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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~ ~':" ~:-:'

CONTENTS

Acknow ledg7lLents. 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 an d ~Ll!"cill .\daw:,ki for suggesting several il1l I' rm'clllcnt s, and to Hcmyb Walas for her painstaking job of typesettin g the continllo usly varying manu script.

Publ ished by Gdallskie W ydawn ictwo O§wiatowe (Gcl a llsk Teachers' Press ) P.O . Box 59, 80-876 Gdallsk 52, Pol an d Cover design by Agnieszka Polak Typ eset by Henryka vValas P;illt.ccl ill Poland by Zaklady Graficzne w Gdallsku

©

Copyright by GdaI1skie Wydawnictwo Oswiatowe, 1993

All rig hts reserved . No part of this publication may be reproduced in any form \vithout the prior p ermission of the publisher. ISBN 83-85G94-02-1

Part A: Phrases Used in Mathematica l Texts Ahstract and introciuction ... .. . . . .... . . . . ...... . . . . . .. ... .... . .. ... .. . 4 Definition ........... . ....... .. ..... . ... . . . ...... . .. . .. . . . .. . .. . .. .. . . , G Notatioll . . . . . . .. ... .. ... . . .... ..... ... . ... ...... .... .. .. ...... . . . ..... 7 P ro perty ... . . . . ................. ..... . . . ..... . . .. . ........ . . . ..... .. . . 8 Assumptio n , cond iti o n, convention . ... . . . ... ... ..... ... . . ..... . ..... . 10 Thcorem: general rem arks .. ... . . ... . .. . . .. .... . ............ .. .. . .. ... 1~ Th eo rcm : illtrociuctofy plfrase . .-:-: . : .. ~-:-......-.. ... .. ..... . .. ....... 1:1 Theorem : formulation . .. . . ..... . ... . ..... .. . ..... . . ... . ......... ..... 1:1 Proof: beginning . . .. . . .. .... .. . . .. ...... . ... .. . ... . . ... . . . ..... . . .. . . 1·1 Proof: argu m e 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 r emarks .... .. .. ... ..................... . ... . . . . 18 n.efcrenccs to the literatu'r e . ... .. ... . . . .. . . . . ... ... . . . . .. . .... .. . .. . . . 19 Acknowledgments ................. .... . . . ... ... . . .. . ... . ... ..... .. ... 20 How t o s horten the paper . . ....... . .... ..... . . ..... .. . . . ... . . .. . ... .. 20 Editorial correspondence .... . . . ..... ....... .. . . ................... . .. 21 R eferee's report .. ... ... .. ... .. . . . .. . . . . ..... ..... .. .. ... . . .. .... . ..... 21 Part B: Selected Problems of En glish Grammar I ndefini te a rt ic le (a, an, - ) .. .. . .... . ... . . ... . . . . .. . ....... ......... :':1 Dcfinit e artic le (t he) .. . . . .. . . ... . . . .. . . . . . . . .. . .... . .. ......... . .. :' I Article ornissio ll .. . . . .. .. . . . .. . . _..... .. . . . .. ..... . .... ........ . ' ..... :>;) 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 Ind ex . .. . .......... . . . ...... .. .... ...... .. . ...... . . .... . ...... . .. ... . 48

r? ",,,:.~,

PART

A:

PHRASES USED IN MATHEM AT ICA L TEXTS ABSTRACT AND INTRODUCTION

\\"' IHove th at in so m e fami li es of compa cta thcre are no u niversa 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 are inter ested in finding .. ... It is natural to try to relate .. .. . to ... ..

We wis u to invest igate ..... Our purpose is to .. ...

T his wo rk was intended as an attcmp t t o motivate (at moti vat ing) The aim of lIlis pap er is to bring together two areas in wh ich .. .. . review some of the st.andard facts on ..... have compiled some basic facts .. summari ze without proofs the rele vant material OIl . . .. . g ive a bri ef expositi on 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 .... . I !i"1 l ilill :1 II III" III i I d ''' 'I I ill II II " dl'\,r lop tlt e theo ry of ... .. wi ll look 1Il 0re closely at .. .. . I NI I/ ,' 11.1 1111', 1 ' '1i11 lI'i li 1)(' c() Ilce m ed with .. .. . / ",, 1111 11 1 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 tin g formub for it is s how n th at .. .. . some of the recent resu lts arc rev iewed in a more genera l sett ing. . some applications arc indica ted . our m a in resu lts arc s tated and proved.

I

con tai ns a brief su mmary (a discuss ion ) of .... . deals witlL (discusscs) _the casc ~~ . .. . ____ is intended to m oti\·ate our in vestigation of ..... Srct ion 4 is d c\·otcd to the study o f .... . provides a d ctailed exposit ion of .. .. . es tablis hes the relation between ... .. presen ts sOllie prelim inari es .

I

\\' . ·11 1t o uch e WI

only a few aspects of the t heory. r estrict our atten tion (the discussion/ourseh·es) to .....

