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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 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 cOIltcllli:'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 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 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: 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 ( ,