the plural:

The existence of partitions of unity may be proved by .. .. . Tbe definition of distributions implies that .. ... ..... , with suitable constants. . ..... , where G and F are differential operators.

3. In definitions of classes of objects (i.e. when there are many objects with the given property) :

A fundamental solution is a fu nction satisfying ... .. We call C a module of ellipticity. A classical example of a constant C such tha t .. ... We wish to find a solution of (6) which is of the form .. ...

in the plural:

Tbe elements of D are often called test functions.

tl t f I points with distance 1 from K

1e se 0 II . . . I a functIOns WIt 1 compact support

The integral may be approximated by sums of the form .... . Taking in (4) ·functions v whicb vanish in U we obtain .... . Let f and g be functions such that .. ...

23

4. In the plural-when you are referring to each element of a class:

Direct sums exist in the category of abelian groups . I II particular, closed sets are Dorel sets. Borel measurable functions are oftell called Borel mappings. This makes it possible to apply fh-results to fUllctions in any Hp.

If you are referring to all elements of a class, you use "the":

The real measures form a Sll bclass of the complex ones .

5. In front of an adiedive which is intended to mean "having t his particular quality":

Tllis map extencls to all of M in an obviolls hshion. A remarkable feature of the solution should be stressed .

S ' ct' 0 1 I gives a condensed exposition of ..... C 1 n describes ill u unified manner the recent results .....

A s imple computation gives ..... Combining (2) and (3) we obtain, with a llew constant C, ..... A more general theory must be sought to account for these

irregularities. The equatioll (3) has a unique solution g for every f.

But: (3) has the unique solutiou 9 = ABf.

DEFINITE ARTICLE (the)

MC',liling "mentioned earlier", "that":

I.et A e X. If aB = 0 for every B intersecting the set A, thell ... .. I ) <'Ii II ' ex p x = L.Xi Ii!. The series can easily be shown to converge.

2. III rront of a noun (possibly preceded by an acijedive) referring to a single, uniquely determined object (e.g. in definitions):

Le t f be the linear form I ~efi'lli~' [/(2). [If there is only one.]

u = 1 in the compact set J{ of a ll. points at distance 1 from L. We denote by B(X) the Banach space of all linear operators in X. .. ... , under the usual boundary conditions. .. .. . , with the nat ural defini tions of addition and multiplication. Using the standard inner product we may identify ~ .. ~ .. ~. ~ _

In th e co nst ru d ion: th e + property (or another charaderistic) + of + obj ect:

'1'1' 0 co ntinuity of f follows from ..... '1'1", (' x i ~ t c nc e of tes t functions is not evident. '1'11( ' 1'<: is it fi xed compact set containing thE; supports of all the fj. '1'1 1(': 11 X is the c entre of an open ball U. Th e intersection of a decreasing family of such sets is convex.

, DILl : Evel Y (\()Il Clllpty open set in IRk is a union of disjoint boxes.

[I ( YO II wish to stress that it is some union .of not too well ~ ; p( !c i(i ,t! objects.]

4. In front of;1 ::lrdinil l number if it embraces all objeds considered:

The two !',rOll[1S have been shown to have the same number of gel) mtors. [Two groups only were mentioned.]

Each of th e th ree products on the right of (4) satisfies .... . [There arc exactly 3 products there.]

5. In front of an ord inal number:

The first Poisson integral in (4) converges to g. The seco nd sta tement follows immediately from the first.

6. In front of surnames used attributively:

the Dirichlet problem I Taylor's formu la the Taylor expansion Bul: [without "the"] the Gauss theorem a Banach space

7. In front of a noun in the plural if you are referrin g to a class of objects as a whole, and not to particular members of the class:

The real measures form a subclass of the complex ones. This class includes the Helsol1 sets.

ARTICLE OMISSION

1. In front of nouns referring to activities:

Application of Definition 5.9 gives (45). Repeated application (use) of (4.8) shows that ..... The'last formula can be derived by direct consideration of ..... A is the smallest possible extension in whic h differen t iation

is always possible. Using integration by parts we obtain .. .. . If we apply induction to (4), we get .. .. . Addition of (3) and (4) gives ..... This reduces the solution to division by Px . Comparison of (5) and (6) shows that ... ..

-~[Nole : In constrilctions with "of" you can also use "the" .]

2. In front of nouns referring to properties if you mention no particular object:

In question of uniqueness one usually has to cOllsider ... .. By continuity, (2) also holds when f = l. By duality we easily obtain the foHowing th eorem. Here we do not require translation invariance. q

25

3. After certain expressions with "of":

a type of cOlv:crgcllce the hypothesis of positivity the method of proof a problem of U!:i'llll'll"~S

the condition of ellipticity the point of i:;cre;LSe

4. In front of numbered obj"cts:

It follows from Thcort~m 7 that ..... Section <1 give.; a cOllcise presentation of .... . Property (iii) is "calkd the tr iangJe inequality. This h;LS been proved in pnrt (a) of the proof.

But: tht~ set of solutions of the form (4.7) To pro,'e the estim ate (5.3) we first extend ..... We thus obtain the inequality (3). [01': inequality (3)] The ;LSymptotic formula (3.G) follows from ..... Since the reg ion (2.9) is in U, we have .....

5. To avoid rEpetition:

t !:e order and symbol of a distribution the ;:::'soci;ltivity and commutativity of A the direct sum and direct product tLl: i!lilCr and outer factors of f [Note the plural.]

S;d: a d.?ficit or an excess

6. In front of surnames in the possessive:

~lillkowsk i 's inequality, but: the Minkowski inequality Fefferman and Stein's famous theorem,

more usual: the famous Fefferm?,n-Stein theorem

7. In some expressions describing a noun, especially after "with" and "of":

an algebra with unit c; an opemtor with domain 1l2; a. solution with

vanislliug Cauchy data; a cube with sides para llel to the axes; a JOIlIilin with smooth boundary; an equation with constant coef­ticient s; a function with compact support;ranclom variables witli zrro expectation

the equa tion of motiou; the velocity of propagation; an element of finite order; a soJu tiou of polynomial growth; a. ball of radius 1; a function of norm p

Bl1t: elements of the form f = ... Let B be a Banach space wit h n weak symplectic form w . Two random variables with a commou distribution.

8. After "to have":

F h;LS I finite nor111. But: F has I n fillite UOf111 not exceedi~g 1.. compact support. n CO IIIP ;lcL support contallled III I .