It is Hot o llr pll r pose to s tudy ..... No atte mpt h:\.S be 'n made h ere to develop .. .. . It is poss ible th a t ..... but we will not develop this point here. A more comp le te th eory lllay be obtained by ..... this t op ic exceeds th e scope of this paper. . .. . However

I

, we w tll not lise tlus fa ct

III

any esse ntIal way.

.) I idea is to. apply T lIe b aS .lc ( malIl . d.... .. .

geometnc mgre !Cut IS .. .. . Th e crucial fact is tbat the norm sa tisfies .... . Our proof invo lves looking at ..... b:\.Sed on the concept of .. ... Th e proof is s imilar in spirit to ..... adapted from ..... This idea goes back at least a5 far as [7)

I

We em phas ize that .. ...

Il is worth p oint ing out that .... . The important point to note here is the forlll of .. .. . The adva ntage of using ..... lies in tlw fact that .... . Th e est imate we obtain in the course of proof sce IHS to be of ind epenrient inte rest. Our theorem provides a natural and intrinsic cil;u acter iza t.i on of. Our proof m akes no appeal to .. ... Our view point 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 probl em is t hat .... . The m a in difficulty in carrying out this construction is th a t ..... ' In this case the method of ..... brea ks down . This class is not well adapted to .... . Poi ntw ise converge nce presen ts a more d elica te p robleI1l . Tbe res ults of this paper were announced witbout proofs in [8). The detai led proofs will appear ill [8) (elsewhere/in a forth coming 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 wh etber 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 ira ble 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
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
Iwllh th e cla.sslcal one for .. ... is ... ..

:\ 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
Our definition agrees w~th the olle givell in [7] if

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

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

1=

.....

requiring 1 to be const;1llt on .. .. . the requirement that 1 be constant on .. ... "[hi;; lIlap is defined by [Note the infinitive.] imposing the following condition: ..... The k7!t;lh uf
= L1

I

is ... .. I , ,t f we la\e se - ... .. , f being the sol u lion of ... .. with f satisfying ..... , W

1Jere

I

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

I

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

II

this coincides with our previously introduced terminology if J{ is convex. this is in agreement with [7] for .....

I

NOTATION

I

_ We \,'ill denote by Z Let us denote by Z - tJleset ..... Let Z denote

Write (Let/Set) 1 = [Not: "Denote

.....

1=

..... "]

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 Here and subsefj uelJ tly, Throughout the proof, In the sefjuel, From now on,

,Istands denotes I I for t Ie map .....

h

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

I

Wit h tIl(! 11ot;1tio11 J = ..... , 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 not.ation, For abbreviation,

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

We abbreviate Fallb to b'. 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

G

t-..("

7

we have jus t d efined 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)

II \~ ill (,, \II Se no confusion if we us e th e sam e lett er to designate " "" '"Ille r o f A and its rest.riction to [\' , \ \'1' ', 11.111 WI i II' the above exp ressio n as II" , "I I/IVII I'x press io ll m
It = .....

' I II" (:II 'I'k illl li l'l'S !
il'IIIIIII OloIl,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 bra ckets), 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' Idt ers i= s mail let. tcrs = lower G\SI! It: t.ters ; (;oLlli r (;1'"11;111) lei LI' I ~ ; sc ripl (ca ll i~ra rhic) let.ters F; s l' cri :ti I{ Ulil l lIl l<'t lle l S If'! Dot " prime " as t. eris k = st. ;ll' ., tilde - , bar - [over a symbo l], ha t - , ve rti cal s troke (vertica l bar) I, s l;Ls h (diagona l s troke/slalll)/, da.5h - , s harp # Dotted line . , , , .. , das hed lin e _____ , wavy line _ _~

..... , tbe constant C being iudependent of .... . [= where ..... , th e s upr emum b eing taken over a ll cubes ..... ... .. , the limit being taken in L,

C is .. .. ,]

is so chosen th a t .. .. , is to be chosen later. is a su itable constant. .... " where C is a conveniently chosen element of ..... invoh'es tbe d e 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. not m erely symmetric but actually se lf-adjoint. not necessarily monotone. both symmetric an d positive- definite. not cont inuons, nor do they satisfy (2). The operators Ai [Not e the inverse word order after "nor" ,] are ne ither symmetric nor positive-d efinite. only nonnegative rather than strictly positive, a.s on e may have expec ted, allY self-adjoi nt operators, possibly even nnbounded . still (no longer) self-adjoint.. not too far from being se lf-adjoint .

I

PROPERTY such that (w it h the proper ty that) [Not : "such an element that " ] with the follow ing properties: ..... sat isfy ing LJ = .... , wit h N J = 1 (with coordin ates J:, 1j , ;::) of norm 1 (of the form .... ,) whose norm is ..... all of whose subsets are ..... by means of whic h 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 eq uations .. ... g;ivclI by LJ = ... .. d epe ndin g olIly on ..... (ind ependent 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 ub le "r" .] gua ranteed by the assumption .... , so obtained

8

Th e

preceding theorem indicated se t [B ut adjectival clauses with above-mentioned group prepositions come after a nou n , e ,g. "llle group de fin ed in Sect ion 1" .] r es ulting region req uired (des ired) element

13otli--X- allCi Yarc finite :- - - - - - - - Neit her X nor Y is fi n ite. X and Yare countable, but neithe r is finite, Neither of the m is finit e. [No te: "Neither" refers to two a lt ernat ives,] None of the fUllctions Fi is finite,

X is not finit e ; nor (n ei th er) is Y. 9

x

is not finite, nor is Y countable.

