Theorem formulation - Ph´ at biê ,
501 If and if , then 501.khi v`a khi, khi ¯d´o
504 Assume that 504.Gi, a thiê´t r`ang ˘
507 Then unlessm= 1 507.Khi ¯d´o tr`u, khim= 1.
508 Then withg a constant satis- fying
508.Khi ¯d´o v´o, ig m ôt h`˘ang sô´ th, oa m˜an
510 In fact, [=To be more precise] 510.Th u, c tê´, [=s˜e l`a ch´ınh x´ac ho, n]
511 Accordingly, [=Thus] 511.Cho nên [=Thus]
512.Cho f 6= 1 bâ´t k`y gi, a s, u, r`˘ang Khi ¯d´o
513 Let P satisfy the hypotheses of
513.Cho P th, oa m˜an gi, a thiê´t Khi d´¯o
514 LetP satisfy the above assump- tions Then
514.Cho P th, oa m˜an gi, a thiê´t trên. Khi ¯d´o
515 LetP satisfyN(P) =l Then 515.Cho P th, oa m˜an N(P) = l Khi ¯ d´o
516.Cho gi, a thiê´t 1-5 ¯d´ung Khi ¯d´o
517 Under the above assumptions, 517.Du,´o,i gi, a thiê´t trên,
518 Under the same hypotheses 518.Du,´o,i nh˜u,ng gi, a thuyê´t n`ay
519 Under the conditions stated above,
519.Du,´o,i nh˜u,ng ¯diê`u ki.ên ¯d˜a ph´at biê, u , o,trên,
520 Under the assumptions of The- orem 2 with "convergent" re- placed by "weakly convergent",
520.Du,´o,i gi, a thiê´t c, ua Ð.inh l´y 2 v´o, i
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 6 Theorem] [M uc l uc]
521 Under the hypotheses of Theo- rem 5, if moreover
521.Du,´o,i gi, a thuyê´t c, ua Ð.inh l´y 5, nê´u ho, n thê´
522 Equality holds in (8) if and only if
522.Ð, ˘ ang th´u, c ¯d´ung tring (8) khi v`a ch, ı khi
523 The following conditions are equivalent:
523.Nh˜u, ng ¯diê`u ki ên sau ¯dây l`a tu, o,ng du¯ , o,ng,
Theorem introductory - Gi´ o ,
524 We have thus proved 524.Ta ¯d˜a ch´u, ng minh ¯diê`u n`ay
525 Summarizing, we have 525.Tô´m l ai, ta c´o
526 We can now rephrase Theorem
526.Ta c´o thê, nh´˘ac l ai Ð.inh l´y 8 nhu, sau.
527 We can now state the analogue of
527.Bây gi`o, ta c´o thê, ph´at biê, u tu, o, ng t u,
528 We can now formulate our main results.
528.Ta c´o thê, ph´at biê, u nh˜u, ng kê´t qu, a ch´ınh c, ua ta.
529 We are thus led to the following strengthening of Theorem 6:
529.Ch´ung ta nhu, v ây d˜ân ¯dê´n ¯d ô d`ai c, ua Ð.inh l´y 6.
530 The remainder of this section will be devoted to the proof of
530.Phâ`n c`on l ai c, ua ¯do an n`ay s˜e d`anh cho ch´u, ng minh c, ua
531 The continuity of A is estab- lished by our next theorem.
531.T´ınh liên t uc c, ua A du¯ ,
.o, c li êt kê b, o, i ¯d.inh l´y sau ¯dây c, ua ch´ung tôi.
532 The following result may be proved in much the same way as
532.Nh˜u,ng kê´t qu, a sau ¯dây c´o thê, du¯ ,
.o,c ch´u,ng minh c`ung m ôt c´ach nhu, Ð.inh l´y 6.
533 Here are some elementary prop- erties of these concepts.
O, dây l`¯ a nh˜u, ng t´ınh châ´t co,b, an c, ua nh˜u, ng kh´ai ni êm n`ay.
534 Let us mention two important consequences of the theorem.
534.Ta kê, ra ¯dây hai h ê qu, a quan tr ong c, ua Ð.inh l´y.
535 We begin with a general result on such operators.
535.Ta b´˘at ¯dâ`u v´o, i m ôt kê´t qu, a tô, ng qu´at trên nh˜u,ng to´an t, u,n`ay.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 6 Theorem] [M uc l uc]
Theorem remarks - Nh˜ u ,
536 This theorem is an extension
(a fairly straightforward gener- alization/a sharpened version/a refinement) of
536.Ð.inh l´y n`ay l`a m ôt m, o, r ông (m ôt tô, ng qu´at h´oa tr u, c tiê´p/phiên b, an h`ınh th´u,c kh´ac/m ôt tinh tê´ho,n)
537 This theorem is an analogue of
537.Ð.inh l´y n`ay tu, o, ng t u, nhu,
(restatement) of in terms of
538.Ð.inh l´y n`ay l`a m ôt ph´at biê, u l ai c, ua trong thu ât ng˜u, c, ua
539 This theorem is analogous to 539.Ð.inh l´y n`ay tu,o, ng t u,v´o,i
540 This theorem is a partial con- verse of
541 This theorem is an answer to a question raised by
541.Ð.inh l´y n`ay l`a m ôt câu tr, a l`o, i câu h, oi ¯du, a ra b, o, i
542 This theorem deals with 542.Ð.inh l´y n`ay liên quan v´o,i
543 This theorem ensures the exis- tence of
543.Ð.inh l´y n`ay kh, ˘ang ¯d.inh s u, tô`n t ai c, ua
544 This theorem expresses the equivalence of
544.Ð.inh l´y n`ay tr`ınh b`ay s u, tu, o, ng du¯ , o, ng c, ua
545 This theorem provides a crite- rion for
545.Ð.inh l´y n`ay ¯du, a ra m ôt tiêu chuâ, n cho
546 This theorem yields information about
546.Ð.inh l´y n`ay l`am tô´t thông tin vê`
547 This theorem makes it legiti- mate to apply
547.Ð.inh l´y n`ay l`am h o, p ph´ap h´oa cho ´ap d ung
548 The theorem states (as- serts/shows) that
548.Ð.inh l´y ph´at biê, u r`˘ang
549 Roughly (Loosely) speaking, the formula says that
549.M ôt c´ach m.anh m˜e, công th´u, c ph´at biê, u r`˘ang
550 When f is open, (3.7) just amounts to saying that
, (3.7) ¯d´ung c´o ngh˜ıa dê¯ , n´oi r`˘ang
551 When f is open, (3.7) just amounts to the fact that
, (3.7) ¯d´ung c´o ngh˜ıa dê¯ , kh, ang ¯˘ d.inh r`˘ang
552 Here is another way of stating
553 Another way of stating (c) is to say:
553.M ôt c´ach kh´ac ph´at biê, u (c) n´oi l`a
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 6 Theorem] [M uc l uc]
554 An equivalent formulation of (c) is:
554.M ôt ph´at biê, u tu,o,ng ¯du,o,ng c, ua (c) l`a
555 Theorems 2 and 3 may be sum- marized by saying that
555.Ð.inh l´y 2 v`a 3 c´o thê, t´om l ai b`˘ang c´ach n´oi r`ang ˘
556 Assertion (ii) is nothing but the statement that
556.Kh, ˘ang ¯d.inh (ii) không l`a g`ı nhu, ng m ênh ¯dê` ph´at biê, u r`˘ang
557 Geometrically speaking, the hy- pothesis is that : part of the conclusion is that
557.N´oi m ôt c´ach h`ınh h oc, gi, a thuyê´t n`ay l`a : m ôt phâ`n c, ua kê´t lu ân l`a
558 The interest of the lemma is in the assertion (that it allows one to )
558.Quan tâm c, ua bô, dê` l`¯ a trong kh, ˘ang ¯d.inh
559 The principal significance of the lemma is in the assertion (that it allows one to )
559.Dâ´u hi êu nguyên l´y c, ua bô, dê` l`¯ a trong kh, ˘ang ¯d.inh
560 The point of the lemma is in the assertion (that it allows one to
560.Ch, ı ra bô, dê` n`¯ ay l`a trong kh, ˘ ang d.inh ¯
561 If we take f = we recover the standard lemma ([7, Theorem
561.Nê´u ta lâ´yf =ta lâ´y l ai bô, dê` co¯ , b, an ([7, Ð.inh l´y 5]).
562 Replacingf by −f, we recover the standard lemma ([7, Theo- rem 5]).
562.Thay f b`˘ang −f, ta ph uc hô`i l.ai bô, ¯ dê` co, b, an ([7, Ð.inh l´y 5]).
563 This specializes to the result of
563.Nh˜u, ng ¯d ˘ac bi.êt h´oa cho kê´t qu, a c, ua [7] nê´uf =g.
564 This result will be needed in Sec- tion 8.
565 This result will prove extremely useful in Section 8.
565.Kê´t qu, a n`ay s˜e ch´u, ng minh cuô´i c`ung c´o ´ıch trong ¯do an 8.
566 This result will not be needed until Section 8.
566.Kê´t qu, a n`ay ¯d˜a không câ`n t´o, i do an 8.¯
567 It is worth pointing out that 567.дang ch, ı ra r`˘ang
568 Theorem 2.1 still holds if 568.Ð.inh l´y 2.1 v˜ân ¯d´ung nê´u
569 Theorem 2.1 remains valid if 569.Ð.inh l´y 2.1 v˜ân c`on hi.êu l u, nê´u
570 Theorem 2.1 remains in force if
570.Ð.inh l´y 2.1 v˜ân c`on hi.êu l u,nê´u
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 6 Theorem] [M uc l uc]
571 It is to be noticed that 571.Câ`n ch´u ´y l`a
572 It is worth noticing that 572.Câ`n ch´u ´y l`a
573 The statement of Theorem 2.1 here is a litle more general than in [10]
573.C´ach ph´at triê, n c, ua Ð.inh l´y 2.1 , o, dây tô¯ , ng qu´at ho, n m ôt ch´ut so v´o,
574 Under hypothesises weaker than those above, one obtains a correspondingly weaker conclu- sion.
574.V´o, i nh˜u, ng gi, a thiê´t nh e ho, n gi, a thiê´t trên, ngu,`o,i ta nh.ân ¯du,
.o, c kê´t qu, a yê´u ho, n tu, o, ng ´u, ng.
575 The results of Theorem 2.1 ex- tend and generalize some of the corresponding results in finite dimensional situations, see Ref
575.Kê´t qu, a c, ua Ð.inh l´y 2.1 m, o, r ông v`a tô, ng qu´at m ôt sô´kê´t qu, a tu, o, ng u´, ng trong tru,`o,ng h.o, p h˜u, u h˜an chiê`u, xem t`ai li êu d˜ân [6].
576 Note that Proposition 2.1 does not hold without the assumption
576.Lu, u ´y r`˘ang M ênh ¯dê` 2.1 không c`on ¯d´ung khi thiê´u gi, a thiê´t
577 The cause of the failure is that
578 Assumption (ii) of Theorem 3.2 is not as restrictive as it might appear We can give a sufficient condition which ensures the sat- isfaction of this assumption.
Giá trị của việc áp dụng điều kiện đúng trong các nghiên cứu không chỉ giúp hiểu rõ hơn về hiện tượng mà còn tạo ra cơ sở vững chắc cho các kết luận Điều này cho phép chúng ta nhận diện và phân tích các yếu tố ảnh hưởng, từ đó đưa ra những giải pháp hiệu quả hơn Việc đảm bảo tính chính xác trong các điều kiện nghiên cứu là yếu tố quan trọng để nâng cao độ tin cậy của kết quả.