26

I \ I

~ I

!

I

rauk 2.

F h;LS cabrdilnality Ie. But : F has a so ute va ue l.

a zero of order at least 2 a t t he origin .

a density g. [U.nless 9 h;LS appeared determinant zero.

9. In expressions with "as" : I earlier; then: F has density g.]

Any rauciom variable can be taken as coordinate variable on Y. Here t is interp reted as area or volume. We show that G is a group with composition as group operation .

But: G is well defined as the integral of f over U.

10. In front of the name of a mathematical discipline:

Tliis idea cQ."rnes fWlll game theory (homologica l algebra).

But: in the theory of distributions

11. Other examples:

\Ve can ass ume that G is in diagon a l for 111.

Then A is deformed in to B bv pushiu<T it at cons t nnt ~rH ... d ,illIl I 1', the integral curves of X .· 0

G is uow viewed ;LS a set, without group structure.

INFINITIVE

1. Indicating aim or int ention:

To prove the theorem, we first let .....

I to study the group of .....

We now apply (5) to deri~e the foll?willg theo re m. _ to obtam an x WIth uorm not exceediug l.

Hr.re an! some examples to show how .....

2. In constructions with "too" and "enough":

This method is too complicated to be used here. This ca.se is import ant enough to be stated s par;ltc ly

3. Indicating that one action leads to another one:

We uow apply Theorem 7 to get N f = O. [= ..... alld w ' !,e t N J IlJ Insert (2) iuto (3) to find that ..... '

4. In constructions like "we may assume lIf to be .. ... ":

\\'e may assume Jt! to be compact. We define ]{ to be the section of Hover S. If we take tbe contour G to lie in U, then .... . We extend f to be homogeneous of degree 1. The .class A is defined by requiring all the functiOllS f to satisfy ..... Partlally order P by declaring X < Y to 111e,.aI1 that .. ...

f.:T . . ~£~~

27

5. In const ructions like "M is assumed to be ... .. ":

is assumed (expectcd/found! considercd/ t akc ll / claimed) to be opell.

will be chosen to contain O. NI can be t ake n to be a constant.

can easily be shown to have ..... [Note: "cas ily" after "can" ] is also found to b e of class S.

T his investigation is likely to produce gooe! results. [= It is very probable it willJ

The close agreemen t of the six elemcnts is unlikely to be a co incidence. [= is probably not]

(I . III t ll C st ru cture "for this to happen":

For liti s to happe n, F must be compact. [= In order tha t this happens]

F r the las t estimate to hold, it is enough to assume ..... Theu for such a map to exist, we Illust have .....

7. As the subject of a sentence: To see that this is not a symbol is fairly easy.

[Or: It is fairly easy to see that .. ... J To choose a point at random in the interval [0, 1J is a concep tual

experiment with an obvious intuitive meaning. '1'0 S ;1Y that u is m<L'<imal means simply that .....

Il/ier expressions with "it":

II. i'j I\ ( ' ( ' (" l!inry (usefu l/very important) to consider .... . I i. /I ".I(n il Sf' Il !:l t o speak of ... .. I i. j· j l.llC ' j('for of ill t erest to look at .....

n Afl er "b," : Our goal (Ill cthou/ approach/ proccdurc/ objective/aim) is t o filld .... . The problem (difficulty) here is to construct .....

9. With nouns and with superlatives. in the place of a relative clause:

The theorem to be proved is the followillg. [= which will be proved] This will be proved by the method to be described in Section 6. For other reasons, to be d iscussed in Chapter 4, we have to .. .. . He was the first to propose a complete theory of .. ...

At fil !l ll,. lll ll (,(! tl'I appears to differ from N. in two major ways: .. ... A III O I ( ~ ~1() Jllii s ticated argument enables one to prove that .....

[Noll' : "cr lab le" requires "one", "us" etc.] H prop osed to study that problem. [Or: He proposed studyillg .. ... J We 111!lku ; net trivially on V . Let f sn ti ;; fy (2). [Not: "satisfies"J \Ve n eed t o conside r the following three cases. \Ye need !lot consider this case separately.

["nee I to" in affirmat ive clauses, without "to" in negative claus s; a lso note: "we only need to consider", but: "we need only consider" J

lNG-FORM

1. As the subject of a sentence (llote the absence of "the"):

R epeating the previous argument and using (3) leads to .... . Since taking symbols commutes with lifting, A is .... . Combining Proposit ion 5 anu Theorem 7 gives .... .

2. After prepositions:

After making a linear transformation, we may assume that ..... In passing from (2) to (3) we have ignored the factor n. In deriving (4) we have made use of .... . On substituting (2) into (3) we obtain ... .. Before making some other estimates, we prove ..... Z enters X without meeting x = O. Instead of using the Fourier method we can multiply .... . In a"ddition to illustrating how our formulas work, it provides .... . Besides being very involved, this proof givcs no info rlll a tion Oll .... . This set is obtained by letting n -> 00.

It is important to pay attention to domains of defin ition when trying to .. .. .

The following theorem is the key to constructing .... . The rcason for preferring (1) to (2) is simply that .... .

3. In certain expressions with "of" : Th ey appear to be the first to have suggesCed the now-accepted

interpretation of ..... - - --- ----- T he idea ~f combining (2) anel (3fc~-tnleI'ionl---:~ --

10. After certa in ve rbs:

28

These proper ties led him to suggest that .... . Th y believe to have discovered .. .. . L::lx claims to h ave obtained a formula for ..... This map turns out to satisp.f .. ...

The problem cOl!sidered there was that of determining WF( u) for ..... \Ye use the t echnique of extending .....

I being very involved.

This method has the disadvantage of requiring that f be positive. [Note the iu fin it ive.]

Actually,S has the much stronger propert~f being COllvex. t".;:? .

29

4. After certain verbs, especially with preposi tions:

We begin by analyzin g (3). \Ve succeeded (were successful) in proving (4).

[Not: "succeeded to proye"] \Ve next tur n t o estimating ..... They persisted in investigating the case .... . \Ve are interested in finding it solution of .... . \Ve were surprised at finding out that .... .

[Or: surprised to find out] Their study resulted in proving the conjecture for .....

__ The suscess of our method will depend on proving t hat ..... To compute the j;Z;:m of~amo"i:Jnts to findirlg---: ... ~ -­We should avoid using (2) here, since .. ...