X is empty

This involves no loss of generality.

[Note the inversioll .]

I;, ,()iJuti~ . ~" IS: !lot. 1

X belongs to )'

\Ve can certainly assume thilt .. ...

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 otherw ise 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 asymp t otic 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 .... . .. ... , where g

is subject to the condition Lg = O. satisfies the cond ition Lg = O. is merely required to be positive.

I

the requirement that g be positive. [Note the infinitive.] Let us orient M by requiring g to be .. ... imposing the condition : .. ... for (provided/whenever/only in case) P unless p = l. (4) bolds

i=

1.

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

~

, , , \

siuce otherwise ..... for .. ... [= because] for if not, we replace .. ... Indeed, .... . .

i\'eit he r 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 ..... is not particularly rest r ictive. is surprisingly mild . admits (rules out/excludes) elements of .. .. . This coLld ition is essential to the proof. cannot be weZlkened (relaxed/improved/om ittedj 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 o n ..... Our lemma does not involve allY assumptions about CUrva llir ' . We bilve been working under the ass umption that .. ... Now suppose that this is no longer so. To study the general case , take .. .. . For the general case, set .... . The map

f

Irca

w ill be viewed (reg;trded/thought of) as a fUlll clO r .... .

From now on we

I

lZll1g .... .

think of L as being constant. reg~rd f as a map from .. ... tZlC\t/y assume that .....

It is und erstood that r i= l. We adopt (a dhere 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 ans wcr to a question raised by ..... d ea ls with ..... (,IISllr s th e existence of ..... ('X P I' sscs th e equivalence of ..... p l'ov' icl cs a c rit e rion for .. .. y i(: ld: ill[ I'mali o n 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 ld y (Loosel y ) spe:tkillg, th e fUl'lIllila says t hat ..... \\,1 . ' len

f IS . opell, (3.7)'.Ill S t tl ,\lll OUIl S

to saying that ... .. to th e fact that ... ..

Here is another way of stating (c): .... . Ano th e r way of stating (c) is to say: .... . II 1\ ('q uiva le nt formu\atiou of (c) is: .... . T I\( :o rc Ill S 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 th a t ..... T he int.e res . t I " fi T IIe pnUClpa Slgru canee 'Ih . t e POIll

1

0

. f I I . I'111 the assertIOn ..... t. Ie emma IS tl t't II t la I a ows olle 0

The theorelll gaius in interest if we realize that ..... till true I'rl we drop the assu;nptioll ... TI Ie tl leo re m is 's1111 1 I . I assumecIt Ila t .... . It .IS onlY I s t l 10 (S I [ wc t;l!
f = ..... f by - J,

I

I we recover th _ e standard clemma ..... [I, Theorem Cl].

'l'l li :; s pec ializes to th e result of [7] if ' ,'Ii i::

I

(, ~; lI l l

f

= g.

be Ileed ed in 1 will pro ve extremely useflll in Section 8. I not be needed until

TH EOREM: INTR ODUCTORY PHRASE \ Ve have thus p rove d ... .. Summarizin g , we have ... ..

rephr ase Theorem 8 as follows. \Ve can llOW state the analogue of ..... 1 formulate our maill results.

We are thu s led to the following strengtheuillg of Theorem G: .... . The re mainder 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 ele mentary 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 ..... Let M be .....

Suppose that .. "' Assume that ..... 1Write .. ...

..... ,

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

Furthermore (Moreover), ..... In fact, ..... [= To be more precise] Accordingly, ..... [= Thus] Gi\'en allY

f i= 1 suppose

Let P satisfy

that .. ... Theu .... .

t he hypotheses of .... . the above assumptious. 1 N(P) = 1.

1

Then ... ..

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

12

1Then

rpi ~,~

,

.' . 13

PROOF: ARGUMENTS

PROOF: BEG INNING

definition, ... ..

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

the defini tion of .... . Iassumptioll, .....

I

I

Out. A =

n.

_

I To his end. consider .. ... f t If I [= For tbis Dllrposc; not: "To this aill l"] Irs COlllpl1 e '1 To do this. tah; ..... t

\!

-

---

straightforward (quite involved). The proof is by induction on n. left to the reader. based on the followillg observatioll.

From this what has already been proved,

I

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

we conclude (deduce/see ) th at .. ... we have (obtain) AI = N. [Note: without "that"] it follows that ..... it may be concluded that .....

I

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

the assertion of the lemm a is false. .....

[hence = from th is ; thus = in this way; therefore = for t hi.it follows that = from the above it fo llows th a t]

I

Conversely (To obtain a contradiction), suppose tlla t .... . () !l t IJe contrary, Sllppose the IClllIlIa were false . Th e n we could fiIlt! .. ...