579 The condition (2) has the advan- tage not only in being weaker, but in giving an explicit charac- terization
579.Ðiê`u ki ên (2) c´o u,u ¯diê, m không ch, ı , o, ch˜ô n´o yê´u ho,n, m`a c`on o,,ch˜ô cho ta s u, d ˘ac tru¯ ,ng tu,`o,ng minh
580 Theorem 1 states one of the most important properties of
580.Ð.inh l´y 1 nêu lên m ôt trong nh˜u, ng t´ınh châ´t quan tr ong nhâ´t c, ua
581 There are certain unanswered questions regarding
581.C´o m ôt sô´ vâ´n ¯dê` c`on m, o, liên quan t´o,i
582 Two interesting questions can be raised for further study
582.Hai câu h, oi th´u v.i c´o thê, d ˘at ra ¯dê¯ , nghiên c´u, u xa ho, n
583 It is an open question whether a weaker condition can be found to imply
583.Vâ´n ¯dê` c`on m, o, l`a li êu c´o thê, t`ım du¯ , o, c ¯diê`u ki ên yê´u ho, n ¯dê, k´eo theo
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 6 Theorem] [M uc l uc]
584 For the method to be of interest, it is essential that
584.Ðê, phu,o,ng ph´ap ¯du,
585 In the interest of simplicity we have avoided the most general presentation and restricted our attention to
585.V`ı muô´n cho ¯do, n gi, an, ta tr´anh c´ach tr`ınh b`ay tô, ng qu´at nhâ´t v`a h an chê´s u,ch´u ´y c, ua ta vê`
586 We now examine how the gen- eral case can be treated.
586.Bây gi`o, ta x´et tru,`o,ng h.o, p tô, ng qu´at ¯du,
.o, c lu ân gi, ai nhu, thê´n`ao.
587 We refer to Demyanov [5] for a more general and detailed dis- cussion of this problem.
587.Xin tham kh, ao Demyanov [5] mong muô´n b`an lu ân tô, ng qu´at v`a chi tiê´t ho, n vê` vâ´n ¯dê` n`ay.
588 We would like to point out an- other consequence of Theorem
2 in passing, even though it has nothing to do with the main topic of this section.
588.Ch´ung tôi muô´n nêu r˜o nhân ¯dây m ôt h.ê qu, a kh´ac c, ua Ð.inh l´y 2, m ˘ac d`u n´o không d´ınh d´ang g`ı dê´n ch¯ , u ¯dê` ch´ınh c, ua ¯do an n`ay.
589 The most successful algorithm to date is due to Newton.
589.Thu ât to´an h˜u, u hi êu nhâ´t cho ¯dê´n nay l`a do Newton ¯du, a ra.
590 The advantage of this algorithm is that it allows the method to be used in the case
590.U, u ¯diê, m c, ua thu ât to´an n`ay l`a , o, ch˜ô n´o cho ph´ep s, u, d ung phu, o, ng ph´ap ¯d´o trong tru,`o,ng h.o, p
591 At about the same time that the paper of E Brown appeared, A.
591.Gâ`n nhu,c`ung m ôt l´uc xuâ´t hi.ên b`ai b´ao c, ua E Brown, th`ı A Black d˜¯a x´et
592 For other essential results, pe- ruse Ref 5 or Ref 8.
592.Vê` nh˜u, ng kê´t qu, a cô´t yê´u kh´ac, h˜ay ¯d oc k˜y t`ai li.êu 5 ho.˘ac t`ai li.êu 8.
593 We conclude this paper with the following comments.
593.Ta kê´t th´uc b`ai b´ao n`ay v´o, i nh˜u, ng nh ân x´et sau.
594 Several comments are called for in connection with
594.M ôt v`ai l`o, i b`ınh lu ân câ`n ¯du,
595 Introductory remarks 595.Nh ân x´et m, o, dâ`u¯
596 Concluding remarks 596.Nh ân x´et kê´t lu.ân
597 We close the paper with 597.Ta kê´t th´uc b`ai b´ao b`˘ang
598 We conclude the paper with 598.Ta kê´t th´uc b`ai b´ao b`˘ang
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 6 Theorem] [M uc l uc]
CH U Ð` Ê 7 Proofs - Ch´ u , ng minh
Proof Arguments - C´ ac l âp lu.ân ch´u ,
599 By definition, 599.Theo ¯d.inh ngh˜ıa,
600 By the definition of 600.Theo ¯d.inh ngh˜ıa c, ua
601 By assumption, 601.Theo g, ai thiê´t,
602 By the compactness of 602.Theo t´ınh ch ˘at ch˜e c, ua
603 By Taylor’s formula, 603.Theo công th´u, c Taylor,
604 By a similar argument, 604.B`˘ang l âp lu.ân tu, o, ng t u,
605 By the above, 605.Theo ¯diê`u , o, trên,
606 By the lemma below, 606.Theo bô, dê` du¯ ,´o,i ¯dây,
607 By continuity, 607.Theo t´ınh liên t uc,
608 Butf =g, which follows from 608.Nhu,ng f = g, n´o ¯du,
609.Nhu, ngf =g nhu, d˜¯a ¯du, o, c mô t, a trong
610 Theorem 4 now shows that 610.Bây gi`o, d.inh l´y 4 ch¯ , ı ra r`˘ang
611 Theorem 4 now yields (gives/ implies)f =
611.Ð.inh l´y 4 cho (¯du, a ra/k´eo theo) f =
612 Theorem 4 now leads tof = 612.Ð.inh l´y 4 d˜ân ¯dê´nf =
614 Since f is compact, we have
615 Since f is compact, it follows thatLf = 0.
615.V`ı f l`a compact, ta suy ra r`˘ang
616 Sincef is compact, we see (con- clude) thatLf = 0.
617 ButLf = 0sincef is compact 617.Nhu,ngLf = 0v`ıf l`a compact.
619 We must haveLf = 0, for other- wise we can replace
619.Ta ph, ai c´o Lf = 0, tru,`o,ng h.o, p ngu,
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 7 Proofs] [M uc l uc]
44 Ch, u ¯dê` 7 Proofs - Ch´u, ng minh
620 Asfis compact we haveLf = 0 620.V`ıf l`a compact ta c´oLf = 0.
621 ThereforeLf = 0by Theorem 6 621.V`ı v âyLf = 0theo Ð.inh l´y 6.
622 ThatLf = 0follows from Theo- rem 6.
622.Ðiê`uLf = 0suy ra t`u, Ð.inh l´y 6.
625 From what has already 625.T`u, nh˜u, ng g`ı ¯d˜a c´o
626 From been proved, 626.T`u, nh˜u, ng ¯diê`u ¯d˜a ch´u, ng minh
627 we conclude (deduce/see) that
627.Ta kê´t lu ân r`˘ang
628 we have (obtain)M =N 628.Ta c´o (nh ân ¯du,
629 it follows that 629.Suy ra r`˘ang
630 it may be concluded that 630.C´o thê, kê´t lu ân r`˘ang
631 According to (On account of) the above remark, we haveM N.
631.Theo ch´u ´y , o, trên, ta c´oM =N.
632 It follows thatM =N 632.Suy raM =N.
634 [hence = from this; thus = in this way; therefore = for this reason; it follows that = from the above it follows that]
635 This givesM =N 635.Ðiê`u n`ay choM =N.
636 We thus getM =N 636.Nhu, v ây ta nh.ân ¯du,
637 The result isM =N 637.Kê´t qu, a l`aM =N.
638 (3) now becomesM =N 638.(3) tr, o,th`anhM =N.
639 This clearly forcesM =N 639.Ðiê`u n`ay r˜o r`ang choM =N.
640 Fis compact, and soM =N 640.Fl`a compact, v`a nhu, v âyM =N.
642 F is compact, and, in conse- quence,M =N.
643 F is compact, and hence bounded,
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 7 Proofs] [M uc l uc]
644 F is compact, which gives (im- plies/ yields) M = N [Not:
645 F =G=H the last equality be- ing a consequence of Theorem
645.F =G=H d¯, ˘ ang th´u,c cuô´i c`ung l`a h ê qu, a c, ua Ð.inh l´y 7.
646 F =G=H which is due to the fact that
648 We conclude from (5) that , hence that , and finally that
649 The equalityf =g, which is part of the conclusion of Theorem 7, implies that
649.Ð, ˘ ang th´u, c f = g, m`a n´o l`a m ôt phâ`n c, ua kê´t lu ân c, ua Ð.inh l´y 7, k´eo theo
650 As in the proof of Theorem 8, equation (4) gives
650.T`u,trong ch´u,ng minh c, ua Ð.inh l´y
8, phu, o,ng tr`ınh (4) cho
651 Analysis similar to that in the proof of Theorem 5 shows that
651.Phân t´ıch tu, o, ng t u, ¯ diê`u n`ay trong ch´u, ng minh ¯d.inh l´y 5 ch, ı ra r`˘ang
652 A passage to the limit similar to the above implies that
652.Ðu,`o,ng lô´i ¯di qua gi´o,i h.an tu,o,ng t u, nhu, trên k´eo theo r`˘ang
653 Similarly (Likewise), 653.Tu, o, ng t u, (Giô´ng nhu,
654 Similar arguments apply to the case
654.Lya lu ân tu, o, ng t u, ´ap d ung cho tru,`o,ng h.o, p n`ay
655 The same reasoning applies to the case
655.C`ung m ôt suy lu.ân k´eo theo tru,`o,ng h.o, p n`ay
656 The same conclusion can be drawn for
656.C`ung kê´t lu ân c´o thê, ph´at biê, u cho
657 This follows by the same method as in
657.Ðiê`u n`ay suy ra b`˘ang c`ung m ôt phu, o, ng ph´ap nhu, trong
658 The termT f can be handled in much the same way, the only dif- ference being in the analysis of
658.Th`anh phâ`nT f c´o thê, n´˘am ¯du,
.o, c b`˘ang c`ung c´ach, ch, ı ¯du, o, c kh´ac nhau trong phân t´ıch c, ua
659 In the same manner we can see that
659.C`ung m ôt kiê, u ch´ung ta c´o thê, thâ´y r`˘ang
660 The rest of the proof runs as be- fore.
660.Phâ`n c`on l ai c, ua ch´u, ng minh th u, c hi ên nhu, tru,´o,c ¯dây.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 7 Proofs] [M uc l uc]
46 Ch, u ¯dê` 7 Proofs - Ch´u, ng minh
661 We now apply this argument again, withI replaced by J, to obtain
661.Ta ´ap d ung l´y lu.ân n`ay m ôt lâ`n n˜u, a, v´o,i I thay cho cho J, nh ân du¯ , o, c
Proof Begin - B´ ˘ at ¯ dâ`u ch´ u ,
662 We (Let us) first prove
662.Tru,´o,c tiên ch´u,ng minh r`˘ang
663 We (Let us) first prove a reduced form of the theorem.
663.Tru,´o,c tiên ch´u,ng minh m.ôt d ang r´ut g on c, ua ¯d.inh l´y.
664 We (Let us) first outline (give the main ideas of) the proof.
664.Ðâ`u tiên ta ph´ac th, ao ch´u, ng minh
665 We (Let us) first examineBf 665.Tru,´o,c tiên ta xem x´etBf.
666 ButA =B To see (prove) this, letf =
666.Nhu, ng A = B Ðê, ch´u, ng minh diê`u ¯¯ d´o, chof =
667 ButA=B We prove this as fol- lows.
667.Nhu, ng A = B Ta ch´u, ng minh diê`u ¯¯ d´o nhu, sau.
668 But A = B This is proved by writingg=
668.Nhu, ng A = B Ðiê`u n`ay ¯du,
.o, c ch´u, ng minh b`˘ang viê´tg=
669 We first computeIf To this end, consider
669.Tru,´o,c tiên ta t´ınhIf Ðê, kê´t th´uc diê`u ¯¯ d´o, ta x´et
670 We first computeIf [=For this purpose; not: "To this aim"]
670.Tru,´o,c tiên ta t´ınhIf [=cho m uc d´ıch n`¯ ay; không d`ung "To this aim"]
671 We first computeIf To do this, take
671.Tru,´o,c tiên ta t´ınhIf Ðê, l`am ¯diê`u ¯ d´o, ta lâ´y
672 We first compute If For this purpose, we set
672.Tru,´o,c tiên ta t´ınh If Cho m uc d´ıch ¯¯ d´o, ta ¯d ˘at
673 To deduce (2) from (3), take 673.Ðê, gi, am (2) t`u,
674 We claim that Indeed, 674.Ta x´ac nh ân th.ât s u, d´¯ung,
675 We begin by proving (by re- calling the notion of )
675.Ta b´˘at ¯dâ`u b`˘ang ch´u, ng minh
676 Our proof starts with the obser- vation that
676.Ch´u, ng minh c, ua ch´ung ta xuâ´t ph´at v´o, i nh ât x´et r`˘ang
677 The procedure is to find 677.Th, u t uc l`a t`ım
678 The proof consists in the con- struction of
678.Ch´u,ng minh b`˘ang xây d u,ng c, ua
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 7 Proofs] [M uc l uc]
679.Ch´u, ng minh tr u,c tiê´p.