[Not: "avoid to usc"] We put off discussing this problem to Section 5. It is wo~th noting that ..... [Not: "worth to note" ] It is worth whi le discussing here this phenomenon.

[Or: worth while to discuss; "worth while" with ing-forms is best avoided as it often leads to errors.]

It is au idea worth carrying out. [Not : "worth while carrying out", nor: "worth to carry out"]

After having finished proving (2), we will turn to ..... [Not: "finished to prove" ]

(2) needs handling with greater carc. One more case merits mentioning here. In [7] he mentions having proved this for I not in S.

5. Present Participle in a separate clause (note that the subjects of the main clanse and the subordinate clause must be the same):

We show that I satisfies (2), thus completing the analogy with .... . Restricting this to R, we can define .. ...

[Not: "Restricting ..... , the lemma follows". The lemma does not restrict !]

The set A, being the union of two continua, is connected.

6. Present Participle describing a noun:

\Ve need only consider paths starting at O. We interpret f as a function with image having support in ..... We regard f as beillg defined on .....

7. In expressions which can be rephrased using "where" or "since" :

30

J is defined to cqual AI, the function I being as in (3). [= where f is ..... J

This is a speci al case of (4), the space X here being B(K) . We construct 3 maps of the form (5), each of them satisfying (8). ..... , the limit being assumed to exist for every x.

1 I

I The ideal is defined by m = . . . , it be ing und erstood that ... . . F be.ing ~onti.nuous,.we can assume th ;1 t ..... [= Since F is ..... ]

(It b~lI1g .m:p~sslble ~o make A and B intersect) [= slllce It IS 1m POSS I ble]

[Do not write "a func tion being an element of X" if you mean "a fu nction which is an element of X". ]

8. In expressions which can be rephrased as "the fact that X is .... ... :

Note tha t M being cyclic implies F is cyclic. The probab ility of X being rational equals 1/2. In addition to f being convex , we requi re that .....

PASSIVE VOrCE

1. Usual passive voice:

This theorem was proved by l\Iilnor in 1976.

In ite ms 2-(j, passive voice str uc tures replace sentences wit h s ub ject .. IV.· .. <J I

imperso nal constructions of otber languat:;es.

2. Replacin g the structu re "we do something":

This identity is establis~ed ,by observing that .': ... This difficulty is avord~d_' above. '-- - - . .

When this is sub~tituted in (3), au analogous description of J( is obtained. . .

Nothing is assumed concerning the expectati~n of X. 3. Replacing the structure "we prove that X is":

M I i5 eas ily shown to have ... .. may be said to be regular if ... ..

This equation is known to hold for .. ...

4. Replacing the construction "we give an object X a stru cture Y" :

Note that E can be given a complex structure by .... . The let ter A is here given a bar to indicate t hat .... .

5. Repla cing the structure "we act on something":

This order behaves well when 9 is acted upon by an opera tor. F can be thought of as .. ... So a ll the terms of (5) a re accounted for. This case is met with in diffraction problems. III the phys ical context already referred to, K is .. .. _ Tile preceding observation, whcn looked at from a more D'eneral

point of view, leads to .... . Co

V '. 31

II Mr. !l ing "which will be (proved etc.),,:

1 klore sta ti ng the result to be provcd, we give ..... Th is i: a special case of convolutions to be introduced in Chapter 8. \V, conclude with two simple lemmas to be lIsed mainly in .....

QUANTIFIERS

T l" l' hAt' I all open subsets of U .

115 Imp les t at con alllS 11 'tI G 1 a y wily = .

I t B b tIll r f I all transforms F of the form .....

~e e 1e co ec Ion 0 all A such that .....

/.' j .] cil' li!l ('d at a ll points of X . 1111 Idl II I 0; fo r a ll m which have ..... ; for all other m; fnl d l h uL it !ill ite ulllll ber of indices i

X cOIlt<1.ins a ll the boundary except the origin. T he ill tegml is taken over all of X.

all extend to a neighbourhood of U. all have their supports in U. are all zero <1.t x. E , F <1.ud G

are all equal.

' 1'1, 1' , f: ex is t fu nctions R, all of whose poles are in U, with .. .. . 1';,11 Ii ()f i.Ill! fo llowing 9 conditions implies all the others. :;,11 1, .111 .,. ,'x is l.s iff a ll the intervals Ar. have .....

1·\ " I'VI'''Y 9 ill X (not iu X) there exists an N .... . 111 11/ ' for all J <1.1ll1 g, for any two maps J ancIg; "every"

i:; fo llowed by a singular noun .] '1'0 t: V'~I'Y J th re co rresponds <1. unique 9 such that .... . Fvcry illvari<l nt subsp<lce of X is of the form .....

[Do lIot write : "Every subspace is llot of the form ..... " if you mean: "No subspace is of the form .. .. . "; "every" must be followed by an affirmative statement.]

Thus f of. 0 at almost every point of X.

Since A " = 0 for each TI, .... . [Each = every, considered separately] Each term ill this series is either itor~l,--~ ~ F is hounded on each bouuded set. j·:.IC" h of t il ese four integrals is finite .

T hese curves arise from ..... , ancI each consists bf .. ... Tilere remain four intervals of length 1/16 each. X ass umes values 0,1, ... ,9, each with probability 1/10. PI , .. . ,Fa me each definecI in the interval [0, 1].

Tilose n d isjoint boxes are translates of each other .

32

If ]( is ll OW lilly (fll ll p:,rL Sllbset of H, there exists .... . [Any W il ld . I 'V I' ! you li ke ; write "for all x", "for every x" if you

just J! II" ; , II I q ililiit ifi r r. ] Every m ';1." 11 1'(: ca ll \)u completed, so whenever it is convenient, we may

assum th i,l /Ill y give ll measure is complete.

T hc r fJ i ~l a !i ubsequence such that ..... Tltm'o cx i !ll!l an x with .... . . . [O/: tll ere ex is ts x, but: there is an x]

Thcre a rc sets satisfying (2) but not (3). T il r is ;t 1iu ique function f such that ..... Ea ·11 J Ii s in zA for some A (at least one A/

exac tly O ll C A/ at most one A). Not that some of the Xu may be repeated.

F has 11 0 fi xed vecto r (no pole) in U. [Or: no poles] F has no limit poi nt in U (hence none in J() . Call a se t dense if its closure contains no nonempty open subset. If no two members of A have an element in common, then .... , No two of the spaces X, Y, and Z are isomorphic. It can be seen that no X has more than one inverse. III other words, for no real x does lim F,J x) exist.