I

LI = o.

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

We have diviued the proof into a sequence of lemmas .

n

ha.'ieLI_= O. it fo llows"that L1 = 0 we see (co llclude) th at

(5)

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

t here existed · an x ..... , If x were no t I l l , it were true that ..... ,

Since .hs

co mpact~ w~

B ut L1 = 0 since I is compact. 'I'Ve bave If = 0, beca use ..... [+ a longer ex planati o n] \Ve must have L1 = 0, fo r o th erw ise we can rep lace ..... As 1 is co mpac t we Iw.ve L1 = O. Therefore LI = 0 by Theorem 6. That L1 = 0 follows from Theorem G.

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

I , contrary to our c l'aJm , t h a t

.....

L1 = o. [Not: "Since ..... , then .... .-']

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 .. ... )

S uppr)se

I =g

By Taylor's formula, ... .. a similar ::l.rgument, ... .. shows that .. ... the above, .. ... yields (gi ves / Theorem 4 now the lemma below, ..... implies) I = co nti nuity, .... . leads to 1 = ... ..

To ~cc (pro':e) this. Id f = ..... \:I\!l:':)\'e th is i\s,f()!:~\:',~~ 1,1,., b proved b~ ,\. )l ld o g - ... .. t

\V

But

t he compactness of .... .

, which follows from ... .. as was described (shown / ment ioned/ noted) in ... ..

Iwe wou ld have .....

tI Id b .lere wou e .....

Assume the formula holds for the degree k; we will prove it for k Assullling (5) to hold for k, we will prove it fo r k + l.

This gives M = N . We thus get AI=N . F is compact, The result is M =N. (3) now becomes M=N. Th is clearly forces M = N.

+ 1.

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.

F

=G=H

I

I (';(!,{) I I ;

and so AI = N. and co nseq ue ntl y M N . and, ill consequ ence , Ai = N. and hence bounded. which gives (im pli es/ yields) 111 = N. [Not: "what gives"]

=

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;:, ,:~~

14

-

15

Th!! 'quality thal .....

f =g.

Repealed app li c:tI.ioll of Lemma 6 enables us to write ..... \Ve now J r r' 'd I Y illduction. \Ve can 1I0 W Pl(l 'ced annlogonsly to the proof of .... .

which is part of the conclusioll of Tlteorem 7, implies .

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), .....

\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 ..... obj ective is to evaluate the integral I. concern will be the behaviour of .....

Similar arguments applyI' It 0 tl Ie case ..... . } 'I Ie same reasoulllg app les Tlte sa me couclusion can be drawn for .... . This foll olV by th e s;].me method as iu .... . T ill! telill T f can be haudled in much the same wav, the only difIereIlce Iwillt'. ill lli e ;]'Dalys is of ..... II I llll! saJllI! mann er we can see that ..... T lt l' rc:; t of lh e proof runs as before. W nolV apply this argument again, with I replaced hy J, to obtain .....

\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 .....

PROOF: CONSECUTIVE STEPS

Consider ..... Define Choose.... . Let Fi x ..... Sel

If = .....

Let. us

.

evaluate .... . compute .... . apply the forl1lllia to ..... suppose for the III0nWIIt. regard s ;L~ fixed and .....

I

In order to get this inequality, it :vill be ll~cessary to .... . IS convelllent to .... . To deal with I J, To estimate the other terIll. For the general C
iN(}l e: The imperative mood is used wheu you

0l'del' the reader to do sO l1l ethi ng, 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. th ese in equnli ties] Replacin" (2) by (3) Letting n co Appl ying (5) 11I 1,!' I,(," ;1111~ i l l g J ,wd 9

'1'111' ill :~ fOrlll is eit.he r th e subject of a sentence ("Adding ..... giIl'ljll ires th e 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(j

1I1 ;\Y

J= .....

now integrat e J.: times to conclude that .... .

I

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

I

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

I

.

V(':I" ), (II'

W,·

PROOF: "IT IS SUFFICIENT TO

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 ... ..

IN,, /,·

Iwe note that .....

~

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

IS

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

clear (e\'i
W I

e

. .... , itS i~ ot<J.~y

to check.

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 .. ... By (21 it is obviolls that]

- !:':2) ,Hakes -it ob '.
Tile

fac:.tO f

Gj I'oses

ilU

prQblel:1

r=

lkC,-lii.-;r.

Gis .. ...

PROOF: CONCLUSION AND REMARKS

..... , wlliciJ : .Vot: .. '.\' h,l.T.. j

proves the theore m. completes the proof. I:stablishes the form ub . is the desired conclusion. is our claim (assertion). [Not : "thesis"] gives (-1) when substituted in (5) (combined with (5)).

..... , and

user

for

..... , wi:!r! 1 is cl,~ar from (3).

\Ve SCI: (check) at O!lC" ::, : ,; . " .. F is ca.;ilv seen (Cilcckc<, to be sm oo th.