680 The proof is by induction onn 680.Ch´u, ng minh b`˘ang quy n ap theon
681 The proof is left to the reader 681.Ch´u, ng minh ¯dê, l ai cho b.an ¯d oc.
682 The proof is based on the follow- ing observation.
682.Ch´u, ng minh d u, a trên co, s, o, nh ân x´et sau ¯dây.
683 The main (basic) idea of the proof is to take
683.Tu, tu,, o, ng co, b, an c, ua ch´u, ng minh l`a lâ´y
684 The proof falls naturally into three parts.
.o,c chia th`anh ba phâ`n.
685 The proof will be divided into 3 steps.
685.Ch´u, ng minh s˜e chia l`am ba bu,´o,c.
686 We have divided the proof into a sequence of lemmas,
686.Ta chia ch´u, ng minh th`anh d˜ay c´ac bô, dê`.¯
687 Suppose the assertion of the lemma is false.
687.Gi, a s, u, kh, ˘ang ¯d.inh c, ua Lemma l`a sai.
688 Suppose , contrary to our claim, that
688.Gi, a s, u, ngu, o, c l ai v´o,i ph´at biê, u c, ua ch´ung ta l`a
689 Conversely (To obtain a contra- diction), suppose that
690 On the contrary, suppose that 690.Ngu,
691 Suppose the lemma were false.
Bô, dê` ¯¯ d˜a sai Khi ¯d´o ta c´o thê, t`ım
692 If there existed an x , we would have (there would be
693 If x were not in B, we would have (there would be )
693.Nê´uxkhông thu ôcB, ta s˜e c´o
694 If it were true that , we would have (there would be )
694.Nê´u ¯diê`u không ¯d´ung l`a , ta s˜e c´o
695 Assume the formula holds for the degreek; we will prove it for k+ 1.
695.Gi, a s, u, công th´u, c ¯d´ung cho b âck, ta s˜e ch´u, ng minh ¯d´ung chok+ 1.
696 Assuming (3) to hold for k, we wrill prove it fork+ 1.
696.Gi, a thiê´t (3) ¯d´ung cho k, ta s˜e ch´u, ng minh ¯d´ung v´o, ik+ 1.
697 We give the proof only for the casen = 3; the other cases are left to the reader.
697.Ta ch, ı ch´u,ng minh cho tru,`o,ng h.o,p n= 3; tru,`o,ng h.o,p kh´ac ¯dê, l ai cho b an ¯d oc.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 7 Proofs] [M uc l uc]
48 Ch, u ¯dê` 7 Proofs - Ch´u, ng minh
698 We only give the main ideas of the proof.
698.Ta ch, ı ¯du,a ra tu,tu,,o,ng ch´ınh c, ua ch´u,ng minh.
Proof Conclusion - Kê´t lu ân c ,
699 , which (This) proves the the- orem.
699 , ¯diê`u ¯d´o ch´u, ng minh ¯d.inh l´y.
700 , which (This) completes the proof.
700 , ¯diê`u ¯d´o ch´u,ng minh ¯dâ`y ¯d, u.
701 , which (This) establishes the formula.
701 , ¯diê`u ¯d´o thiê´t l âp lên công th´u,c.
702 , which (This) is the desired conclusion.
702 , ¯diê`u ¯d´o r´ut ra kê´t lu ân.
703 , which (This) is our claim
703 , ¯diê`u ¯d´o l`a kh, ˘ang ¯d.inh c, ua ch´ung ta.
704 , which (This) gives (4) when substituted in (5) (combined with (5)).
704 , (Ðiê`u n`ay ) cho (4) khi thay thê´v`ao (5) ( kê´t h o, p v´o, i (5 ) )
705 , and the proof is complete 705 , v`a ch´u, ng minh ¯dâ`y ¯d, u.
706 , and this is precisely the as- sertion of the lemma,
706 , v`a ¯d´o l`a ch´ınh x´ac kh, ˘ang ¯d.inh c, ua bô, dê`.¯
707 , and the lemma follows 707 , v`a bô, ¯dê` suy ra.
709 , and f = g as claimed (re- quired).
709 , v`af =gnhu, d˜¯a nêu (nhu, dô`i¯ h, oi).
710.Ðiê`u n`ay tr´ai v´o, i gi, a thiê´t c, ua ch´ung ta.
713 ., which contradicts the maxi- mality of
713 , n´o tr´ai v´o, i t´ınh c u, c ¯d ai c, ua
715 The proof forGis similar 715.Ch´u, ng minh c, uaGtu, o, ng t u,
716 Gmay be handled in much the same way.
716.Gc´o thê, th u, c hi ên theo c`ung m ôt c´ach.
717.Xem x´et tu,o, ng t u,´ap d ung choG.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 7 Proofs] [M uc l uc]
718 The same proof works (remains valid) for
718.Ch´u, ng minh nhu, v ây th u, c hi ên cho
719 The same proof still goes (fails) when we drop the assumption
719.Ch´u, ng minh c`ung c´ach v˜ân c`on d´¯ung khi m`a ta b, o gi, a thiê´t
720 The method of proof carries over to domains
720.Phu, o, ng ph´ap ch´u, ng minh d u, a trên miê`n x´ac ¯d.inh
721 The proof above gives more, namely /is
721.Ch´u, ng minh trên cho ho,n thê´, ta d ˘at tên ¯
722 A slight change in the proof ac- tually shows that
722.Nhâ´n m anh s u, thay ¯dô, i ch´u, ng minh ch, ı ra r`˘ang
Proof Sufficient - Ch´ u , ng minh ¯ diê`u ki ên ¯d , u
ng minh ¯ diê`u ki ên ¯d , u
723 It is clear (evi- dent/immediate/obvious) that
723.R˜o r`ang ( hiê, n nhiên / ngay l âp t´u, c / r˜o r`ang )
724 It is easily seen that 724.N´o c´o thê, d˜ê d`ang nh`ın thâ´y r`˘ang
725 It is easy to check that 725.N´o râ´t d˜ê d`ang ¯dê, kiê, m tra xem
726 It is a simple matter to 726.N´o l`a m ôt vâ´n ¯dê` ¯do, n gi, an ¯dê,
727 We see (check) at once that 727.Ch´ung tôi nh`ın thâ´y (kiê, m tra) m ôt lâ`n m`a
728 F is easily seen (checked) to be smooth.
728.F c´o thê, d˜ê d`ang nh`ın thâ´y ( kiê, m tra) ¯du,
729 , which is clear from (3) 729 , дo l`a r˜o r`ang t`u,
730 , as is easy to check 730 , Nhu, l`a d˜ê d`ang kiê, m tra
731 It follows easily (immediately) that
731.Sau m ôt c´ach d˜ê d`ang ( ngay l.âp t´u,c) m`a
733.B`˘ang ch´u, ng l`a ¯do, n gi, an (tiêu chuâ, n / ngay l âp t´u, c )
734 An easy computation (A trivial verification) shows that
734.M ôt t´ınh to´an d˜ê d`ang (A x´ac minh tâ`m thu,`o,ng) ch, ı ra r`˘ang
735 (2) makes it obvious that [= By
735.(2) l`am cho n´o r˜o r`ang r`˘ang [(2) r˜o r`ang l`a ]
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 7 Proofs] [M uc l uc]
50 Ch, u ¯dê` 7 Proofs - Ch´u, ng minh
736 The factorGf poses no problem becauseGis
736.C´ac yê´u tô´Gf không gây ra vâ´n dê` v`ı¯ G
Proof Easily - Ch´ u ,
ng minh ¯ diê`u ki ên ¯d , u
737 It suffices to show (prove) that
737.N´o ¯d, u ¯dê, ch, ı ra ( ch´u, ng minh ) m`a
738 It suffices to make the following observation.
738.N´o c˜ung ¯d, u ¯dê, l`am cho c´ac nh ân x´et sau ¯dây
739 It suffices to use (4) together with the observation that
739.N´o ¯d, u ¯dê, s, u, d ung (4) c`ung v´o,i vi êc nh.ân x´et
740 It is sufficient to show (prove) that
740.N´o l`a ¯d, u ¯dê, hiê, n th.i ( ch´u, ng minh ) m`a
741 It is sufficient to make the fol- lowing observation.
741.N´o l`a ¯d, u ¯dê, l`am cho c´ac nh ân x´et sau
742 It is sufficient to use (4) together with the observation that
742.N´o l`a ¯d, u ¯dê, s, u, d ung (4) c`ung v´o, i vi êc quan s´at
743 We need only consider 3 cases:
743.Ch´ung ta ch, ı câ`n xem x´et 3 tru,`o,ng h o,p :
744 We only need to show that 744.Ch´ung tôi ch, ı câ`n ¯dê, ch´u, ng minh r`˘ang
745 It remains to prove that (to exclude the case when )
745.N´o v˜ân c`on ¯dê, ch´u, ng minh r`˘ang ( ¯dê, lo ai tr`u, c´ac tru,`o,ng h.o, p khi )
746 What is left is to show that 746.Nh˜u, ng g`ı c`on l ai l`a ¯dê, ch´u, ng minh r`˘ang
747 We are reduced to proving (4) for
747.Ch´ung tôi ¯dang gi, am ¯dê, ch´u,ng minh (4)
748 We are left with the task of de- termining
748.Ch´ung tôi ¯d˜a b, o qua v´o, i nhi êm v u x´ac ¯d.inh
749 The only point remaining con- cerns the behaviour of
749.Ðiê, m duy nhâ´t c`on l ai liên quan dê´n h`¯ anh vi c, ua
750 The proof is completed by show- ing that
.o,c ho`an th`anh b`˘ang c´ach cho thâ´y r`˘ang
751 We shall have established the lemma if we prove the follow- ing:
751.Ch´ung ta s˜e thiê´t l âp bô, dê` nê´u¯ ch´ung tôi ch´u,ng minh nh˜u,ng ¯diê`u sau ¯dây :
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 7 Proofs] [M uc l uc]
752 If we prove that , the asser- tion follows.
752.Nê´u ch´ung tôi ch´u,ng minh r`˘ang , Kh, ˘ang ¯d.inh sau.
753 The statement O(g) = l will be proved once we prove the lemma below.
.o, c ch´u, ng minh khi ch´ung tôi ch´u,ng minh
Proof Property - Ch´ u ,
754 the (an) element such that (with the property that)
754.c´ac phâ`n t, u, (m ôt ) nhu, v ây m`a (v´o, i c´ac t´ınh châ´t m`a )
755 the (an) element with the fol- lowing properties:
(m ôt ) v´o,i c´ac thu ôc t´ınh sau :
758 the (an) element of norm 1 (of the form )
758.c´ac phâ`n t, u, d.inh m´u¯ , c 1 (m ôt ) ( h`ınh th´u, c )
759 the (an) element whose norm is
760 the (an) element all of whose subsets are
760.c´ac phâ`n t, u, (m ôt ) tâ´t c, a c´ac t âp con c´o
761 the (an) element by means of which g can be computed
761.c´ac phâ`n t, u, (m ôt ) phu, o, ng ti ên trong ¯d´o g c´o thê, ¯ du, o, c t´ınh
762 the (an) element for which this is true
763 the (an) element at which g has a local maximum
(m ôt ) m`a t.ai ¯d´o g c´o c u, c ¯d ai ¯d.ia phu, o,ng
764 the (an) element described by the equations
.o, c mô t, a b, o, i c´ac phu, o, ng tr`ınh
765 the (an) element given byLf
766 the (an) element depending only on (independent of )
766.c´ac phâ`n t, u, (m ôt ) ch, ı ph u thu ôc v`ao (không ph u thu ôc )
767 the (an) element not inA 767.c´ac phâ`n t, u,
768 the (an) element so small that
768.c´ac phâ`n t, u, (m ôt ) qu´a nh, o m`a ( d¯, u nh, o )
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 7 Proofs] [M uc l uc]
52 Ch, u ¯dê` 7 Proofs - Ch´u, ng minh
769 the (an) element as above (as in the previous theorem)
769.c´ac phâ`n t, u, (m ôt ) nhu, trên (nhu, trong ¯d.inh l´y tru,´o,c )
770 the (an) element so obtained 770.c´ac phâ`n t, u, (m ôt ) ¯dê, thu ¯du,
771 the (an) element occurring in the cone condition
771.c´ac phâ`n t, u, ( m ôt ) x, ay ra trong diê`u ki ên h`ınh n´on¯
772 the (an) element guaranteed by the assumption
773 the (an) element we have just defined
(m ôt ) ch´ung tôi ¯d˜a d.inh ngh˜ıa¯
774 the (an) element we wish to study (we used in Chapter 7)
( m ôt ), ch´ung tôi muô´n nghiên c´u, u ( ch´ung tôi s, u, d ung trong Chu, o, ng 7 )
775 the (an) element to be defined later [=which will be defined]
.o, c x´ac ¯d.inh sau n`ay [=s˜e ¯du,
776 the (an) element in question 776.c´ac phâ`n t, u, (m ôt ) trong câu h, oi
777 the (an) element under study
778 ., the constant C being inde- pendent of [=whereCis ]
778 , H`˘ang sô´C d ôc l.âp c¯ , ua [No, iC ]
779 ., the supremum being taken over all cubes
.o, c th u, c hi ên trên tâ´t c, a c´ac h`ınh khô´i
780 ., the limit being taken inL 780 , Gi´o, i h an ¯du,
781 ., whereCis so chosen that 781 ,Cl`a l u, a ch on m`a
782 ., whereCis to be chosen later 782 ,Cl`a ¯du,
783 ., where C is a suitable con- stant.