[Note the inversion after the negat ive clause.] If there is no bounded functional such that .... .

.. .. . provided none of the SUillS is of the form .... . Let Au be a sequence of positive integers none of which is one less than

a power of two. If there is an f such that .... . , we put .. ... If there are (is) none, we

define ..... N one ofthese are (is) possible .

Both f and 9 are obtained by .... . [Or: f and 9 are both obtained]

For both Gee and analytical categories, ..... C behaves covariantly with respect to maps of both X and G. We now apply (3) to both sides of (4). Both (these/the) conditions are restrictions only au .. ...

[Note: "the" aft er "both"] C lies on no segment both of whose endpoints are in J( . Two consecutive elements do not belong both to A

01' both to B . Both its sides are convex . [01': Its sides are both convex.] Bane! C are positive numbers, not both O. Choose points x in /11 and y in N, both close to z, and ..... We now show how this method works ill 2 cases .

In both, C is .. ...

33

In either c;ese, it is clear that ..... [= In both cases] Each f can be ex pres~('d ill either of the forms (1) and (~).

[= in any of the two fOi"!;lS]

TL(~ den~ity of X + Y is gi\"Cll by either of the two illte~r:ll ~. The t\\·o da.sscs coiuciJe if X is cOlllpact. In tbat case we write C(X) for

either of tuem. Either f or !l must be bou nded .

Ld 11 and v be two distributions neither of which is . .... [Use "neither" WhCll there arc iwo alternatives.]

This is true for nei th er of the two functions. Neithel' statement is true. In neither case CUI f be smooth.

[:\ote- tllp. inversion after a negative cbuse. ] lIe proposes two coutiitions, but neither is satisfactory.

NUMBER, QUANTITY, SIZE

1. Cardinal numbers: A aud B are also F-funct. ions, any two of A , B, and C being

independent.

t 1 I .. j 'tl I all entries zero except the kth which is one 18 lllll tl-Illl ex WI 1 the last k entries zero

This shows that there are no two points a and b such that .... . There are three that the reader must remember. [= three of them] \\'e have defined A, B, and C, and the three sets satisfy .... . For the two maps defined in Section 3, .....

[;'The" if only two maps are defined there.] R is concentrated at the n points Xl, . .. , Xu defined above.

for at least (at most)' one k; with norm at least equal to 2

Tbere are at most 2 such Tin (0,1) . There is a unique map satisfying (4). (~) !la.s a unique solution g for each f ·

But: (4) has the unique solution 9 = ABf· (-n has one and only one solution. Precisely T of the intervals are closed. In Example 3 only one of the Xj is positive. If p = 0 then there are an ad itional m arcs.

2. Ordinal numbers:

34

The first two are simpler than the thil-d . Let Si be the first of the remaining Sj . The nth trial is the last. X 1 appears at the (k + 1 )th place.

3. Fractions:

The gain up to and in -' lll dill h( til , n th trial is ..... The elements of th e third a lld fOllrth rows are in I .

[Note the pluraL] F has a zero of at leas t t.hir·d order a t x.

Two-thirds of its diameter is covered by .... . Bu t: Two-thirds of the gamblers are ruined _

G is half the Sllm of the positive roots. [Note: Only "half" can be used with or without "of".]

On the average, about half the list will be tested. J contains an interval of half its length in which .... .

F is greater by a half (a third). -'--The- other- pla);er is- haW(one third tas- fast:­

We divide J in half. All sides were increa.sed by the same proportion. About 40 percent of the energy is dissipated . A positive percentage of summands occurs in all the k

parti tions.

4. Smaller (greater) than :

great er (less) than k. much (substantially) greater than k.

n is no greater (smaller) than k. greater (less) than or equal to k.

[Not : "greater or equal to"] strictly less than k.

All points at a distance less than K from A satisfy ·(2) . We thus obtain a graph of no more than kedges.

TI . t I I fewer elements than f( has . 11S se la.s no fewer than twenty elements.

F can have no jumps exceeding 1/4. The degree of P exceeds that of Q. find the density of the smaller of X and Y . The smaller of the two satisfies ... .. F is dominated (bounded/estimated/majorized) by .....

5. How much smaller (greater):

25 is 3 greater than 22. 22 is 3 less than 25. Let an be a sequence of positive integers none of which IS one less

than a power of two. The degree of P exceeds that of Q by at least 2. f is grea ter by a half (a third). C is less t han a third of the distance between .....

/. ........ t. _(:~

35

Within J, the functiail f varies (ascillates) by less t hali l.

The upper anel lawer limits af f differ by at most l. \Ve thus have iu A one element too many. O n applying this argument k more times, we abtain ... .. T his met had is recently less and less used. A success ian af more and more refined discrete models.

6. How many times as great: twice (t en timeslone third) as long as; half as big as T he langes t edge is at mast 10 times as lang as the shartest ane. A has twice as many elements as B has. .J mllt.;t iIl S a su binterval of half its length ill which ..... 1\ 11 :\.'; f')IIl' lillies the rad ius of B. Till' d iall ll'lcr of L is 11k times (twice) that of M.

r. MIJi Lip l :

T in; k-fold illtegratia ll by parts shows that .. ... F cavers Ai twofold.

AI is oouncbl by a Illultiple of t (;c constant tillles t). This distance is less than a constant mt~l tiple of d. G acts 011 H as a multiple, say n, of V.

n Most, I as t, grea test, smallest:

36

" It ; l ~; tit most (the fewest) points when .... . I II 1I11 "d, r :l.'c. it turns out that .. ... "'I wit, Ill' t.1t!! t.heorems presellted here are original.

T it " fI' Ol)r:: a l l ! , for tit m os t part , o.nly sketched. Mwd. pl' (J !) lIbly, Lt is lIlet had will prove useful in .. ... \Vhitt Ill os t ill terest us is whet her .....

T Lte leas t such constant is called the norm of f· This is the least useful of the faur theorems. The method described abave seems to be the least camplex. T hat is the least one can expect. Tue elements of A are comparatively big, but least in nu mber. Nane of those proofs is easy, and John 's least of all.

T he best estimator is a- linear-combinatian U- such that-var U is smallest possible.