1;"\"(,

the proof is complete. this is precisely the assertion of the lemma . 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). , • C,1 1 • [N a t : 'w ., It'''] . ..... '.VU1 15 .IlllpOSSJ'ble. la IS ...... '.vLie!! con t r:\dicts the maximality of ..... .... .. :l cOill.r'ldiction. The proof for G is similar.

ll oll ly the fact that ... ..

the existence of only the ri ght-han d derivati\·c.

f = 11 it

is no longer true that .. ... the argument breaks down .

The proof strongly dep ended 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] Thi s 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 d efinition of ..... ) the complete bi bliograpby)

TIH~ bes t general reference here TIlP. sUllldard wo rk Oll .. ... The cl:Lssical wo rk Iwre

I.. IS .. ...

wa~ proved hy L lX [t>]. This can be found ill

Lax [7, eh. 2] .

is due to Strang [8]. goes back 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] .

I

The l1letbod of rroof carries over to dom aills .... .

a rece nt account of the theory a treatment of a more genera l case a fu ll er (thorough) treatmell t we refer the reader to [7]. For a dee rer discussion of .... . direct constructions along more cl assical lines yet anothe r method

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

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

C may be handled in much the same way. Silllilar considerations app ly to C. TI

Ie Silme proo

18

I

f works (remains valid) for .... . still goes (fails) when we drop theassumptioll .....

0.. 7 ~.:

19

' I'hl! lll aill results of this paper were announced in [7]. Silll il:tr results havebecn obtained independently by pub lished in [7] .

LiLX

and arc to be

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) = .....

ACKNOWLEDGMENTS

I

T he a uthor :vishes to ~xpress his thauks (gratitude) to ..... IS greatly Illdebted to .. ... his <1.ctive interest iu the publication of tllis paper. suggestiug the problem and for many stimulating conversations. for 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 ill g Lit pr para tion of tile paper. Thi s is p<1.r t of th e a uth or's Ph .D. thesis, written uuder the supervision of .... . a t th e Uni versity of ..... The author wishes to tlt a nk the University of .. ... , where the paper WiLS written, for financial s upport (for the invitation and. hospitality).

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 t o 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 --> . . . . . •

I would like to submit I am submitting

, lI era l rul es: J. n.eme mber: you are writing for an expert. Cross out a ll th at is trivial or routine. ~. Avoid repeti ti on: do not repeat the assumptions of a theorem at th e beginning of its proof, or a complicated conclusion at the end of the proof. Do not repeat th e ~s uillptions of a previous theorem in the statement of;\ next on e (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 formul as : is e
Phrases you can cross out: We denote by IR the set of all re:tl numbers. lYe have the following lemma. T he followin g lem ma will be useful. .. .. . the follow illg inequ a lity is s<1.tisfied: P h r
20

Tb eo rem 2 a ll cl Th eorem 5 "" Theorems 2 <1.ud 5 This follow s frolI l (1), (5), (6) and (7) --> This follows from (4)- (7) For deta ils!i ' [:31. [4J a ud [5] ..... For details see [3]-[5]

EDITORIAL CORRESPONDENCE

HOW TO SHORTEN THE PAPER

I.e'/. Ill' <1.11 a rb iLrary but fi xed positive numb er""" Fix 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 co mpute """ \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 illIl P of( ,I) ,13y (4) Oy 11!lalio ll (d) , lIy (4)

c;

>0

Ithe enclosed manuscript " ..... "

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 prese nt ed are original.

21

The paprer is a good piece of work considerable attention.

Oll

a subject that attracts

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

PART

B:

SELECTED PROBLEMS OF ENGLIS H GRAMMAR INDEFINITE ARTICLE (a, an, - )

III

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

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 __ _ , ., ~ I "re I , c· f kintrrcstilig. 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.

I

1• '). as stated . T ueorClll - IS fit Ise.III t h'IS genera j'Ity.

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

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 bound ed functi on. This is easily seen to be an equivale nce relation. Theorem 7 bas been extended to a class of bound a ry val\l e jll'Oh ll'III '; The transitivity is a conseque nce of the fac t th at .... . 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 Tbe ..... , ..... ,

existence of partitions of unity may be proved by .. .. . 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 th a t .. ... We wish to find a solution of (6) which is of th e form .. ...

in the plural: Tbe elements of D are often called test functions. tl 1e se t

0f

points with distance 1 from K

. . . I compact support WIt Ia II functIOns 1

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 .. ... 22

23

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

4. In the plural-when you are referring to each element of a class: Direct sums exist in the category of abelian g roups . I II particular, closed sets are Dorel sets. Borel measurable functions are oftell called Borel mappings. Th is 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 C

1

I

1 gives a condensed exposition of ..... 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.

[I ( YO II wi s h 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 .... . [The re arc exactly 3 products there.] 5. In front of an ord in a l number: The first Poisson integral in (4) converges to g. The seco nd sta tement follows immediately from the first. 6. In front of surn a mes used attributively: the Dirichlet problem th e Taylor expansion th e Gauss theorem

Bul:

ITaylor's formu la [without "the"] 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.