783 ,Cl`a m ôt h`˘ang sô´ph`u h o, p
784 ., where C is a conveniently chosen element of
784 , C l`a m ôt yê´u tô´ thu.ân ti.ên du¯ , o, c l u, a ch on c, ua
785 ., whereC involves the deriva- tives of
785 ,Cliên quan ¯dê´n c´ac d˜ân xuâ´t c, ua
786 ., whereC ranges over all sub- sets of
786 ,Cdao ¯d ông trên tâ´t c, a c´ac t âp h o, p con c, ua
787 ., whereC may be made arbi- trarily small by
787 ,C c´o thê, du¯, o, c th u, c hi ên nh, o t`uy ´y
788 The operators Ai have (share) many of the properties of
A c´o râ´t nhiê`u t´ınh châ´t c, ua
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 7 Proofs] [M uc l uc]
789 The operatorsA i have still bet- ter smoothness properties.
789.To´an t, u,Av˜ân c`on t´ınh châ´t tro,n.
790 The operators A i lack (fail to have) the smoothness properties of
790.To´an t, u,Ac´o t´ınh châ´t tro,n ho,n.
791 The operatorsA i still have norm
791.To´an t, u,Av˜ân c´o chuâ, n 1.
792 The operatorsAi are not merely symmetric but actually self- adjoint.
Ai không ch, ı ¯do, n thuâ`n l`a ¯dô´i x´u, ng nhu, ng th u, c s u, t u, liên h o, p
793 The operatorsAi are not neces- sarily monotone,
Ai không nhâ´t thiê´t ph, ai ¯do, n ¯di êu ,
794 The operatorsA i are both sym- metric and positive-definite.
794.C´ac to´an t, u,A i dô´i x´¯ u, ng v`a t´ıch c u, c x´ac ¯d.inh
795 The operatorsA i are not contin- uou nor do they satisfy (2).
795.C´ac to´an t, u,A i không continuou c˜ung không l`am h o th, oa m˜an (2).
796 The operators A i are neither symmetric nor positive-definite,
796.C´ac to´an t, u,A i không ph, ai l`a ¯dô´i x´u, ng v`a c˜ung không t´ıch c u, c x´ac d.inh ,¯
797 The operatorsAi are only non- negative rather than strictly pos- itive, as one may have expected.
Ai ch, ı không âm ho, n l`a nghiêm ch, ınh t´ıch c u, c , l`a m ôt trong nh˜u, ng c´o thê, mong ¯d o, i
798 The operators Ai are any self- adjoint operators, possibly even unbounded.
Ai bâ´t k`y nh`a khai th´ac t u, liên h o, p , th âm ch´ı c´o thê, b.i ch.˘an
799 The operators Ai are still (no longer) self-adjoint,
Ai v˜ân c`on (không c`on ) t u, liên h o, p ,
800 The operatorsAiare not too far from being self-adjoint.
Ai không ph, ai l`a qu´a xa l`a t u, liên h o, p
801 The preceding theorem 801.Ð.inh l´y tru,´o,c
802 The indicated set 802.C´ac thiê´t l âp ch, ı ¯d.inh
803 The above-mentioned group 803.C´ac nh´om n´oi trên
804 The resulting region 804.Khu v u, c kê´t qu, a
805 The required (desired) element 805.(Mong muô´n) yêu câ`u yê´u tô´
806 BothX andY are finite 806.C, a hai YX v`a l`a h˜u, u h an.
807 NeitherX norY is finite 807.Không X c˜ung không Y l`a h˜u,u h an.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 7 Proofs] [M uc l uc]
54 Ch, u ¯dê` 7 Proofs - Ch´u, ng minh
808 XandY are countable, but nei- ther is finite.
.o,c, nhu,ng không ph, ai l`a h˜u, u h an
809 Neither of them is finite [Note:
"Neither" refers to two alterna- tives.]
809.Không m ôt c´ai n`ao t`u, ch´ung l`a h˜u, u h an [Ch´u ´y: "Không " ¯dê`c.âp dê´n hai l u¯ , a ch on thay thê´ ]
810 None of the functions F, is fi- nite.
810.Không m ôt h`am c, uaFl`a h˜u, u h an.
811 X is not finite; nor (neither) is
811.X không ph, ai l`a h˜u, u h an , c˜ung không ( không) l`aY
812 X is not finite, nor is Y count- able [Note the inversion.]
812.X không ph, ai l`a h˜u, u h an, c˜ung không ph, ai l`aY dê´m ¯¯ du,
813 Xis empty; so isY 813.X l`a r˜ông, nhu, ng Y không nhu, v ây
814 Xis empty, butY is not 814.X l`a r˜ông, nhu, ngY không
815 Xbelongs toY; so doesZ 815.X n`˘am trongY, nhu, ng Z không nhu, v ây
816 Xbelongs toY, butZdoes not 816.X n`˘am trongY, nhu, ngZ không
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 7 Proofs] [M uc l uc]
CH U Ð` Ê 8 Proof steps C´ ac bu ,´o,c ch´u,ng minh
823 Let us evaluate 823.H˜ay ¯dê, ch´ung tôi ¯d´anh gi´a
824 Let us compute 824.H˜ay cho chugns ta t´ınh
825 Let us apply the formula to 825.H˜ay ´ap d ung công th´u, c cho
826 Let us suppose for the moment
826.Ta gi, a s, u, th`o,i ¯diê, m n`ay
827 Let us regardsas fixed and 827.Ta xemsnhu, cô´ ¯d.inh v`a
828 Adding g to the left-hand side 828.Thêmgv´ao bên tr´ai
830 Writing (Taking)h=Hf 830.Ta ¯d ˘ath=Hf
833 Combining these [E.g these in- equalities]
837 Interchangingf andg 837.Ðô, i ch˜ô cho nhauf v`ag
840 we conclude (deduce/see) that
840.ch´ung tôi kê´t lu ân ( suy ra / nh`ın thâ´y)
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 8 Proof steps] [M uc l uc]
56 Ch, u ¯dê` 8 Proof steps - C´ac bu,´o,c ch´u,ng minh
841 we can assert that 841.ch´ung ta c´o thê, kh, ˘ang ¯d.inh r`˘ang
842 we can rewrite (5) as 842.ch´ung ta c´o thê, viê´t l ai (5)
843 We continue in this fashion ob- taining (to obtain)f =
843.Ch´ung tôi tiê´p t uc theo c´ach n`ay c´o ¯du,
844 We may now integrate times to conclude that
844.Bây gi`o, ch´ung ta c´o thê, t´ıch h o, p th`o, i gian ¯dê, kê´t lu ân r`˘ang
845.L ˘ap ¯di l.˘ap l.ai ´ap d ung Bô, dê`6 cho¯ ph´ep ch´ung tôi viê´t
846 We now proceed by induction 846.Bây gi`o,ch´ung ta tiê´n h`anh b`˘ang quy n ap.
847 We can now proceed analo- gously to the proof of
847.Bây gi`o, ch´ung ta c´o thê, tiê´n h`anh
Tu, o, ng t u, ch´u, ng minh
848 We next claim (show/prove that)
848.Ch´ung tôi yêu câ`u ph´at biê, u ( hiê, n th.i / ch´u, ng minh r`˘ang )
849 We next sharpen these results and prove that
849.Tiê´p theo ch´ung ta l`am s´˘ac n´et nh˜u, ng kê´t qu, a n`ay v`a ch´u, ng minh r`˘ang
850 Our next claim is that 850.Tuyên bô´ tiê´p theo c, ua ch´ung tôi l`a
851 Our next goal is to determine the number of
851.M uc tiêu tiê´p theo c, ua ch´ung tôi l`a x´ac ¯d.inh sô´lu,
852 Our next objective is to evaluate the integralI.
852.M uc tiêu tiê´p theo c, ua ch´ung tôi l`a ¯d´anh gi´a thiê´uI.
853 Our next concern will be the be- haviour of
853.Quan tâm tiê´p theo c, ua ch´ung tôi s˜e l`a h`anh vi c, ua
854 We now turn to the casef 6= 1 854.Bây gi`o, ch´ung ta chuyê, n sang tru,`o,ng h.o, pf ne1.
855 We are now in a position to show [=We are able to]
855.Ch´ung tôi hi ên ¯dang , o, m ôt v.i tr´ı dê¯ , ch´u, ng minh [=Ch´ung tôi c´o thê,
856 We proceed to show that 856.Ch´ung tôi tiê´n h`anh ¯dê, ch´u, ng minh r`˘ang
857 The task is now to find 857.Nhi êm v u l`a bây gi`o, ¯ dê, t`ım
858 Having disposed of this prelimi- nary step, we can now return to
858.Sau khi x, u, l´y so, b ô bu,´o,c n`ay , ch´ung ta c´o thê, quay tr, o, l ai
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 8 Proof steps] [M uc l uc]
859 We wish to arrange thatf be as smooth as possible [Note the in- finitive.]
859.Ch´ung tôi muô´n s´˘ap xê´pf du¯ ,
860 We are thus looking for the fam- ily
860.Ch´ung tôi xem nhu, v ây, cho m ôt h o
861 We have to construct 861.Ta ph, ai xây d u, ng
862 In order to get this inequality, it
862.Ðê, c´o ¯du, o, c s u, bâ´t b`ınh ¯d, ˘ ang n`ay , n´o:
863 will be necessary to 863.s˜e l`a câ`n thiê´t ¯dê,
864 is convenient to 864.thu ân ti.ên ¯dê,
865 To deal withIf, we note that 865.Ðê, dô´i ph´¯ o v´o,iIf , ch´ung tôi lu,u ´ y r`˘ang
866 To estimate the other term, we note that
866.Ðê, u,´o,c t´ınh th`o,i h.an kh´ac , ch´ung tôi lu, u ´y r`˘ang
867 For the general case, we note that
867.Ðô´i v´o, i tru,`o,ng h.o, p chung, ch´ung tôi lu, u ´y r`˘ang
868 We are now in a position to prove the following theorem.
868.Bây gi`o, ch´ung ta ¯d˜a s˜˘an s`ang ¯dê, ch´u, ng minh ¯d.inh l´y sau ¯dây.