T he expected waiting time is smallest if ..... L is the smallest number such that ..... F has the smallest norm amal1g all f such that ..... [( is th e largest of the functians which occur in (3). T here exists a smallest algebra with this praperty. Find the second largest clement in the list L.

9. Many, few, a nUl11 be r of:

T h re are [N te the

plural.]

a large number of illustrations. only a finite number of f with Lf = l. a small number of exceptians. an infinite number of sets ..... a negligible number of points with .....

Ind c is the number of times that c winds around O. \Ve give a numb cl' of results concerning ..... [= same] This may happeu in a number of cases. They carrespond to the values of a countable number of invariants . .. ... far all n except a finite number (for all but finitely many n). Q cantains all but a countable number of the t . There are only countably many elements fJ af Q with dam fJ = S.

The thearem is fairly genera!. There are, however, numerous exceptions.

A variety of other characteristic functians can be constructed iu this way.

There arc few p.xceptians t.o this rlll (, . [= not lllany] Few of varia liS existing proofs are coust.rllctivc. He accounts for all the majar achievements in topology

over the last few years. The generally accepted point of view in tuis domain af

science seems to be changing every few years . There are a few exceptions to this rule. [= some] MallY interesting examples are knawn. vVe now describe

a few of these. Only a few of those results have been published befare. Quite a few of them are naw widely used.

[= A considerable number]

10. Equ ali ty, difference:

A equals B 01' A is equal to B The Laplacian af 9 is 4r > O.

[Not: "il. is equal B"] Then T is about kn.

The inverse af FG is GF. The norms af f and 9 coincide.

__ Y' ll a:'~1C s~n-.:e number of z~ros al151_poles ~n U.

F and G differ by a linear term (by a scale factar) . The differen tial af J is different from O. Each memher of G other than the identity mapping

is ..... F is nat identically O. Let a, band c be distinct complex numbers. Each w is pz far precisely Tn distinct values of ::.

r.:;r "'~.K 37

functiolls which are eq,I;11 :l.e. ,HC indistinguish:Lble as felr a.s illtegra tion is concerned.

11. Numbering: Exercises :2 to 5 furuish l,tbcr applic;1tions of this tt>cllli:'lIW.

lAma.: Exercises :2 th;'ough 5] ill the third and fourth ro\\'s from line 16 onwards in Jines Hi-10 the next-to-bst CO!Ulllll

the deri\-ati,,'cs \lp to order k the odd-numbered terms

the la.st par;1graph but one of the pre\'ious proof

TI ~t . ~-- . 1-11 in the (i,j) entry <lnd .zero (·lsewhcrc

Ie 111<l flX WILI!I . ' t' \ ' . (\' ') :1 entlws zero except ur, -) at ' .)

Tl' . I hinted ilt in Sectio!!s 1 ;1nd :2. liS IS quoted on page 3G of [.1].

HOW TO AVOID REPETITI ON

1. Repetition of nouns :

38

Note that the continuity of f implies that of g. The passage from Riemann's theory to that of Lebesgue is .. .. . The diameter of F is about twice that of C, His method is similar to that used in our previous paper. The nature of this singularity is the same as that which f has at

x = O. Our results do 'Ilot follow from those obtained by L<LX ,

Olle can check that the metric on T is the one wc have jus t described. It follo\\'s that 5 is the union of two disks. Lct D be the one that

contains .... . The cases p = 1 and p = 2 will be the ones of interes t to us. We prow a uniC[ueness result, similar to those of the preceding section .

Each of the functions on the right of (2) is one to wuich .....

F has many points of continu ity. Suppose x is one, In addition to a contribution to WI, there may be one

to W2 .

\ .Ve !lOW IHOV<! t.hat the constant pq cannot be replaced by a smaller one.

Consider the differences between these integrals anel the corresponding ones with f in place of g,

On account of the estimate (2) a nd s imilar ones which call be ..... The geodesics (4) are the only ones that realize the distance between

their endpoints .

We may replace A and B by whicheve r is the larger of the two. [Not : "the two ones"]

This inequality applies to cond itional expectations as well as [0

ordinary ones.

One has to examine the equ at ions (4) . If these ha\'e no solutions then ..... '

D yields operators D+ and D-. These are formal adjoillts of each other. .

This gives rise to the maps Fi . All the other maps are suspensions of these.

F is the sum of A, B, C and D. The last two of these are zero.

Both f a nd' g-;:'lfe connected, but the latteris- in addition compact. [The latter = the second of two objects]

Doth AF and BF were first cons idered by 13(ln<lch, but onlv the for­mer is referred to as the Banach map, the latter being "called the Hausdorff map. '

\\'e have thus proved Theorems 1 and 2, the latte r wit hon t using .. ...

Si nce the vectors Ci are orthogonal to th is last space, ..... As a consequencc of this last result, ..... Let us consider sets of the type (1), (2), (3) and (4) .

These last two are called .... .

We shall now describe a general situation in which the last-mentioned functionals occur naturally.

2, Repetition of adjectives. adverbs or phrases like "x is .... , .. :

ff f and g are mcasurable functions, then so are f + 9 and f . g. The union of mea.surable sets is a measurable set; so is thc complement

of every measurab le set. Tue group C is compact and so is its image under f. It is of the same fundamental importance in analysis as is the

construction of .....

F is bounded but is not necessarily so after division by C.

Show that there are many such Y, There is only one such series for each y . Such an h is obtained by .....

3. Repe tition of verbs:

A geodesic which meets blvI does so either transversally or ..... Th is wi ll ho ld fo r x > 0 if it does for x = O. T he iu t grit l might not converge, but it does so after .... . No te th a t w have not required that ..... , and we shall not do so ex 'cpt

wli '11 exp licitly stated. ~ ,,~ ''i'

30

\Ve will show below that the wave equation call be put in thi s form, as can many other systems of cqu<1.tions. "

The elements of L are not ill 5, as they are in the proof of .....

'I . I ~ petition of whole sentences:

The same is true for j in place of g. The same being true for j, we can ..... [= Since the same ..... ] The same holds for (applies to) the adjoint map.

\Ve shall assume that this is the case. Such was the case in (2) . The L2 theory has more symmetry than is the case

in L1. Then ei ther ..... or ..... In the latter (former) case , ... ..

1.'(1 1 k this is no longer b·ue. 'f 1,iB is Hot true of (2) .

This is not so ill other queuing processes. If this is so, we may add .... . If fi ELand if F= h + ... + fn then FE H, and eve ry

F is so obtained.