DEFINITE ARTICLE (the) ARTICLE OMISSION

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

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

I ) <'Ii II

2. III rront of a noun (possibly preceded by an acijedive) referring to a single, uniqu ely 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 tio ns of addition and multiplication . Us ing the standard inn er product we may identify ~..~..~.~ _

In th e co nst ru d ion: th e o bj ect:

+ property (or another charaderistic) + of +

'1'1' 0 co ntinuity of f follows from ..... '1'1", (' x i ~ t c nc e of tes t functions is not evident.

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 int e rsection of a decreasing family of such sets is convex. '1'11('1'<: is

it

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 ... .. -~ [Nol e :

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 a problem of U!:i'llll'll"~S the condition of ellipticity

a zero of order at least 2 rauk 2. a t t he origin . y F h;LS cabrdilnalit Ie. But : F has a density g. a so ute va ue l. [U.nless 9 h;LS app eared determinant zero. I earlier; then: F has density 9. In expressions with "as" :

the hypothesis of positivity the method of proof 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:

I \

tht~

set of sol utions 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 form ula (3.G) follows from ..... Since the reg ion (2.9) is in U, we have .....

I

~ I

!

I

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.]

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 discipli ne: 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 t o B bv pushiu
1. Indicating aim or int ention:

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 1l 2 ; 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 equ a 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

= ...

Let B be a Banach space wit h n weak symplectic form w . Two random variables with a commou distribution. 8. After "to have":

Icompact support. But: F has Inn COfillite UOf111 not exceedi~g 1.. IIIP ;lcL support contallled

F h;LS finite no r111.

III

26

,illIl I1',

INFINITIVE

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

Bl1t: elements of the form f

g.]

I.

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

I

2. In constructions with "too" and "enou gh": 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. Insert (2) iuto (3) to find that .....

[= ..... alld w ' !,e t N J

IlJ

'

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 th at .. ... f.:T .. ~£~~

27

5. In const ructions like "M is assumed to be ... ..": is ass umed (ex pectcd/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 "ca n" ] 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 tll C st ru cture "for this to happen": For liti s to ha pp e n, F must be compact. [= In order th a t this happens] F r th e las t estimate to hold, it is enough to assume ..... The u for s u ch 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
Il/ier expressions with "it": II. i'j I\ ( ' ( ' (" l!i nry (usefu l/very important) to consider .... . I i. /I ".I( n il Sf' Il !:l t o s p eak of ... .. I i. j·j l.llC ' j('for of ill t erest to look at .....

n

At fil !l ll,. lll ll (,(! tl'I appears to differ from N. in two major ways: .. ... A

III O I ( ~ ~1() Jllii s ti ca ted

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 n eed !lot co nsider this case separately. ["nee I to" in affirmat ive clauses, without "to" in negative claus s; a lso no te: "we only need to consider", but: "we need only consider" J lNG-FORM

1. As the subject of a se ntence (llote the absence of "the"): R epeating the previous argument and using (3) leads to .... . Since taking symbols commutes with lifting, A is .... . Combinin g 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 d efin ition when trying to .. .. . The following theorem is the key to constructing .... . The rcason for preferring (1) to (2) is simply that .... .

Afl er "b," : Our goal (Ill cthou/ a pproach/ 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 clau se: 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 .. .. . 3. In certain expressions with "of" : He was th e first to propose a complete theory of .. ... Th ey appear to be the first to have suggesCed the now-accepted - - - - - - - - - - T he id ea ~f combining (2) anel (3fc~-tnleI'ionl---:~ -interpretation of ..... The problem cOl!sidered th ere was that of determining WF( u) for ..... \Ye use the t echnique of extending ..... 10. After certa in ve rbs: Th ese proper ties led him to suggest that .... . being very involved . This method has the disadvantage of requiring that f be positive. Th y believe to have discovered .. .. . [Note the iu fin it ive.] L::lx claims to h ave ob tained a formula for ..... Actually,S has the much stronger propert~f being COllvex. This map turns out to satisp.f .. ...

I

t".;:?

28

.

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 th at ..... [Not: "worth to no t e" ] 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: "finis hed t o 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" :

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 exi st for every x. 30

1 I I

The ideal is defined by m = . . . , it be ing u nd erstood that ... .. F be.ing ~on ti.nuous,.we ca n assu me th;1 t ..... [= Since F is .....] (It b~lI1g .m:p~ss lbl e ~o make A an d B intersect) [= slllce It IS 1m POSS Ible] [Do not write "a func tion being an element of X" if you mean "a fu nction which is a n element of X". ] 8. In expressions which can be re phrased as "the fact that X is .... ... : Note th a t M being cyclic implies F is cyclic. The probab ility of X being rational equals 1/2. In addition to f b eing 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 re place sente nces wit h s ub ject .. IV.· .. imperso nal constructions of otber languat:;es.

<J I

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

Ii5mayeasbeily said shown to have ... .. 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": Thi s ord er behaves well when 9 is act ed 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.),,: 1klore st a ti ng the result to be provcd, we give ..... Th is i: a special case of convolutions to be introduced in Chapter 8. \V, co ncl ude with two simple lemmas to be lIsed mainly in ..... QUANTIFIERS

I

" l' hAt' all open subsets of U . T l11 5 Imp les t at con alllS a 11 y wily 'tI G = 1.