869 We shall begin with showing that
869.Ta s˜e b´˘at ¯dâ`u b`˘ang vi êc ch, ı ra r`˘ang
870 We begin by establishing 870.Ta s˜e b´˘at ¯dâ`u b`˘ang vi êc thiê´t l âp
871 We first observe that 871.Tru,´o,c tiên ta nh.ân x´et r`˘ang
872 We proceed now to establish the fundamental result
872.Bây gi`o,ta chuyê, n sang thiê´t l âp kê´t qu, a co,b, an
873 Evidently, Obviously, Clearly 873.D˜ê thâ´y, Hiê, n nhiên, R˜o r`ang.
874 It is obvious that 874.Hiê, n nhiên l`a
875 It is plain that 875.R˜o r`ang l`a
876 It is clear that 876.R˜o r`ang l`a
877 It is easy to prove (show, see, check, verify, ) that
877.D˜ê d`ang ch´u, ng minh(ch, ı ra, nh ân thâ´y, kiê, m tra, th, u, l ai) l`a
878 It is easily seen that 878.D˜ê thâ´y r`ang ˘
879 It is easily be seen that 879.C´o thê, d˜ê thâ´y r`˘ang
880 As an immediate consequence of
880.Nhu, m ôt h.ê qu, a c, ua tr u,c tiê´p c, ua Ð.inh l´y 1 ta c´o
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 8 Proof steps] [M uc l uc]
58 Ch, u ¯dê` 8 Proof steps - C´ac bu,´o,c ch´u,ng minh
881 Lemmas 5 and 6 immediately give the first assertion of the
881.C´ac bô, dê` 5 v`¯ a 6 cho ngay ¯diê`u kh, ˘ang ¯d.inh ¯dâ`u tiên c, ua Ð.inh l´y n`ay.
882 The Theorem now follows from applying Corollary 2.
.o, c suy ra b`˘ang c´ach ´ap d ung H.ê qu, a 2.
883 The first part of this Proposition is essentially Lemma 2 in [.].
883.Phâ`n ¯dâ`u c, ua m ênh ¯dê` n`ay th u, c châ´t l`a Bô, dê` 2 trong [.].¯
884 The second part of the Proposi- tion is obtained from
884.Pâ`n th´u, hai c, ua m ênh ¯dê` nh.ân du¯ , o, c t`u,
885 We invoke Theorem 8 to deduce that
885.Ta d u, a v`ao Ð.inh l´y 8 ¯dê, suy ra r`˘ang
886 By virtue of Lemma 2.1 one can
886.Do Bô, dê` 2.1, ta c´¯ o thê,
887 In view of the lemma above, there is
887.Do Bô, dê` trên, ta c´¯ o
888 Taking Lemma 3 into account we get
888.C´u ´y Bô, dê` 3 ta nh ân ¯du¯ ,
889 Taking account of Lemma 3, we get
889.C´u ´y Bô, dê` 3 ta nh ân ¯du¯ ,
890 It follows from Fubini’s Theo- rem applied to indicator func- tion that
890.T`u, ¯d.inh l´y Fubini ´ap d ung cho h`am ch, ı ¯d.inh suy ra r`˘ang
891 This Theorem can be proved by repeated application of Lemma
891.Ð.inh l´y n`ay c´o thê, ch´u, ng minh b`˘ang c´ach ´ap d ung Bô, dê` 1.2 l ˘ap¯ l ai nhiê`u lâ`n.
892 We can now combine the results of Theorem 5.1 with the method of and obtain the following theorem.
892.Bây gi`o, ta c´o thê, kê´t h o, p c´ac kê´t qu, a c, ua ¯d.inh l´y 5.1 v´o, i phu, o, ng ph´ap c, ua v`a thu ¯du,
893 Arguing as in we can 893.L´y lu ân nhu, trong ta c´o thê,
894 It follows readily from that 894.T`u,
895 To do this we shall utilize the re- sults obtained in
895.Ðê, l`am vi êc ¯d´o ta s˜e d`ung c´ac kê´t qu, a nh ân ¯du,
896 That fact follows from (2) and the observation that∆ttends to
.o,c suy ra t`u,(2) v`a t`u, nh ân x´et r`˘ang∆ttiê´n t´o,i0.
897 Equality (6) follows from replac- ingαbyψin (3).
897.Ð, ˘ ang th´u,c (6) suy ra t`u,(3) b`˘an c´ach thayαb, o,iψ.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 8 Proof steps] [M uc l uc]
898 This follows from (1) withalpha in place ofβ.
898.Ðiê`u ¯d´o suy ra t`u, (1) v´o,i alpha thay choβ.
899.Vi êc thay (3) v`a (4) v`ao (2) cho ta
901 which together with (2.1) im- plies that
901 ¯diê`u ¯d´o c`ung v´o, i (2.1), cho ph´ep suy ra r`˘ang
902 we now apply (1.3) withAplay- ing the role of
902.Bây gi`o,´ap d ung (1.3) v´o,iAd´¯ong vai tr`o c, ua
903 DefiningQi in term of Fi, as Q is defined in term ofF, one has
903.Ð.inh ngh˜ıaQ1theoFi, nhu,
Qd˜¯a du¯ , o, c ¯d.inh ngh˜ıa theoF, ta c´o
904 In exactly the same way, but re- placingQbyA and interchang- ing the role ofx, y, we have
904.B`˘ang ch´ınh c´ach â´y, nhu, ng thayQ b`˘angAv`a ¯dô, i vai tr`o c, uax, yta c´o
905 In the same way as in 905.C˜ung b`˘ang c´ach nhu, trong
906 By an argument analogous (sim- ilar) to the previous one We get
906.B`˘ang l âp lu.ân tu, o, ng t u,
(giô´ng) l âp lu.ân trên Ta nh.ân ¯du,
907 By an argument analogous to that used for the proof of The- orem 1
907.B`˘ang l âp lu.ân tu, o, ng t u, nhu, l âp lu ân d`ung ¯dê, ch´u, ng minh Ð.inh l´y
908 In the case the proof is analo- gous to the one above.
908.Trong tru,`o,ng h.o, p c´ach ch´u, ng minh tu, o, ng t u, nhu, trên.
909 According to the method of 909.Theo phu,o,ng ph´ap c, ua
910 The main tool for our proof will be the following Lemma.
910.Công c u ch´ınh c, ua ch´ung ta trong ch´u, ng minh s˜e l`a Bô, dê` sau.¯
911 The main tool we shall use to de- rive our results utilizes the fol- lowing concepts.
911.Công c u ch´ınh m`a ch´ung ta s˜e d`¯ung ¯dê, suy ra c´ac kê´t qu, a s, u, d ung c´ac kh´ai ni.êm sau.
912 Assume the contrary, that 912.Gi, a s, u, ngu, o, c l ai l`a
913 In the contrary case 913.Trong tru,`o,ng h.o, p ngu, o, c l ai
914 Otherwise 914.Trong tru,`o,ng h.o, p ngu, o, c l ai
915 Were this false, there would ex- ist
915.Nê´u ¯diê`u ¯d´o sai, th`ı s˜e tô`n t ai
916 If it were not so, then we should have
916.Nê´u không nhu, thê´ th`ı ch´ung ta s˜e c´o
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 8 Proof steps] [M uc l uc]
60 Ch, u ¯dê` 8 Proof steps - C´ac bu,´o,c ch´u,ng minh
917 If this were not so, there would be
917.Nê´u không nhu,thê´th`ı s˜e c´o
918 This contradicts the hypothesis 918.Ðiê`u ¯d´o tr´ai v´o, i gi, a thiê´t.
919 Which contradicts 919.Ðiê`u ¯d´o tr´ai v´o, i
920 Which contradicts with 920.(m`a) ¯diê`u ¯d´o mâu thu˜ân v´o,i
921 We thus arrive at a contradic- tion.
921.Nhu, thê´ ch´ung ta ¯di t´o, i mâu thu˜ân.
922 from which the contradiction would arise
922.m`a t`u, d´¯o mâu thu˜ân s˜e x, ay ra.
923 The assumption that now leads to a contradiction by the same reasoning used in the proof of (8).
923.Gi, a thiê´t bây gi`o,d˜ân t´o,i mâu thu˜ân b, o, i ch´ınh l âp lu.ân ¯d˜a d`ung trong ch´u, ng minh (8).
924 Since (7) is clearly violated (13) holds
924.V`ı r˜o r`ang (7) b.i vi ph.am, nên
925 Since , one of the conditions
925.V`ı nên m ôt trong nh˜u, ng ¯diê`u ki ên (3), (5), (7) v`a (9) ph, ai không ¯d´ung.
926 But this shows that , a contra- diction.
926.Nhu, ng ¯diê`u ¯d´o ch, ı ra r`˘ang mâu thu˜ân.
927 Since{αn}must satisfy (3) 927.V`ı{αn}ph, ai th, oa m˜an (3).
928 Since is arbitrary chosen, we get
.o, c ch on t`uy ´y nên ta c´o
929 This is incompatible withAbe- ing disjoint fromD
929.Ðiê`u ¯d´o mâu thu˜ân v´o, i vi êc A không c´o ¯diê, m chung v´o, iD.
930 That (2)=⇒(1) is trivial, so 930.Ðiê`u (2) =⇒ (1) l`a tâ`m thu,`o,ng, v ây
931 Consequently, Therefore, Hence 931.Cho nên, T`u, d´¯o, Do ¯d´o.
932 Since , it follows that 932.V`ı nên suy ra r`˘ang
933 Observe, further, that 933.Sau n˜u, a, ta nh ân x´et r`˘ang
934 We prove first the necessity of the condition.
934.Tru,´o,c hê´t ta ch´u,ng minh t´ınh câ`n c, ua ¯diê`u ki ên.
935 We now turn to the proof of suf- ficiency.
935.Bây gi`o, ta quay sang ch´u, ng minh t´ınh ¯d, u.
936 The "if" part 936.Phâ`n "nê´u"
937 The "only if" part 937.Phâ`n "ch, ı nê´u"
938 Conditions (1) through (5) 938.C´ac ¯diê`u ki ên t`u,
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 8 Proof steps] [M uc l uc]
939 Assuming the condition (1) ful- filled, we shall show that
939.Gi, a s, u, diê`u ki ên (1) ¯du¯ ,
.o,c th, oa m˜an, ta s˜e ch, ı ra r`˘ang
940 Now, by the above 940.Bây gi`o,
, qua nh˜u, ng ¯diê`u ki ên trên
941 as shown above 941 nhu, d˜¯a ¯du, o,c ch´u,ng minh , o, trên.
942 It has already been shown that 942 ,
943 We have thus proved that 943.Nhu, thê´ ch´ung ta ¯d˜a ch´u, ng minh ¯ du, o, c l`a
944 We have remarked before on the fact that
944.Ta ¯d˜a nh ân x´et, o, trên vê`s u, vi êc l`a
945 Using the results just obtained we
945.S, u, d ung c´ac kê´t qu, a v`u, a thu ¯du,
946 By the results just mentioned 946.Qua c´ac kê´t qu, a v`u, a kê, t´o, i.
947 With this fact in hand, we can prove the following proposition.
947.C´o s˜˘an s u, ki ên ¯d´o, ta c´o thê, ch´u, ng minh m ênh ¯dê` sau ¯dây.
948 The result above shows , the next theorem in addition shows that
948.Kê´t qu, a , o, trên cho thâ´y , c`on d.inh l´y tiê´p sau ¯dây s˜e ch¯ , ı thêm l`a
949 The next observation shows that
949.Nh ân x´et tiê´p theo cho thâ´y r`˘ang
950 By translating if necessary we may assume thatx= 0.
950.B`˘ang t.inh tiê´n nê´u câ`n c´o thê, gi, a thiê´t r`˘angx= 0.
951 In view of the preceding remark, we can limit attention to meth- ods
951.Do nh ân x´et trên, ta ch, ı câ`n gi´o, i h an s u, quan tâm t´o, i c´ac phu, o, ng ph´ap
952 We shall confine our attention to the case, where
952.Ta s˜e h an chê´s u, ch´u ´y v`ao tru,`o,ng h o, p m`a
953 It suffices to treat the case where
953.Ch, ı câ`n x´et tru,`o,ng h.o,p m`a.
954 Then, by taking a subsequence, if necessary, we can assume that
954.Khi â´y, b`˘ang c´ach lâ´y d˜ay con nê´u câ`n, ta c´o thê, gi, a thiê´t.
955 By passing to an appropriate subsequence and renumbering if necessary, let us assume that the sequence is convergent.
955.B`˘ang c´ach chuyê, n qua d˜ay con th´ıch h o, p v`a ¯d´anh sô´ l ai nê´u câ`n, ta h˜ay gi, a s, u, l`a d˜ay h ôi t u.