We would like to ... .. If U is open, this can be done. Ou 5 , this gives the ordinary topology of the plane. N()I.(~ that this is not equivalent to ... ..

INote t he differeuce between " this:' and "it" : you. say "it i.s :lOt <' <I \I iva.I nt to" if you are refernng to some olJject exphcl tly (1 1i' lI l, io Ii CcI ill lhe preceding seutence.]

I" I",·. t.Il1 ' s t. a t. ed (rl'sired/cla imed) properties.

WORD ORDER

General r emarks: The normal order is: subject + verb + d irect ohject + ,,,h 'erbs ill

the order manner-place-t illle. Adverbial cla uses Cnn also be pbceJ at the beginning of a sentence, and SOllie adverbs

a iways come between subject anJ verb. Subject almost always precedes ve"rb, except

in questio ns and some negat ive clauses.

1. ADVERBS - - -1,1. B lween subject and verb. but after "be"; in compound tenses after

firs l :wxiliary

• I I 'quell yadverbs:

40

This has a lready been proved in Section 8. Th is result will now be derived computationally. Every measurable subset of X is again a measure space. Vie first prove a reduced form of the theorem .

There has since been little systematic work Oil .... .

It has recently been pointed out by Fix that .... . It is sometimes difficult to ..... This usually implies further conclusions about f. It often does not matter whether .. ...

• Adverbs like "also", "therefore", "thus":

Our presentatiou is therefore organized ill such a way that .... . The sum in (2), though form ally infinite, is therefore actually finite . Oue must therefore also introduce the class of ... .. C is connected and is therefore not the union of ... ..

These properties , with the exception of (1), also hold for t.

\Ve will also leave to the reader the verification that ... .. It will thus be sufficient to prove that .... . (2) implies (3), since one would otherwise obtain J,; = O.

The order of several topics has accordingly been changed .

• Emphatic adverbs (clearly, obviously, etc.):

It would clearly have been sufficient to assume that ..... F is clearly not au I-set. Its restriction to N is obviously just f. This case must of course be excluded. The theorem evidently also holds if x = O.

The crucial assumption is that the past history in no way infl uences .....

\Ve did not really have to use the existence of T. The problem is to decide whether (2) rea lly follows

from (1). The proof is 110W easily completed . The maximum is actually attained at some point of AI.

\Ve then actually have ..... [= \Ve have even more] At present we will merely show that ..... A stronger result is in fact true. Throughout integration theOl")" one inevitably encounters 00 .

--nut H ltself call equallY well be a menlDer- or-S. ------

lb. After verb-most adverbs of manner:

\Ve conclude similarly that .... . Oue sees immediately that .... . Much relevant information can be obtained directly from (3) . This difficulty disappears entirely if ..... This method was used implici t ly ill random walks.

r-;r ~~~

41

Ie. After an object if it is short:

\ \<.: will pnwe \.lIe t ilt:(lrl' ll! din~ctly widll'llt \!.,ing the lclllllla. iJul: \\'e will prm'(' din'ctly a tlieorcm st:1:ing t!\:1t .....

This is trl\e for evcry ~Cql!i"llC t ~ thllt ~!Jril!k..:; to J: lIicely. Denne F!J an:llo::;ollsly as lLt~ lil:lit of .. ... (2) deline's g 1II1aIllbig\!Oll~ly [or e\Try g'.

ld . At the beginning-adverbs referrin& to the whole sentence:

Incidentally, we h:1\'c 110\\' construCtcd . ... . Actuallv. Theorem :3 "i\'CS lllore. llamely .... . Finally,' en shows (h a'~ j =!l. [:\'at: "At bst"] Ncv-erl he kss, it turns t)ut that -:- .. .. -~----~ ------- ---- -- --

Next, let V Lc the \'('ct()[ Sp:1CC of .... .

T'.Iore precisely, Q consists of .... . Explicitly (Intuitively), this means that .... . Needless to say, the bounded ness of f was assumed only

for simplicity. Accord ingly, either f is asymptotically dense or .....

Ie. In fro nt of adjectives-adverbs describing them:

a slowly varying function prohabilistically significant problems a method better suited for deal ing with .....

F and G arc similarly obtained from H. F has a rectangularly shaped graph. Three-quarters of this aren. is covered by subsequently

chosen cubes. [Note the singular. ]

If. "on ly"

\Vc need t lj(~ ()p(~nness only to provc the foll owing. It reduccs to the statement that only for tbe distribution F do the

m;lp~ Fi satisfy (2). [Note the inversion.] III thi:, cbapter we will be concerned only with .....

In (3) the Xj assume the values 0 aile! 1 on ly. If (iii) is required for finit e unions ollly, then .....

We Ilced oilly require (5) to hold fo r b \Il1decl sels. The prouf of (2 ) is simibr , n.ud will o lll y be illdicated briefly. To pro\'c (3). it only remains to ver ify .....

2. ADVERBIAL CLAUSES

2a . At the beginning:

42

In testing the character of .. . .. , it is sO II It'lilllt':1 difli 'ult to ..... For n = 1,2, . . . , consider a famil y o f ....

?b. At the end (normal position):

The n.verages of Fll become small ill small neighbourhoods of I.

2e. Between subject and verb, but afte r fi rst auxiliary-only sh ort clauses:

The observed values of X will on the average cluster around .. ... This could in principle imply an n.dvantage . f or simplicity, we will for the time being accept as F oniy C? milps .

Accordingly we are in effec t dealing with .. . . . The knowledge of f is at best equivalent to ... . . The stronger res ult is in fact true.

-- - It is ill all respects similar to matrix mu ltiplicatioll .

2d. Between verb and object if the latter is long:

It suffices for our purposes to assume ..... To n. given density on the line there corresponds on the circle the

density given by .. . ..

3. INVERSION AND OTHER PECULIARITIES

3a. Adject ive or past partici ple after a noun:

Lct Y be the complex X witb the origin r emoved. T heorems I and 2 combined give a theorem ..... We uow show thn.t G is in the symbol cln.ss indicated. We conclude by the pn.rt of the theorem already proved that ..... The bilinear form so defined extends to ... . . Then for A sufficiently small we hn.ve .... . 13y queue length we mean the number of customers present including

the,customer being served. The description is the same with the roles of A and [J reversed .