I

ll r f all transforms F of the form ..... I~e t B b e tI1e co ec Ion 0 all A such that ..... /.' j.]

cil' li!l ('d at a ll points of X .

I

1111 Idl II 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 th e boundary except the origin. T he ill tegml is t a ken over all of X. all extend to a neighbourhood of U. all have their supports in U. E , F <1.ud G are all zero <1.t x. are all equal. ' 1'1, 1' , f: ex is t fu ncti ons 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 th e intervals Ar. have .....

X (not iu X) there exists an N .... . 11 11/ ' for all J <1.1ll1 g, for a ny 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 .... . Fvcr y illvari
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 comp leted, so whenever it is convenient, we may assum th i,l /Ill y give ll meas ure is complete. T hc r fJ i ~l a !i ubsequence such that ..... Tlt m'o cx i !ll!l an x with .... . . . [O/: tll ere ex ists x, but: there is an x] Th cr e a r c 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 tl y O ll C A/ at most one A). Not th at some of the Xu may be repeated.

F has 11 0 fi xed vecto r (no pole) in U. [Or: no pol es] 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 ca n be see n 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 co nvex.] 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 tu em. Either f or !l must be bo u nded . Ld 11 and v be two distributions neither of which is ..... [Use "neither" WhCll there arc iwo alternatives.] This is true for n ei th er of the two functi ons. 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. t1

I ..

18 lllll tl-Illl

j

ex

'tl WI

1

I all entries zero except the kth which is one 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 conce ntrated 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: 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.

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. 3. Fractions: 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- pl a);er is- haW(one third tas- fast:We divide J in half. All sides were increa.sed by the same proportion. About 4 0 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.

I

. se t Ila.s fewer elements than f( has . TI11S no fewer than twenty elem ents. F can have no jum ps exc eeding 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 th an 25. Let an be a sequence of positive integers none of which 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 h a n a third of the distance between .....

IS

one less

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

34

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 rece ntly less and less used. A s uccess ian af more and more refined discrete mod els. 6. How many tim es as great: twi ce (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 twic e as m a ny elements as B has. .J mllt.;tiIl S a su binterval of half its leng th ill which ..... 1\ 11 :\.'; f')IIl' lillies the rad ius of B. Till' d iall ll'lcr of L is 11k tim es (twice) that of M.

r. MIJi Lip l

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

9. Man y, few, a nUl11 be r of: a large number of illustrations. T h re are only a finite number of f with Lf = l. [N te the a small number of exceptians. plural.] an infinite number of sets ..... a negligible number of points with ..... Ind c is th e numb e r 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

The thearem is fairly genera!. Th ere 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 describ e 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]

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

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

"'I

for tit m os t part , o.nly sketched. Lt is lIlet had will prove useful in .. ... \Vhitt Ill os t ill terest us is whet her .....

T it " fI' Ol)r::

a l l!,

Mwd. pl' (J !) lI bl y,

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 thatvar 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 th e functians which occur in (3). T here exists a smallest algebra with this praperty. Find the second largest clement in the list L.

10. Equ ality, difference: A equals B 01' A is equal to B [Not: "il. is equal B"] The Laplacian af 9 is 4r > O. 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 linea r 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 id entically O. Let a, band c be distinct complex numbers. Each w is pz far precisely Tn distinct values of ::.

r.:;r "'~.K

36

= S.

37

functiolls which are eq,I;11 :l.e. illtegra tion is concerned.

,HC

indistinguish:Lble as felr a.s

11. Numbering: Exercises :2 to 5 furuish l,tbcr applic;1tions of this tt>cllli:'lIW. lAma.: Exercises :2 th;'ough 5] the deri\-ati,,'cs \lp to order k ill the third and fourth ro\\'s the odd-numbered terms from lin e 16 onwards in Jines Hi-10 the next-to-bst CO!Ulllll the la.st par;1graph but one of the pre\'ious proof

1-11

~t . ~-- . in the (i,j) entry
Tl'

.

liS IS

Iquoted hinted in Sectio!!s 1 ;1nd :2. on page 3G of [.1]. ilt

HOW TO AVOID REPETITI ON

1. Repetition of nouns : 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 pap er. 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
We may replace A and B by whicheve r is the larger of the two. [Not : "the two ones"] This inequality applies to cond itio nal 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 F i . 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 co ns idered by 13(ln
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 o r ..... Th is wi ll ho ld fo r x > 0 if it does for x = O. T he iu t gritl migh t not converge, but it does so after .... . No te th a t w have not required that ..... , and we shall not do so ex 'c pt wli '11 exp licitly stated. ~

,,~

''i'

38

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 .....

There has since been little systematic work Oil .... . It has recently been pointed out by Fix that .... . It is sometim es difficult to ..... This usually implies further conclusions about f. It often does not matter whether .. ...