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62 Ch, u ¯dê` 8 Proof steps - C´ac bu,´o,c ch´u,ng minh
956 Proceeding to a subsequence, if necessary, we may assume that
956.Chuyê, n qua d˜ay con, nê´u câ`n ta c´o thê, gi, a thiê´t r`˘ang
957 Dropping to a subsequence, if necessary, (9) shows that
957.R´ut l ai m ôt d˜ay con, nê´u c´o (9) ch, ı ra r`˘ang
958.Suô´t thay d˜ay conn 0 ta c´oMn 0 ≤ 2.
959 TakingCto be the class of all se- quences of form (1) we obtain
959.Lâ´y C l`a l´o, p tâ´t c, a c´ac d˜ay d ang
960 We pickàso large that 960.Ta lõ´y raà¯d, u l´o,n sao cho
961 Given anyα≥0, we can choose 961.Cho tru,´o,c α ≥ 0, ta c´o thê, ch on
962 For the validity of Lemma 1 it is enough to suppose merely that α≥1.
962.Bô, dê` 1 c´¯ o hi êu l u, c, ch, ı câ`n gi, a thiê´t l`aα≥1.
963 Without loss of generality we may assume that
963.Không mâ´t t´ınh tô, ng qu´at ta c´o thê, gi, a thiê´t r`˘ang
964 We shall omit the easy proof of the following properties of
964.Ta s˜e b, o qua ch´u,ng minh d˜ê d`ang c´ac t´ınh châ´t sau ¯dây c, ua
965 For the final assertion of the proposition, observe that
965.Ðô´i v´o,i ¯diê`u kh, ˘ang ¯d.inh cuô´i c`ung c, ua m ênh ¯dê`, h˜ay lu, u ´y r`˘ang
966 The problem is now reduced to proving that
966.L´uc n`ay vâ´n ¯dê` r´ut l ai ch, ı c`on ch´u,ng minh r`˘ang
967 To complete the proof it remains to show that
967.Ðê, ho`an th`anh vi êc ch´u, ng minh ch, ı c`on ph, ai ch, ı ra r`˘ang
968 There remains the question of suitable choice of
968.C`on l ai vâ´n ¯dê` l`a ch on th´ıch h o, p
969 To this end, we remark 969.Nh`am m uc ¯d´ıch ¯d´o, ta ch´u ´y ˘
970 It is worth noticing that 970.дang ghi nh ân l`a
972 We attempt to show that 972.Ta s˜e g´˘ang ch, ı ra r`ang ˘
973 We infer that 973.Ta kê´t lu ân r`˘ang
974 We then have the desired prop- erties.
974.Khi â´y ta c´o nh˜u, ng t´ınh châ´t mong muô´n.
975 Sinceβ is finite, the proposition follows.
976 This completes (concludes) the proof.
976.Ðiê`u ¯d´o ho`an th`anh (kê´t th´uc) vi êc ch´u,ng minh.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 8 Proof steps] [M uc l uc]
977 The proof is complete 977.Ch´u, ng minh ¯d˜a xong.
978 The proof is straight-forward 978.Ch´u, ng minh l`a t´u, c kh´ac.˘
979 The proof parallels that of Theo- rem 10 and will be omitted.
979.Ch´u, ng minh giô´ng nhu, dô´i v´¯ o, i Ð.inh l´y 10 v`a s˜e không tr`ınh b`ay, o, dây.¯
980 as was to be shown 980 ¯d´o ch´ınh l`a c´ai câ`n ch´u,ng minh.
981 We use our results to derive an important extension of Theorem
981.Ta s, u, d ung kê´t qu, a c, ua m`ınh ¯dê, suy ra m ôt m, o, r ông quan tr ong c, ua Ð.inh l´y [].
982 a result which we shall have opportunity to use later.
982 m ôt kê´t qu, a m`a ta s˜e c´o s, u, d ung sau n`ay.
983 The function f, besides being measurable, is summable.
983.H`amf không nh˜u, ng ¯do ¯du,
984 The function Lis defined to be constant except for jumps at the pointsai(i= 1,2, , N).
.o, c x´ac ¯d.inh l`a h`am const tr`u, nh˜u, ng bu,´o,c nh, ay c´ac diê¯ , mai(i= 1,2, , N).
985 The function attains the lower bound of its values on this set.
985.H`am â´y ¯d at c.ân du,´o,i c, ua c´ac gi´a tr.i c, ua n´o trên t âp n`ay.
986 f may not be defined on all of
986.f c´o thê, không x´ac ¯d.inh trên to`an b ôX.
987 We now wish to remove these restrictions.
987.Bây gi`o, ta muô´n b, o ¯di c´ac h an chê´ d´¯o.
988 The proof is by a standard com- pactness argument
988.Ch´u, ng minh b`ang l âp lu.ân quen˘ thu ôc d u, a trên t´ınh compact.
989 If, as is usually the case, A is in- vertible, then
989.Nê´u nhu,thu,`o,ng hay x, ay raAkh, a ngh.ich, th`ı
990 Observe, in passing, thatK 1 and
K 2 lie in opposite half-spaces determined byH.
990.Nhân ¯dây ta nh ân x´et r`˘angK 1 v`a
K 2 n`˘am trong c´ac n, u, a không gian ¯ dô´i nhau x´ac ¯d.inh b, o, i.
991 The proof is by induction on the dimension of the subspace.
991.Ch´u, ng minh b`˘ang quy n ap theo th´u, nguyên c, ua không gian.
992 As the next step, we claim that
992.Bu,´o,c tiê´p theo, ta kh, ˘ang ¯d.inh r`˘ang
993 We contend that 993.Ta kh, ˘ang ¯d.inh r`˘ang
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64 Ch, u ¯dê` 8 Proof steps - C´ac bu,´o,c ch´u,ng minh
994 We now set out to prove this as- sertion, along the way we shall establish a few other equivalent properties.
994.Bây gi`o, ta b´˘at ¯dâ`u ch´u,ng minh diê`u kh¯ , ˘ang ¯d.inh ¯d´o, d oc ¯du,`o,ng ta s˜e thiê´t l âp m ôt sô´t´ınh châ´t tu, o, ng du¯ , o, ng kh´ac.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 8 Proof steps] [M uc l uc]
CH U Ð` Ê 9 References - T` ai li êu tra c´u , u
995 (see for instance [7, Th 1]) 995.(xem v´ı d u ch, ˘ang h an [7, Th 1])
996 (see [7] and the references given there)
996.(xem [7] v`a tra c´u,u ¯du,a ra, o, dây)¯
997 (see [Ka2] for more details) 997.(xem [Ka2] v´o, i chi tiê´t ho, n)
998 (see [Ka2] for the definition of
998.(xem [Ka2] cho ¯d.inh ngh˜ıa c, ua )
999 (see [Ka2] for the complete bib- liography)
999.(xem [Ka2] cho t`ai li êu tham kh, ao ¯dâ`y ¯d, u)
1000 The best general reference here is
1000 T`ai li êu tra c´u, u tô, ng quan nhâ´t , o, dây l`¯ a
1001 The standard work on is 1001 Công vi êc chuâ, n trên l`a
1002 The classical work here 1002 Công vi êc cô, diê¯ , n , o, dây¯
1003 This was proved by Lax [8] 1003 Ðiê`u n`ay ¯d˜a ch´u, ng minh b, o, i Lax[8].
1004 This can be found in Lax [7, Ch.
1004 Ðiê`u n`ay c´o thê, t`ım trong Lax [7, Ch.2].
1005 This construction is due to
1005 C´ach xây d u, ng n`ay suy ra t`u, Strang[8].
1006 This construction goes back to the work of (as far as [8]).
1006 C´ach xây d u, ng n`ay tr, o, l ai công tr`ınh c, ua
1007 This construction was motivated by [7].
1007 C´ach xây d u,ng n`ay ¯d˜a ¯du,
1008 This construction generalizes that of [7].
1008 C´ach xây d u, ng n`ay m, o, r ông kê´t qu, a c, ua [7].
1009 This construction follows [7] 1009 C´ach xây d u, ng n`ay suy ra [7].
1010 This construction is adapted from [7] (appears in [71).
1010 C´ach xây d u, ng n`ay ¯du,
1011 This construction has previously been used by Lax [7].
1011 C´ach xây d u,ng n`ay ¯d˜a ¯du,
1012 For a recent account of the the- ory we refer the reader to [7]
1012 Ðô´i v´o, i m ôt kê´t qu, a gâ`n ¯dây c, ua l´y thuyê´t n`ay , ch´ung tôi m`o,i ¯d ôc gi, a xem [7 ]
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66 Ch, u ¯dê` 9 References - T`ai li êu tra c´u, u
1013 For a treatment of a more gen- eral case we refer the reader to
1013 Ðô´i v´o, i th u, c hi ên m ôt tru,`o,ng h.o,p tô, ng qu´at ho,n , ch´ung tôi m`o,i ¯d ôc gi, a xem [7 ]
1014 For a fuller (thorough) treat- ment we refer the reader to [7]
1014 Ðô´i v´o, i m ôt kê´t qu, a ¯dâ`y ¯d, u ho, n ( k˜y lu,˜o,ng) , ch´ung tôi m`o,i ¯d.ôc gi, a xem [7 ]
1015 For a deeper discussion of we refer the reader to [7]
1015 Ðô´i v´o, i m ôt cu ôc th, ao lu ân sâu s´˘ac ho, n vê` ch´ung tôi m`o, i ¯d ôc gi, a xem [7 ]
1016 For direct constructions along more classical lines we refer the reader to [7]
1016 Ðô´i v´o, i c´ac công tr`ınh tr u, c tiê´p c`ung nhiê`u d`ong cô, diê¯ , n , ch´ung tôi m`o, i ¯d ôc gi, a xem [7 ]
1017 For yet another method we refer the reader to [7]
1017 Ðô´i v´o, i phu, o, ng ph´ap kh´ac , ch´ung tôi m`o, i ¯d ôc gi, a xem [7 ]
1018 We introduce the notion of , following Kato [7].
1018 Ch´ung tôi gi´o, i thi êu c´ac kh´ai ni.êm , theo Kato [7]
1019 We follow [Ka] in assuming that
1019 Ch´ung tôi theo [ Ka ] trong gi, a d.inh r`˘ang ¯
1020 The main results of this paper were announced in [7].
1020 C´ac kê´t qu, a ch´ınh c, ua nghiên c´u, u n`ay ¯d˜a ¯du,
1021 Similar results have been ob- tained independently by Lax and are to be published in [7].
1021 Kê´t qu, a tu, o, ng t u, d˜¯a ¯du, o, c ¯d ôc l.âp b, o, i Lax v`a s˜e ¯du,
1022 The author proves the interest- ing result that
1022 T´ac gi, a ch´u, ng minh kê´t qu, a th´u v.i
1023 The proof is short and simple, and the article well written.
1023 B`˘ang ch´u, ng l`a ng´˘an v`a ¯do, n gi, an , v`a b`ai viê´t c˜ung b`˘ang
1024 The results presented are origi- nal.
1024 C´ac kê´t qu, a tr`ınh b`ay ban ¯dâ`u.
1025 The paper is a good piece of work on a subject that attracts considerable attention.
1025 B`ai viê´t n`ay l`a m ôt m, anh tô´t công vi êc vê`m ôt ch, u ¯dê` thu h´ut s u, ch´u ´y ¯d´ang kê,
1026 I am pleased to recommend it for publication in Studia Math- ematica.
1026 Tôi h`ai l`ong ¯dê, gi´o, i thi êu n´o cho công bô´ trong Studia Mathemat- ica
1027 It is a pleasure to recommend it for publication in Studia Mathe- matica.
1027 дo l`a m ôt niê`m vui ¯dê, gi´o, i thi êu n´o cho công bô´ trong Studia Mathematica
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 9 References] [M uc l uc]
1028 I strongly recommend it for pub- lication in Studia Mathematica.
1028 Tôi ¯dê` ngh.i công bô´ trong Studia Mathematica
1029 The only remarkIwish to make is that condition B should be formulated more carefully.
1029 Nh ân x´et duy nhâ´t I muô´n th u, c hi ên l`a ¯diê`u ki.ênBcâ`n ¯du,
.o,c xây d u, ng câ, n th ân ho,n
1030 A few minor typographical er- rors are listed below.
1030 M ôt v`ai l˜ôi nh, o ¯d´anh m´ay ¯du,
.o, c li êt kê du,´o,i ¯dây.