3b. Direct object or adjectiva l clause placed fal'ther than usual- wh en they are long:

We must add to the right side of (3) the probability that .. .. . This is equivalent to defining in tbe z-plane a density with ... .. Denote for the moment by f the element sat isfying .. .. . F is the r es triction to D of the unique linear Dlap .... . Tbe probabi li ty at birtb of a. iifetime exceeding t is n.t most ... ..

3e. In version in so me negat ive cl auses:

\Ve do not assu me that ... .. , nor do we n.ssnme n. priori that .... . N eithel' is the problem simplified by assuming f = g. The "if" part now fo llows from (3), since at no point can S exceed

tIte large r o f X n.nd Y. Tile fac t th a t for no X does Fx coutain y impli~s that .... .

/-:;r ';:>-~-;

43

III 11 0 case does the absence of a reference imply any cbim to originali ty 011 my part.

3d . In version-other examples:

F is compact a nd so is G. If f, 9 arc measurable. then so are f + 9 and f . g .

o 1 £ f - 1 I can one expect to obtain ..... n y or - does that limit exist.

3e. Adj ective in front of "be" -for emphasis:

By far the most important is the case where ..... wluch more subtle are the following results of John. l ';sse ll t ia l to the proof are certain topological properties of III.

• f .)l liJj 'c t on ling sooner than in some other languages:

/':q lt nlity occurs ill (1) iff f is cons tant. Til na tural ques tion arises whet llCr it is possible to ..... III the foll owing app li cations use will be made of .... . Recently proofs have been cOllstructed which use .... .

3g. Incomplete clause at the beginning or end of a sentence:

Put differently, the moments of arrival of the lucky customers con­st itute a rellcwal process.

Ttnthet· than discuss tllis in full generality, le t us look at ..... I L is important that the tails of F and G are of comparable magnitude,

" Ht:lt lII ent 1l1 ade more precise by the following inequalities .

WHERE TO INSERT A COMMA

C:C1Lc,.,,1 r uks: Do no t over-use com mas-English usage requires th em less often than in many other languages. Do not use commas around a clause th~t defines (li­mits, makes more precise ) some part of a sentence. Put commas before and after non-d efining clauses (i .e. ones which can be left out without damage to the sense) . Put a comma where its lack Illay lead to ambiguity, e.g. betwee n two sy mbols.

l. Comma not required :

44

We s llall now prove that f is proper. The fact that f has radi2..1 limits was pro"ecl in [4J. It is reasonable to ask wllether this Ilolds for 9 = 1.- - --­AI is til e se t of all maps which take values in V. There is a polynomial P such that P f = g. T he clement given by (3) is of the form (5). Let 1\1 be tile manifold to whose boundary f maps IC Tak an element all of whose powers are in S.

F is called proper if G is dense . There exists a D such that DxyH wuellever HxyG.

F(x) = G(x) for all x E X. Let F be a nontrivial continuous lillear operator in V.

2. Com ma required:

The proof of (3) depends on the notion of !If-space, which has already been used in [4].

We will use the map H, which bas all the properties required . There is only one such f, and (4) defines a map from .....

In fact, we can do even better. In this sect ion, however, we will not use it explicitly. Moreover, F is countably additive. Fiualiy, (d) and ( e) are consequences of ( 4). Nevertheless, he succeeded in proving that .... . Conversely, suppose that ..... Consequently, (2) takes the form ... .. In particular, f also satisfies (1) .

Guidance is also given, whenever necessary or helpful on further reading. '

This observation, when looked at from a more general point of view leads to ... .. '

It follows that f, being COllvex, cannot satisfy (3) . If e = 1, which we may assume, then ..... \Ve can assume, by decreasing k if necessary, that ..... Then (5) shows, by Fubini's theorem , that .....

Put this way, the question is not precise enough. Being open, V is a union of disjoint boxes.

This is a special case of (4), the space X here being IJ(K) .

- In [2], X is assumed to be compact. for all x, G(x) is convex.

[Comma between two symbols.] In the context already referred to, K is the complex field.

[Comma to avoid ambiguity.]

3. Comma optional:

By Theorem 2, there exists an h such that ..... For z near 0, \~e have ..... If /; is sm~oth,the-Il M is compact. Since h is smo-oth, A1 is compact.

It is possible to u~e (4) here, but it seems preferable to ..... This gives (3)! because (sinc-e) we may assume .....

Integrating by parts, we obtain ..... To do this! put ..... -

45

X, Y, nlJd L nrc compact. X = :1;'G. \I·lien! F is defilled by ... . . Thus (ll~!lr e/Thl!rdor e }-, we itn\"(' .... .

SOME TYPI CAL ERRORS

L Spelling errors: Spelling; should be cOIJ.ji~tellt. eiilIl!r Dritisu or .-\lllcriciln t hrougliouL:

[Jr.: colom, llci,~h1Jo11r. celltre, fibre. labelled, I'1()ddlillg Amer'.: co lor , llei:.,.;llboL ccnter , fiber, labe led, tllodelillg

;lll lluir.l'd appronc!l ~ a unified npproilch a ;\ f s \:ch thnt ~- ;lll .\I such that

[U~,! "a" or " <lU" a ccording to rrollllnciation .]

2. Grammatical errors:

4G

Let f denotes ....... Let f denote Mos t. of them is ~ i\ Jos t of them are There is ;t finite number of...,... There are a finite number of In 196-1 La...>: has slImvll ...,... III 19G4 La...x showed

[Usc the past tense if a date is givell.]

The Taylor 's fo rmuh ...,... Taylor's formula [Or: the Taylor formu la] The sertion 1 ---; Sect ion 1 Such lllilp exists ....... Such a map exists [But: for every such mar] In the C;loSe ,\1 is compact ...,... In case AI is compact

[Or: In the case where AI is compact] In case of smooth norms""'" III the case of smooth llorms We are ill the position to prove""'" \Ve are in a position to prove

F i ~ ,'q ual G ....... F is equal to G [Or: F cquals G] F is grl';ller or eqllal to G ...,... F is greater thall or equal to G

Continuous in t.he point x ...,... Continuous at x Disjoint with B ...,... D isjoin t from B Eqllivalent with B ...,... Equivalent to B Indcpendent on B -v-+ Independent of B

[But: depending on B] Similar as B ...,... Similar to B

Simi larly to Sec. 2 As (J ust as} in Sec. 2

Similarly as in Sec. 2 -v-+ As is the case in Sec. 2 In much the same way as

in Sec. 2

In the end of Sec. 2 ...,... At the end of Sec. 2 On Fig. 3 -v-+ In Fig. 3

Sillce f = 0 thCll M is closcd ---; Si nce f = 0, M is closed [Or: Since /=0, wc conclude t hat M is closed]

.. ... as it is shown in Sec. 2...,... ..... , as is shown ill Sec. 2 J:: ',u)' function being an e lement of X is convex

--+ [,'cry fUllction which is an element of X is convex S.-,ttillg 11 =]/, the equation can be .. .. .