'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 cas e 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 tru e 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 plan e. N()I.(~ t hat this is not equivalent to ... .. INote t he differeuce between " this:' and "it" : you. say "it i.s :lOt <'
I",·. t.Il1 '

s t. a t. ed (rl'sired/cl a imed) properties. WORD ORDER

General r emarks: The normal order is: subject

+

verb

+

d irect ohject

+

,,, h 'erbs ill

the order manner-plac e-t illle. Adverbial cla uses Cnn also be pbceJ at the beginning of a sentence, and SOllie adverbs a iways come betwee n 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 :w xiliary • I I 'qu ell yadverbs:

Thi s has a lr eady been proved in Sectio n 8. Th is res ult will now be deri ved computationally. Every measurable subset of X is again a measure space. Vie first prove a reduced form of the theorem .

• Adverbs like "also", "therefore", "thus": Our The Oue C is

presentatiou is ther efo re organized ill such a way that .... . sum in (2), though form ally infinite, is therefore actually finite . must therefore also introduce the class of ... .. connected and is therefore not the union of ... .. These properties , wit h the exception of (1), also hold for t. \Ve will also leave to the reader th e 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 max imum 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 .... . M uch 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

~~~

40

41

Ie. After an object if it is sh ort:

\ \<.: 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,' shows (h a'~ j =!l. [:\'at: "At bst"] Ncv-erl he kss, it tur ns t)ut that -:-.. .. -~----~ - - - - - - - - - - - - - - Nex t, let V Lc the \'('ct()[ Sp:1CC of .... .

en

T'.Iore precisely, Q consists of .... . Explicitly (Intuitively), this means that .... . Needl ess 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 suit ed 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 ow ing. It reduccs to the statement that only for tbe distributio n F do th e 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, the n ..... 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: 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 claus es: The observed values of X will on the a verage cluster around .. ... This could in principle imply an n.dvantage . f or simplicity, we will for the time being a ccept as F oniy C? milps .

-- -

Accordingly we are in effec t dealing with .. ... Th e k nowledge 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 circl e the density given by .. ... 3. INVERSION AND OT HER 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 reverse d . 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 e ithel' 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 fa c t th a t for no X does Fx coutain y impli~s that .... .

/-:;r

';:>-~-;

42

43

III

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

11 0

3d . In ve rsion-other examples:

F is compact a nd so is G.

If f, 9 arc measurable. then so are f

o

1 £ n y or

f - 1 -

+ 9 and f . g .

Ican one expect to obtain ..... does that limit exist.

By far the most important is the case where ..... wlu ch more subtle are the following results of John. l';sse ll t ia l to th e proof are certain topological properti es of III. o n lin g soo ner than in some other languages:

/':q lt nlity occ urs ill (1) iff f is cons tant. Til n a 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 impo rtant that the tails of F and G are of comparable magnitude, " Ht:lt lII e nt 1l1 ade more precise by the following inequalities .

WHERE TO INSERT A COMMA

mits, makes more precise ) some part of a sentence. Put commas before and after non-d efining clauses (i .e. on es 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 mbo ls.

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. - 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:

---

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

44

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) .

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

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 (li-

l. Comma not required : We s llall now prove that f is proper. The fact that f has radi2..1 limits was pro"ecl in [4J. It is reasona ble to ask wllether this Ilolds for 9 = 1.AI is til e se t of all maps which take values in V. Th ere is a polynomial P such that P f = g. T he clement give n 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.

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 .....

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

• f .)l liJj 'c t

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

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 ..... Int egrating 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 ERROR S 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 " : 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 m uch the same way as in Sec. 2

f = 0 thCll M is closcd ---; Si nce f = 0, M is closed [Or: Since /=0, wc conclude t hat M is cl osed ] .. ... as it is shown in Sec. 2...,... ..... , as is shown ill Sec. 2

Sillce

J:: ', u)' function being an e leme nt of X is co nvex --+ [,'cry fUllction which is an element of X is convex S.-,ttillg 11 =]/, the equation can be .. .. . --+ Se tting 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 te me nt} [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 comp letes 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 5. Other exa m ples :

f

--+

We denote tbis m a p by

f

have (obtain} that [J is .... . ....... We sec (conclllcie/cleullce/fimi/ infer} that [J is .... . \Ve arc done --+ The proof is completc,

W(!

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

4G

1'(

INDEX 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 fe w, :.1 7, 47 a ll, 32 a lso , 41 a nu m be r 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, 4 7 :tv id , :.10

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

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

d 'n o te , 7

largest, :.16 last but one, :.18 lat ter, :.19, 40 lC
d e pe nd in g o n , tl d ill('l' , :Hi , 'J7 c1i njo in t fro lll , 4G d is t in c t , :)7

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

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

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 in g, 26 , 38

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

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

48

"obtain that" ~15 16, 4, of, 25, 26, 29' . one, 23, 38 only, 29, 42 ordinal numb e rs, 25, 34, 38 p aragraph, 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 th e, 24 the one, :.Itl th e reforc~ , 1;;, ·11, '1
ullion, 25 unique, 24, 34 unlike ly , 28

u pTo , :35 , :>8 - - - - - what, 15, 18, 47 w hi ch, 15, Hi , 47 with,2 6 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