1031 I have indicated various correc- tions on the manuscript.
1031 Tôi ¯d˜a ch, ı ra ¯diê`u ch, ınh kh´ac nhau trên b, an th, ao
1032 The results obtained are not par- ticularly surprising and will be of limited interest.
1032 C´ac kê´t qu, a thu ¯du,
.o, c không ph, ai l`a ¯d ˘ac bi.êt ¯d´ang ng.ac nhiên v`a s˜e du¯ , o, c quan tâm h an chê´.
1033 The results are correct but only moderately interesting.
1033 Kê´t qu, a l`a ch´ınh x´ac nhu, ng ch, ı v`u, a th´u v.i
1034 The results are rather easy mod- ifications of known facts.
.o, c s, u, a ¯dô, i kh´a d˜ê d`ang c´ac s u, ki ên ¯du, o, c biê´t ¯dê´n
1035 The example is worthwhile but not of sufficient interest for a re- search article.
1035 V´ı d u l`a ¯d´ang gi´a nhu, ng không quan tâm ¯dâ`y ¯d, u cho m ôt b`ai viê´t nghiên c´u, u
1036 The English of the paper needs a thorough revision.
1036 Anh ng˜u, c, ua b`ai b´ao câ`n m ôt phiên b, an k˜y lu,˜o,ng
1037 The paper does not meet the standards of your journal.
1037 Giâ´y không ¯d´ap ´u, ng c´ac tiêu chuâ, n c, ua t ap ch´ı c, ua b an
1038 Theorem 2 is false as stated 1038 Ð.inh l´y 2 l`a sai nhu, ¯ d˜a nêu
1039 Theorem 2 is false in this gener- ality.
1039 Ð.inh l´y 2 l`a sai chung chung n`ay
1040 Lemma 2 is known (see ) 1040 Bô, dê` 2 ¯¯ du,
1041 Accordingly, I recommend that the paper be rejected.
1041 Theo ¯d´o, tôi ¯dê` ngh.i b.i t`u, chô´i giâ´y
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 9 References] [M uc l uc]
M ôt sô´quy t´˘ac ¯d oc k´y hi.êu
C´ ac k´ y hi êu công th´u ,
1045 × 1045 multiplication sign (sign of multiplication)
1046 : 1046 division sign (sign of division)
1051 ' 1051 congruent to , is isomorphic to
1052 α=β 1052 alpha equals beta, alpha is equal to beta
1053 α6=β 1053 alpha is not beta, alpha is not equal to beta
1055 α±β 1055 alpha plus or minus beta
1056 α > β 1056 alpha is greater than beta
1057 αβ 1057 alpha is substantially greater than beta
1058 α < β 1058 alpha is less than beta
1059 αβ 1059 alpha is substantially less than beta
1060 α2> αn 1060 alpha second is greater than alpha n-th
1065 α 00 1065 alpha double prime, alpha second prime
1067 α 1067 alpha vector, the mean value of alpha
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 10 Ð oc k´y hi.êu] [M uc l uc]
1071 α 1 1071 alpha first, alpha sub one, alpha suffix one
1072 α n 1072 alpha n-th, alpha sub n, alpha suffix n
1073 f c 0 1073 f prime sub c, f prime suffix c, f suffix c prime
1074 α 00 2 1074 alpha second double prime, alpha double prime sec- ond
1078 α+β =γ 1078 alpha plus beta is gamma, alpha plus beta equals gamma, alpha plus beta is equal to gamma, alpha plus beta makes gamma
1079 (α+β) 2 1079 alpha plus beta all squared
1080 α−β =γ 1080 alpha minus beta is gamma; alpha minus beta leaves gamma
1081 (2x−y) 1081 bracket two x minus y close the brackets
1083 5×5 = 25 1083 five times five is twenty five, five multiplied by five equals twenty five.
1084 S =vãt 1084 S is equal to v multiplied by t; S equals v times t.
1085 α= β γ 1085 alpha is equal to the ratio of beta to gamma.
1086 β γ =α 1086 beta divided by gamma is alpha; beta by gamma equals alpha.
1087 αβ 2 β =αβ 1087 alpha beta square (divided) by beta equals alpha beta.
∞ = 0 1088 alpha by infinity is equal to zero.
∞ = 0 1089 alpha by infinity is equal to zero.
1090 x±p x 2 −y 2 y 1090 x plus or minus square root of x square minus y square all over y.
1091 α β =a b 1091 the ratio of alpha to beta equals the ratio of a to b; alpha to beta is as a to b.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 10 Ð oc k´y hi.êu] [M uc l uc]
70 Ch, u ¯dê` 10 M ôt sô´quy t´˘ac ¯d oc k´y hi.êu
1097 0.5 1097 o [ou] point five; zero point five; nought point five.
1098 0.002 1098 o [ou] point o [ou] o [ou] two; zero point zero zero two.
1100 0.0000001 1100 o [ou] point six noughts one.
1102 15.505 1102 fifteen point five o [ou] five
1103 x 2 1103 x square; x squared; the square of x; the second power of x; x to the second power; x raised to the second power.
1104 x 3 1104 x cube; x cubed; x raised to the third power.
1106 x −n 1106 x to the minus n-th power.
1107 √ α 1107 the square root of alpha.
1108 √ 3 α=β 1108 the cube root of alpha is beta.
1109 √ 5 α 2 1109 the fifth root of alpha square.
1110 alpha equals to the square root of (capital) R square plus x square.
2xb 00 1111 the square root of alpha first plus capital A divided to xb double prime.
1112 df dx 1112 df over dx; the first derivative of f with respect to x.
1113 d 2 f dx 2 1113 the second derivative of f with respect to x d two f over d x square.
1114 d n f dx n 1114 the n-th derivative of f with respect to x.
1115 partial d two f over partial d x square plus partial d two f over partial d y square equals zero.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 10 Ð oc k´y hi.êu] [M uc l uc]
1117 Rβ α 1117 the intergral from alpha to beta; integral between limits alpha and beta.
Rx x 0 F dx 1118 d over dx of the integral fromx0to x of capital F dx.
1119 E =P1 αβ 1119 capital E is equal to the ratio of the product P1 to the product alpha beta.
= √ n α m 1120 alpha to the m by n-th power equals the n-th root of alpha to the m-th power.
1121 R dx pc 2 −y 2 1121 the integral of dx divided by the square root of c square minus y square.
1122 alpha plus beta over alpha minus beta is equal to c plus d over c minus d.
1123 V equals u square root of sin square alpha minus co- sine square alpha.
1124 tangent alpha equals tangent beta divided by l.
1125 α 3 = log c d 1125 alpha cubed is equal to the logarithm of d to the base c.
1126 x+a in round brackets to the power p minus the r-th root of x all (in square brackets) to the minus q-th power.
1127 open round brackets capital D minus r first close the round brackets open square and round brackets cap- ital D minus r second close the round bracket by y close the square brackets.
The equation for the force \( f_{sub v} \) is expressed as \( f_{sub v} = \frac{m \omega^2 \alpha^2}{[r p^2 m^2 + R(R + \omega^2 \alpha^2)/(rp)]} \) This formula incorporates variables such as mass \( m \), angular velocity \( \omega \), and radius \( r \), providing a comprehensive relationship between these physical quantities in a mechanical system.
1129 the absolute value of the quantity f sub j of t one minus f sub j of t two
Pn i=1aij(t)1130 maximum over j of the sum from i equals one to i equaqls n of the modulus ofaij of (t), where j runs from one to n.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 10 Ð oc k´y hi.êu] [M uc l uc]
72 Ch, u ¯dê` 10 M ôt sô´quy t´˘ac ¯d oc k´y hi.êu
K´ y hi êu chuyên ng`anh
Logic
1132 ∃xF(x) 1132 there exists an x such that F(x) holds.
1137 A⇐⇒B 1137 A and B are logically equivalent (Equivalence)
Set
1138 x∈X 1138 element x is a member of the set X (element x be- longs to the set X)
1145 A/R 1145 Set of equivalence classes of A with respect to an equivalence relation R
Aλ 1147 Cartesian product of the A sub lambda
1148 {x|p(x)} 1148 Set of all element x with the property p(x)
1149 {Aλ}λ∈Λ 1149 Family with index set Lambda
1153 lim supA n 1153 Supperior limit of the sequence of sets A sub n
1154 lim infA n 1154 Inferior limit of the sequence of sets A sub n
−→Aλ 1155 Inductive limit of A sub lambda
1156 lim←−Aλ 1156 Projective limit of A sub lambda
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 10 Ð oc k´y hi.êu] [M uc l uc]
Order
Algebra
1166 detA 1166 Determinant of a square matrix A
1167 trA 1167 Trace of a square matrix A
1169 In 1169 Unit matrix of degree n
1170 A⊗B 1170 Kronecker product of two matrix A and B
1171 M ∼=N 1171 Two algebraic systems M and N are isomorphic
1172 M/N 1172 Quotiont space of an algebraic system M by N
1173 dimM 1173 Dimension of a linear space M
1178 (¯a,¯b) 1178 inner product of two vectors¯aand¯b
1179 M ⊗N 1179 tensor product of two modules M and N
1180 hom(M, N) 1180 Set of all homomorphisms from M to N
1181 ΛM 1181 Exterior algebra of a linear space M
Topology
1182 an →a 1182 sequence a sub n converges to a
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 10 Ð oc k´y hi.êu] [M uc l uc]
74 Ch, u ¯dê` 10 M ôt sô´quy t´˘ac ¯d oc k´y hi.êu
1183 sequence a sub n converges monotically decreasingly (increasingly) to a
1184 liman 1184 limit of a sequence a sub n
1185 lim supan 1185 superior limit of a sequence a sub n
1186 lim infan 1186 inferior limit of a sequence a sub n
1189 d(x, y) 1189 distance between two point x and y
Function
1192 gradϕ 1192 gradient of a function varphi
D(x1, x2, , xn)1194 Jacobian determinat of(u1, u2, , un)with respect to
1196 Rez 1196 real part of a complex number z
1197 Imz 1197 Imaginary part of a complex number z
1198 argz 1198 Argument of a complex number z
Probability
1203 E(X) 1203 Mean (expectation) of a random variable X
1205 ρ(X, Y) 1205 Correlation coefficient of two random variables X and Y
1206 E(X|Y) 1206 Conditional expectation of random variable X under the condition Y
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 10 Ð oc k´y hi.êu] [M uc l uc]
M ôt sô´quy t´˘ac ng˜u , ph´ ap
Ch´ u ´ y m ao t`u , x´ ac ¯ d.inh
1 Ð ai t`u, ngh˜ıa l`a "mentioned earlier", "that":
LetA⊂X IfB = 0for everyBintersecting the setA, then
The series can easily be shown to converge.
o, ng duy nhâ´t ¯d˜a x´ac d.inh (ho ˘ac trong ¯d.inh ngh˜ıa)¯ :
Letf be the linear formf →(g, F).
Letf be the linear form denned by (2) [If there is only one.] u= 1in the compact setKof all points at distance 1 fromL.
We denote byB(X)the Banach space of all linear operators inX. , under the usual boundary conditions.
, with the natural definitions of addition and multiplication.
Using the standard inner product we may identify
3 Trong câ´u tr´uc: the+t´ınh châ´t (ho ˘ac ¯d ˘ac t´ınh kh´ac )+of+¯dô´i tu,
The continuity off follows from
The existence of test functions is not evident.
There is a fixed compact set containing the supports of all thef j Thenxis the centre of an open ballU.
The intersection of a decreasing family of such sets is convex.
Ngo ai l.ê: Every nonempty open set inR k is a union of disjoint boxes.
4 дu, ng tru,´o, c sô´lu, o, ng nê´u n´o bao gô`m tâ´t c, a ¯dô´i tu,
The two groups have been shown to have the same number of generators.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 11 Quy t´˘ac ng˜u, ph´ap] [M uc l uc]
76 Ch, u ¯dê` 11 M ôt sô´quy t´˘ac ng˜u, ph´ap hu, ng: Two groups only were mentioned.]
Each of the three products on the right of D) satisfies hu, ng: There are exactly 3 products there.]