--+ Setting n = p, we can ... .. [Because we seL]

3. Wrong word used:

\Ve now gi ve few examples [= not many] ....... vVe now give a few examples [= some]

SUlllming (2) and (3) by sides...,... Summi ng (2) and (3) In the first paragraph...,... In the first section

_ ... .. , \Wich_Rro\,CS gur_thesi!? ______ _ ...,... . .... , \\·!Jich proves our assertion (concl usion/sta tement}

[Thesis = dissertation] For n big enough...,... For 11 la rge enough To this ;"tim ....... To this end At first, note that -v-+ Firs t, note that At last, wc obtain -v-+ Finally, we obtain For every two elements...,... For any two elements .. ... , what completes the proof...,... ..... , wbich completes the proof ...... what is impossible -v-+ ..... , which is impossible

4 . Wrong wo rd order:

The described above condition ---; The condition described above

The both conditions...,... Botb conditions, Both the conditions Its both sides...,... Both its sides

The three first rows...,... The first three rows '. The two following sets...,... The followin g two se Ls

This map we denote by f --+ We denote tbis ma p by f 5. Other exa m ples :

W(! have (obtain} that [J is .... . ....... We sec (conclllcie/cleullce/fimi/ infer} that [J is .... .

\Ve arc done --+ The proof is completc,

1'(

It, I,l l, 2:) , 1\6 1\(cHdi nr, ly , 1:1 ,\f \ 1I .d ly , I V, 4 1 adject i v, I clauses 9 Ildverb i. 1 cla us es : 42 ad ve rbs , ,to a few, :.1 7, 47 a ll, 32 a lso , 41 a nu m ber of, 37, 46 a ny, ::13 as , 15 , 18 , 2 7 .s ic., :JO , H III [i,,;l, 4 7 III I. LS ( , <1 2, 47

:tv id , :.10

b ecause, 15 being, g, 30, 47 both, :13, 47 brackets, 8

cardin:tl numbers, :J.1 case , 40, 46 co nl ra d ic ti o n, 11, Hl

d 'no te , 7 de pend ing o n, tl d ill('l' , :Hi , 'J7 c1i njo in t fro lll , 4G d is t in c t , :)7

each, ::\2 eithe r , 34 en<1.b le, 29 en o ug h, tl, 27 eq ua l, 37, 46 e rro rs , 46 every, 'J2 , 4 7

f,'w , :l7, 4 7 f,·\V,;r, :,5 [i ,, :t lly . '12 , 47 fin is h , :50 k- fo ld , :J6 fo llowing, 13, 19, 20, 47 fo r , 11, 2tl forn1e r , 39, 40 for s hort, 7 fr a ctions , :.15

48

INDEX

generality, 10 greater, :.15

half, :.15 have , 2G "have that", 15. 16, .\7 hence, 15, 4G

if necessary, 11 imper:ttive, Hi in a position, 17, 45 independent of, 8, 4G

induct ion , loI in fact, 13 infinitive, 10, 17, '27, 20 introduction, 4 inversion, 9, 10, :.1:.1, 42, 43 it, Itl, 19, 28, 40 it follows that, 15

largest, :.16 last but one, :.18 lat ter, :.19, 40 lC <lSt , :H, :3(j less, :.IS let, 46 likely, 28

matrices, :l8 more , :.I(j mos t, :.14, :.Ib, 46 multiple, :.16

need, 17, 29 neither, 9, :.14, 4:.1 nex t-to-last, :.18

no, :.1:.1 no greater, 35 none, :.13 nor, 9, 10, 43 number ing, 26 , 38

"obtain that" ~15 16, 4, of, 25, 26, 29' .

one, 23, 38 only, 29, 42 ordinal numb ers, 25, 34, 38

paragraph, 4, 47 participles, 30 percent, 35

print, 8

same, IG, 40 say, 11 second largest, :.I(j

section, 4, 47 short-cu (s, 20 shortly, 7 similar, 16, 46 similarly, 46 since, 15, 47 sm:tller, 35 smallest possible, :l(j so is, 10, 39 SOlne, 33 succeed, 30 such , :.19 , 46 such that, 8

th<1.t, 38 the, 24 the one, :.Itl th e reforc~ , 1;;, ·11, '1<i the re is, :\:l t hese , :j<J thesis, 18, 20, ,17 the two, :H , :J9

this, 40 th is 1<1.5t, :lV those, :li'! thus, 15,41,46 t o be defi ned, 9, 28

too, 27 t o thi s end, H, 47 twice as long iIS , 36 t wo-thirds , 35 typefaces, 8

ullion, 25 unique, 24, 34 unlikely , 28

u p To, :35 , :>8------

what, 15, 18, 47 w hich, 15, Hi , 47 with,26 wOTth,30 worth while, :10

GDA.:'\rSK TE.:\CH£RS·

PRESS

The _booklet is intended to provide practical. help for authors - ofrnatIierriatic8]- papers. It will -be - useful both as a guide for beginners and as a reference book for experienced Writers.

The first pa...r-t of the booklet provides a useful collection of ready-made sentences and expressions occurring . in · mathematical papers. ' The examples are divided into sections according to their use (in introductions, definitions, theorems, proofs, comments, references to the lilerature, ackrlOwledgments, editorial correspon­dence and referee's reports). Typical errors are also pointed out.

The second part concerns selected problems of English grammar and usage, most often encountered by mathematical writers. Just as in the first part, an abundance of examples are presented. all of them taken from the actual mathematical texts.

The index enables the reader to find many particular pieces of information scattered throughout the text.

Jerzy Trzeciak, formerly of Polish Scientific Publishers, is now the senior copy editor at the Institute of M?-thematics, Polish . Academy of Sciences. He is responsible for journals including Studia Mathematica; Fundarri.enta Mathematicae. Acta Arithmetica and others.

ISBN 83-85694-02-1

;If