5 дu, ng tru,´o, c m ôt sô´th´u, t u,: The first Poisson integral in (4) converges tog.
The second statement follows immediately from the first.
6 дu,ng tru,´o,c tên d`ung nhu, m ôt thu.ôc t´ınh: the Dirichlet problem the Taylor expansion the Gauss theorem
Nhu,ng : Taylor’s formula [thiê´u "the"] a Banach space
7 дu,ng tru,´o, c m ôt danh t`u,sô´ nhiê`u nê´u ta muô´n ch, ı m ôt l´o,p c´ac ¯dô´i tu,
o,ng nhu, to`an thê,
, ch´u, không m ôt phâ`n t, u, ¯dô´i tu, o,ng:
The real measures form a subclass of the complex ones.
This class includes the Helson sets.
11.2 Ch´ u ´ y m ao t`u , không x´ ac ¯ d.inh
1 Thay cho sô´m ôt "one":
The four centres lie in a plane.
A chapter will be devoted to the study of expanding maps.
For this, we introduce an auxiliary variablez.
2 C´o ngh˜ıa l`a th`anh viên c, ua m ôt l´o, p c´ac ¯dô´i tu,
o, ng nhu,"some", "one of":
ThenDbecomes a locally convex space with dual spaceD 0
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 11 Quy t´˘ac ng˜u, ph´ap] [M uc l uc]
The right-hand side of (4) is then a bounded function.
This is easily seen to be an equivalence relation.
Theorem 7 has been extended to a class of boundary value problems. The transitivity is a consequence of the fact that
Let us now state a corollary of Lebesgue’s theorem for
After a change of variable in the integral we get
We thus obtain the estimate with a constantC. trong sô´nhiê`u th`ı không:
The existence of partitions of unity may be proved by
The definition of distributions implies that
, where G and F are differential operators.
3 Trong c´ac ¯d.inh ngh˜ıa nh˜u, ng l´o, p ¯dô´i tu,
o, ng (ngh˜ıa l`a khi tô`n t ai râ´t nhiê`u dô´i tu¯ ,
A fundamental solution is a function satisfying
We callCa module of ellipticity.
A classical example of a constantCsuch that
We wish to find a solution of (6) which is of the form
4 trong sô´nhiê`u th`ı thôi:
The elements ofDare often called test functions. the set of points with distance 1 fromK the set of all functions with compact support
The integral may be approximated by sums of the form
Taking in (4) functionsvwhich vanish inU we obtain
Letf andgbe functions such that
5 Trong sô´nhiê`u khi ta tham chiê´u t´o, i m˜ôi phâ`n t, u, c, ua l´o,p:
Direct sums exist in the category of abelian groups.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 11 Quy t´˘ac ng˜u, ph´ap] [M uc l uc]
78 Ch, u ¯dê` 11 M ôt sô´quy t´˘ac ng˜u, ph´ap
In particular, closed sets are Borel sets.
Borel measurable functions are often called Borel mappings.
This makes it possible to applyH2-results to functions in anyHp.
Nê´u ta ch, ı t´o, i tâ´t c, a phâ`n t, u, c, ua l´o, p th`ı d`ung "the":
The real measures form a subclass of the complex ones.
6 Tru,´o, c m ôt t´ınh t`u, m`a n´o hu,o´, ng theo ngh˜ıa "c´o ¯d.inh lu,
This map extends to all of M in an obvious fashion.
A remarkable feature of the solution should be stressed.
Section 1 gives a condensed exposition of
Section 1 describes in a unified manner the recent results
Combining (2) and (3) we obtain, with a new constantC,
A more general theory must be sought to account for these irregularities. The equation (3) has a unique solutiongfor everyf.
But: (3) has the unique solutiongf.
Ch´ u ´ y b ,
1 Tru,´o, c m ôt danh t`u, ch, ı t´o, i ¯dang hi.ên h`anh:
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 which differentiation is always possible.
Using integration’by parts we obtain
If we apply induction to (4), we get
This reduces the solution to division byP x.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 11 Quy t´˘ac ng˜u, ph´ap] [M uc l uc]
Comparison of (5) and (6) shows that
2 Tru,´o,c c´ac danh t`u,ch, ı t´o,i t´ınh châ´t nê´u ta không kê, ra ¯dô´i tu,
In question of uniqueness one usually has to consider
By continuity, (2) also holds whenf = 1.
By duality we easily obtain the following theorem.
Here we do not require translation invariance.
3 Sau m ôt biê, u th´u, c bâ´t k`y v´o, i "of": a type of convergence a problem of uniqueness the condition of ellipticity the hypothesis of positivity the method of proof the point of increase
It follows from Theorem 7 that
Section 4 gives a concise presentation of
Property (iii) is called the triangle inequality.
This has been proved in part (a) of the proof.
Nhu, ng: the set of solutions of the form (4.7)
To prove the estimate (5.3) we first extend
We thus obtain the inequality (3) [Or: inequality (3)]
The asymptotic formula (3.6) follows from
Since the region (2.9) is inU, we have
5 Ðê, tr´anh l ˘ap l ai t`u,tiê´p sau: the order and symbol of a distribution
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 11 Quy t´˘ac ng˜u, ph´ap] [M uc l uc]
80 Ch, u ¯dê` 11 M ôt sô´quy t´˘ac ng˜u, ph´ap the associativity and commutativity ofA the direct sum and direct product the inner and outer factors off [Note the plural.]
Nhu, ng: a deficit or an excess
Minkowski’s inequality, but: the Minkowski inequality
Fefferman and Stein’s famous theorem, more usual: the famous Fefferman-Stein theorem
6 Trong m ôt sô´m.ênh ¯dê` mô t, a m ôt danh t`u,
In mathematical contexts, we often encounter terms such as "an algebra with unit e," "an operator with domain H²," and "a solution with vanishing Cauchy data." Additionally, we refer to "a cube with sides parallel to the axes" and "a domain with smooth boundary." Important concepts include "an equation with constant coefficients," "a function with compact support," and "random variables with zero expectation." The dynamics of systems can be described by "the equation of motion" and "the velocity of propagation." We also consider "an element of finite order" and "a solution of polynomial growth," along with geometric aspects like "a ball of radius 1" and "a function of norm p."
But: elements of the formf =
LetBbe a Banach space with a weak symplectic formw.
Two random variables with a common distribution.
F has a finite norm not exceeding 1.
F has a compact support contained inI.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 11 Quy t´˘ac ng˜u, ph´ap] [M uc l uc]
CH U Ð` Ê 12 K´ y hi êu trong LaTeX
K´ y hi êu trong môi tru ,`o,ng v˘an b ,
> \textgreater \dots ł \l Ł \L ứ \o ỉ \O ồ \aa Å \AA ò \ss
SS \SS ổ \ae ặ \AE œ \oe Œ \OE
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 12 l ênh k´y hi.êu trong L A TEX] [M uc l uc]
82 Ch, u ¯dê` 12 K´y hi êu trong LaTeX
K´ y hi êu trong môi tru ,`o,ng to´an
K´ y hi êu chung
⊃ \supset ã \cdot \centerdot © \copyright z \maltese
Ch˜ u , c´ ai Hy L ap
α \alpha β \beta γ \gamma δ \delta λ \lambda ω \omega ψ \psi χ \chi ρ \rho
\epsilon κ \kappa π \pi φ \phi σ \sigma θ \theta υ \upsilon ξ \xi τ \tau ι \iota η \eta ζ \zeta à \mu ν \nu
$ \varpi ϕ \varphi ς \varsigma ϑ \vartheta Γ \Gamma
Ω \Omega Π \Pi Φ \Phi Ψ \Psi Σ \Sigma Θ \Theta Υ \Upsilon Ξ \Xi z \digamma ג \gimel k \daleth Γ \varGamma
Ω \varOmega Π \varPi Φ \varPhi Ψ \varPsi Σ \varSigma Θ \varTheta Υ \varUpsilon Ξ \varXi
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 12 l ênh k´y hi.êu trong L A TEX] [M uc l uc]
Binary Operations and Relations
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 12 l ênh k´y hi.êu trong L A TEX] [M uc l uc]
84 Ch, u ¯dê` 12 K´y hi êu trong LaTeX
⊀ \nprec \npreceq \precnapprox \precnsim \nsucc \nsucceq \succnapprox
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 12 l ênh k´y hi.êu trong L A TEX] [M uc l uc]
T âp h o ,
= \sqsupset w \sqsupseteq c \Supset + \nsupseteq ) \supsetneq
Quan h ê tam gi´ac v`a n ,
( \leftharpoonup ) \leftharpoondown \leftrightharpoons \upharpoonleft \downharpoonleft
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 12 l ênh k´y hi.êu trong L A TEX] [M uc l uc]
86 Ch, u ¯dê` 12 K´y hi êu trong LaTeX
Bâ´t ¯ d , ˘
≯ \ngtr \nleqslant \ngeqslant \nleq \ngeq \nleqq \ngeqq
≫ \ggg \lnsim \gnsim \lnapprox \gnapprox l \lessdot m \gtrdot
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 12 l ênh k´y hi.êu trong L A TEX] [M uc l uc]
K´ y hi êu m˜ui tên
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 12 l ênh k´y hi.êu trong L A TEX] [M uc l uc]
88 Ch, u ¯dê` 12 K´y hi êu trong LaTeX
To´ an t ,
Nh˜u ,
This article covers a range of mathematical functions and symbols, including trigonometric functions such as sine (sin), cosine (cos), and tangent (tan), as well as their inverses like arcsine (arcsin) and arccosine (arccos) It also discusses hyperbolic functions such as sinh, cosh, and tanh, alongside logarithmic functions like log and lg Additionally, the article touches on limits, including lim, lim inf, and lim sup, as well as modular arithmetic represented by m mod n Other important concepts include determinants (det), greatest common divisors (gcd), and homomorphisms (hom) This comprehensive overview serves as a valuable reference for mathematical notation and operations.
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 12 l ênh k´y hi.êu trong L A TEX] [M uc l uc]
Nh˜ u ,
ng k´ y hi êu kh´ac
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 12 l ênh k´y hi.êu trong L A TEX] [M uc l uc]
90 Ch, u ¯dê` 12 K´y hi êu trong LaTeX
Ch˜ u , c´ ai trong môi tru ,`o,ng to´an
c´ ai trong môi tru ,`o,ng to´an
OP QRST U V W XY Z abcdef ghijklmnopqrstuvwxyz
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 12 l ênh k´y hi.êu trong L A TEX] [M uc l uc]
OP QRST U V W XY Z abcdef ghijklmnopqrstuvwxyz
(Ðu, a v`ao \usepackage{mathrsfs} tru,´o,c \begin{document})
Nh˜ u ,
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 12 l ênh k´y hi.êu trong L A TEX] [M uc l uc]
92 Ch, u ¯dê` 12 K´y hi êu trong LaTeX p \ulcorner x \llcorner
⇓ \Downarrow m \Updownarrow ˜ a \tilde{a} ˆ a \hat{a} ˇ a \check{a}
Nh˜ u , ng dâ´u châ´m châ´m
In mathematical notation, various types of dots are used to convey specific meanings: lower dots (ã ã ã) and center dots indicate different operations, while diagonal dots ( ), vertical dots (…), and fill dots (….) serve distinct purposes in equations Additionally, multiplication is represented by a specific dot (ã ã ã), and there are unique dots for integrals (ã ã ã) and binary operations (ã ã ã), each contributing to the clarity and functionality of mathematical expressions.
\dotsc dots after commas \dotso other dots
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 12 l ênh k´y hi.êu trong L A TEX] [M uc l uc]
L ênh biê´n ¯dô ,
i theo c˜ o , z}|{abc \overbrace{abc} abc
√abc \sqrt{abc} abcf \widetilde{abc} abcc \widehat{abc} abc \overline{abc} abc \underline{abc}
←→ abc \overleftrightarrow{abc} abc−→ \underrightarrow{abc} abc←− \underleftarrow{abc} abc←→ \underleftrightarrow{abc}
Nguy˜ên H˜u, u Ðiê, n [Ch, u ¯dê` 12 l ênh k´y hi.êu trong L A TEX] [M uc l uc]